# Projective contact log varieties

Adrian Langer

###### Abstract

We study contact structures on smooth complex projective varieties with a
simple normal crossing divisor, generalizing some well-known results
concerning the non-logarithmic case. In particular, we describe the structure
of elementary log contractions of such log varieties and we construct the
corresponding contact structures.

Address:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097
Warszawa, Poland
e-mail<EMAIL_ADDRESS>
## Introduction

Let $X$ be a smooth complex projective variety. A _contact structure_ on $X$
is given by a subbundle ${\mathcal{F}}$ of the tangent bundle $T_{X}$ such
that the pairing ${\mathcal{F}}\times{\mathcal{F}}\to
T_{X}/{\mathcal{F}}={\mathcal{L}}$ induced by the Lie bracket is everywhere
non-degenerate. In this case $X$ has odd dimension $2r+1$ and we can dually
consider an ${\mathcal{L}}$-valued $1$-form $\theta\in
H^{0}(X,\Omega_{X}^{1}\otimes{\mathcal{L}})$. This form gives a contact
structure if and only if $\theta\wedge(d\theta)^{\wedge r}$ (which is a well-
defined section of $\Omega_{X}^{2r+1}\otimes{\mathcal{L}}^{r}$) has no zeroes.

Contact structures first appeared in real geometry and later they were heavily
studied in complex geometry. One of the motivations in the complex case came
from C. LeBrun and S. Salamon, who proved that twistor spaces of positive
quaternion-Kähler manifolds are contact Fano varieties (see [LS94]). They also
conjectured that every such variety is homogeneous. Despite some recent
progress (see [BW22] and [ORCW21]), this conjecture is still open. One of the
main classification results for contact complex varieties is due to the
combined results of [KPSW00] and [Dem02]. It says that a projective contact
variety is either the projectivization of the tangent bundle of a smooth
projective variety or it is Fano with $b_{2}=1$.

The main aim of this paper is to study generalization of this result to the
logarithmic case. We assume that all varieties are defined over a fixed
algebraically closed field $k$ of characteristic $0$ and we consider contact
structures on a projective snc pair, i.e., a pair consisting of a smooth
projective variety $X$ and a simple normal crossing divisor $D$. In this case
a contact structure is a subbundle ${\mathcal{F}}$ of the logarithmic tangent
bundle $T_{X}(-\log D)$ such that the pairing
${\mathcal{F}}\times{\mathcal{F}}\to T_{X}(-\log
D)/{\mathcal{F}}={\mathcal{L}}$ induced by the Lie bracket is everywhere non-
degenerate. We usually forget about ${\mathcal{F}}$ and we think about the
contact structure as the projection $T_{X}(-\log D)\to{\mathcal{L}}$. Dually
one can consider an ${\mathcal{L}}$-valued logarithmic $1$-form $\theta\in
H^{0}(X,\Omega_{X}^{1}(\log D)\otimes{\mathcal{L}})$. This form is contact if
and only if $\theta\wedge(d\theta)^{\wedge r}$ has no zeroes. $(X,D)$ is
called _contact_ if it admits a contact structure (or equivalently a contact
form). Such structure is a natural compactification of a contact structure on
the quasi-projective variety $U=X\backslash{\mathop{{\rm Supp\,}}}D$.

It is natural to expect that similarly to the usual case, one can classify
projective contact snc pairs. It is easy to see that the projectivization of
the logarithmic tangent bundle of a projective snc pair is contact (see
Proposition 6.1). The contact form in this case is given by a log version of
Liouville’s $1$-form (cf. [Arn89, Appendix 4 D] for the classical case). One
of the main surprises of the theory is that there exist contact projective snc
pairs with $b_{2}\geq 2$ for which the contact structure is not related to
such contact structures. A posteriori this fact has the following explanation.
It is well-known in differential geometry and PDEs that higher jet bundles of
a given vector bundle on a manifold carry a canonical contact form
corresponding to the Cartan distribution (see, e.g., [Sau89]). This contact
structure has a somewhat different meaning as it is given by a distribution
${\mathcal{F}}\subset T_{X}$ for which $T_{X}/{\mathcal{F}}$ is a vector
bundle of possibly higher rank than $1$. But in case of the first jet bundle
of a line bundle it gives the usual contact structure (see, e.g., [MS98,
Example 3.44] for the case of the trivial line bundle). The corresponding
contact form is not homogeneous with respect to fiber coordinates, so this
does not give rise to a contact form on the projectivized jet bundle. However,
we show that it extends to a contact form on the natural log variety
underlying certain projective bundle compactifying this jet bundle. This
explains how log contact varieties of dimension $(2r+1)$ can appear as
projective bundles over $r$-dimensional varieties. In general, for any
projective snc pair $(Y,B)$ of dimension $d\leq r$ and any collection of
$(r+1-d)$ line bundles on $Y$, we give an explicit construction of a contact
snc pair of dimension $(2r+1)$ describing it as a projective bundle over $Y$
(see Proposition 6.3). In particular, classification of projective contact snc
pairs is much more complicated than that of usual contact varieties.

What is surprising from this point of view is that our construction describes
essentially all new contact structures. More precisely, we prove the following
theorem, which is a special case of Theorem 5.14. The
$c_{1}^{\log}({\mathcal{M}})$ in this theorem denotes the image of the first
Chern class $c_{1}({\mathcal{M}})\in H^{1}(Y,\Omega_{Y}^{1})$ in
$H^{1}(Y,\Omega_{Y}^{1}(\log B))={\mathop{{\rm Ext\,}}}^{1}(T_{Y}(-\log
B),{\mathcal{O}}_{Y})$.

######  Theorem 0.1.

Let $(X,D)$ be a contact projective snc pair of dimension $2r+1$. Then
$-(K_{X}+D)$ is not nef and $X$ admits an elementary log contraction
$\varphi:X\to Y$. If the the exceptional locus of $\varphi$ is not contained
in the support of $D$ then either $(X,D)$ is log Fano with $b_{2}(X)=1$ or $Y$
is smooth and there exists a simple normal crossing divisor $B$ on $Y$ such
that one of the following occurs:

  1. 1.

$\dim Y=r+1$ and $(X\to Y)\simeq({\mathbb{P}}(T_{Y}(-\log B))\to Y$) and
$D=\varphi^{-1}(B)$,

  2. 2.

$\dim Y=r+1-s$ for some $1\leq s\leq r$ and there exists $s$ line bundles
${\mathcal{M}}_{1},...,{\mathcal{M}}_{s}$ on ${Y}$ such that $(X\to
Y)\simeq({\mathbb{P}}({\mathcal{E}})\to Y)$, where we have an extension

$0\to\bigoplus_{i=1}^{s}{\mathcal{M}}_{i}{\longrightarrow}{\mathcal{E}}\longrightarrow{\mathcal{C}}\to
0,$

in which ${\mathcal{C}}$ is the extension of $T_{Y}(-\log B)$ by
${\mathcal{O}}_{Y}^{\oplus s}$ given by
$-\bigoplus_{i=1}^{s}c_{1}^{\log}({\mathcal{M}}_{i})$. In this case $D$ is the
sum of $\varphi^{-1}(B)$ and $s$ divisors determined by the cokernels of
${\mathcal{M}}_{i}\to{\mathcal{E}}$.

Our main contribution is description of log contractions and construction of
contact structures in the corresponding cases. The first part of the above
theorem is a simple corollary of the result of F. Touzet [Tou16] (see
Corollary 4.12) generalizing earlier result of J.-P. Demailly [Dem02] to the
log case. In fact, this result shows that $K_{X}+D$ is not pseudoeffective. So
by [BDPP13, Theorem 2] there exist on $X$ strongly moving curves $C$ with
$(K_{X}+D)\cdot C<0$. In this case the log Bend and Break conjecture [KM99,
Conjecture 1.11] of S. Keel and J. McKernan predicts that the quasi-projective
variety $U=X\backslash{\mathop{{\rm Supp\,}}}D$ is ${\mathbb{A}}^{1}$-uniruled
(in the sense of, e.g., [CZ19, Definition 1.2]). In case $(X,D)$ admits an
elementary log contraction which is not an isomorphism on the open part
$U=X\backslash{\mathop{{\rm Supp\,}}}D$, this fact follows easily from the
above result.

We also prove a far reaching generalizations of Theorem 0.1 that describe a
precise structure of all log contractions of contact snc pairs. Unlike in the
case of contact varieties, there also appear birational log contractions. In
this case the whole exceptional locus is contained in the support of $D$ and
the target is often singular. But divisorial contractions can also lead to snc
pairs, e.g., we can obtain a contact snc pair by blowing up the fiber of
${\mathbb{P}}(T_{Y}(-\log B))\to Y$ over a point that does not belong to the
support of $B$ (see Example 7.6).

To study log contractions of contact snc pairs we need to study more general
snc pairs with so called _semi-contact structure_ (see Definition 4.5). A
semi-contact structure is defined as a contact structure on some special Lie
algebroid generalizing a Picard Lie algebroid in the sense of A. Beilinson and
J. Bernstein [BB93, Definition 2.1.3]. Such structure allows us to describe
restriction of the contact structure on the pair $(X,D)$ to the strata of the
stratification of $X$ induced by the components of $D$. An obvious advantage
of this approach is that it allows for inductive arguments on the dimension,
whereas the usual contact structures appear only in odd dimensions.

The classification of contact varieties with $b_{2}\geq 2$ in [KPSW00] used
Mori’s theory and, more precisely, a careful study of contractions and
rational curves on contact varieties. So it is natural to approach the
classification of contact snc pairs using log contractions and the non-proper
analogue of rational curves suggested by S. Keel and J. McKernan in [KM99]
(see also [CZ19]). However, many of analogues of usual results are not
available in the log case and to prove Theorem 0.1 we need to find appropriate
formulations working in the log case. In particular, we need a relative
version of Keel–McKernan’s log bend-and-break [KM99, Corollary 5.4], a log
version of the Ionescu–Wiśniewski inequality and a log version of the rank
computation for ${\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X)$ (see [Kol96,
Chapter II, Proposition 3.10] for the usual case). We also need several
results concerning Lie algebroids and generalized Atiyah extensions that we
use quite extensively in the paper. With these at hand, we try to mimic the
proof from [KPSW00] but this gets quite complicated and we run into various
additional problems that are usually resolved by careful study of various loci
and using induction on the dimension.

Note that a contact structure can be more generally defined on a log smooth
log variety in the sense of Fontaine–Illusie–Kato (see [Kat89] and [Ogu18]).
In this case contact structure is defined by the same data as usually and the
only difference is that we replace a smooth variety with a log smooth log
variety. Such structures also appear as natural compactifications of contact
structures on quasi-projective varieties. In this paper we restrict to contact
snc pairs but analogues of our results can be also proven in the logarithmic
case. Unfortunately, this requires quite a lot of additional work that would
obscure the main ideas, so we decided to first explain the classical case, and
deal with the logarithmic case elsewhere.

As pointed out above, log contractions of contact snc pairs (or log smooth log
varieties) that are not of fiber type often lead to singular varieties. The
study of singular contact log varieties and contact log Fano log varieties
with $b_{2}=1$ is also left to another paper. We also leave out possible
applications of the above theory in complex or real geometry.

In this paper we deal with algebraic varieties (although ocassionally we also
use analytifications of such varieties). This approach in no way restricts
generality of the obtained results as complex projective manifolds are well
known to be algebraic. It should also be said that much of the paper could be
written in the language of complex manifolds but we prefer to use the language
of algebraic geometry. Whereas to many readers this might seem unnecessary
(and it can make the paper more difficult to read), there are also
mathematical reasons to do so. One of the reasons is that in some cases our
study leads to singular (projective) varieties and singularities in complex
geometry are not as rigid as singularities of algebraic varieties. However,
the most important reason is that we want to show that contact projective
(log) varieties have a nice local algebraic structure. This might be quite
surprising as it is well-known that Darboux’s theorem describing local
structure of a symplectic or contact manifold does not work in algebraic local
coordinates.

### Relation to other work

Contact structures are well-known to be related to symplectic structures.
Rather recently, differential geometers started to study a generalization of
symplectic structures under the name of b-symplectic manifolds. These
structures first appeared in [GMP14] and they are described in the language of
b-geometry refering to Melrose’s calculus for differential operators on real
manifolds with (smooth) boundary (see [Mel93]). Complex analogues of
symplectic structures in case of complex manifolds with boundary have also
been heavily studied under the name of log symplectic manifolds (see, e.g.,
[CG18], [Got16] and [Pym17]).

Even more recently, b-contact manifolds were defined in [MO23] (see also
[BDM+19]). The definition is the same as in the case of usual manifolds but it
replaces manifolds with b-manifolds (tangent bundle with b-tangent bundle
etc.). Our contact snc pairs are an obvious analogue of such structures in
complex geometry but it seems they have not been studied till now. Let us also
note that our constructions work in the real case giving a new construction of
(compact) b-contact manifolds that compactify $1$-jet bundles (see Appendix
B). It seem that this construction has not been noticed before.

The structure of the paper is as follows. In Section 1 we generalize Wahl’s
theorem on ample line bundles of a tangent bundle to the logarithmic case and
we recall some results on the logarithmic cone and contraction theorems. Then
in Section 2 we generalize several results on rational curves on smooth
projective varieties to the log case. Then we generalize contact and
symplectic structures to the Lie algebroid case and prove a few auxiliary
results. Then in Section 4 we introduce special Lie algebroids generalizing
Picard algebroids and study contact structure on such Lie algebroids. The main
result is proven in Section 5. In Section 6 we show some constructions of
contact snc pairs. Then in Section 7 we study contact projective snc pairs
with two elementary log contractions and we show that there are many such
varieties. In Appendix A we show classification of semi-contact snc pairs in
small dimensions. In Appendix B we show how analogues of the constructions
from Section 6 produce new b-contact b-manifolds.

### Notation

In the whole paper, we assume that our varieties are defined over an
algebraically closed field $k$ of characteristic $0$. A variety is always
irreducible. Since all structures in this paper are $k$-linear, by abuse of
notation we write $\Omega_{X}^{1}$ for the sheaf of Kähler differentials
$\Omega_{X/k}$ and $T_{X}$ for the sheaf
$T_{X/k}=Der_{k}({\mathcal{O}}_{X},{\mathcal{O}}_{X})$ of $k$-linear
derivations of ${\mathcal{O}}_{X}$. For a coherent ${\mathcal{O}}_{X}$-module
${\mathcal{E}}$ we write ${\mathcal{E}}^{*}$ for
${\mathop{{\mathcal{H}}om}}_{{\mathcal{O}}_{X}}({\mathcal{E}},{\mathcal{O}}_{X})$.
All tensor operations on ${\mathcal{O}}_{X}$-modules are performed over
${\mathcal{O}}_{X}$ unless otherwise stated explicitly.

A vector bundle on $X$ is a locally free coherent ${\mathcal{O}}_{X}$-module
of finite rank.

##  1 Preliminaries

In this section we generalize Wahl’s characterization of projective spaces
[Wah83] to log varieties, recall the logarithmic cone (and contraction)
theorem and prove some auxiliary results on log extremal curves.

###  1.1 Logarithmic version of Wahl’s theorem

######  Theorem 1.1.

Let $(X,D)$ be a projective snc pair and let ${\mathcal{L}}$ be an ample line
bundle on $X$. Let $n=\dim X\geq 1$. If $H^{0}(X,T_{X}(-\log
D)\otimes{\mathcal{L}}^{-1})\neq 0$ then either
$(X,{\mathcal{L}})\simeq({\mathbb{P}}^{1},{\mathcal{O}}_{{\mathbb{P}}^{1}}(2))$
and $D=0$ or
$(X,{\mathcal{L}})\simeq({\mathbb{P}}^{n},{\mathcal{O}}_{{\mathbb{P}}^{n}}(1))$
and $D$ is an arrangement of $s\leq n$ hyperplanes in a general position
(i.e., it forms an snc divisor).

###### Proof.

The proof is by induction on $n$ with $n=1$ being clear. Since $T_{X}(-\log
D)\subset T_{X}$, the first part follows immediately from Wahl’s theorem
[Wah83, Theorem 1]. So we can assume that
$(X,{\mathcal{L}})=({\mathbb{P}}^{n},{\mathcal{O}}_{{\mathbb{P}}^{n}}(1))$ and
$D\neq 0$. For any irreducible component $D_{i}$ of $D$ we have a short exact
sequence

$0\to T_{D_{i}}(-\log(D-D_{i})|_{D_{i}})\to T_{{\mathbb{P}}^{n}}(-\log
D)|_{D_{i}}\to{\mathcal{O}}_{D_{i}}(-(D-D_{i}))\to 0.$

By assumption we have a nonzero map ${\mathcal{L}}\to
T_{{\mathbb{P}}^{n}}(-\log D)$. Its restriction to $D_{i}$ gives a nonzero map
${\mathcal{L}}|_{D_{i}}\to T_{{\mathbb{P}}^{n}}(-\log D)|_{D_{i}}$. Since
there are no nonzero maps from ${\mathcal{L}}|_{D_{i}}$ to
${\mathcal{O}}_{D_{i}}(-(D-D_{i}))$, we have an induced non-zero map
${\mathcal{L}}|_{D_{i}}\to T_{D_{i}}(-\log(D-D_{i})|_{D_{i}})$. Using the
induction assumption we see that $D_{i}$ is a hyperplane and
$(D-D_{i})|_{D_{i}}$ consists of at most $(n-1)$ hyperplanes in
$D_{i}\simeq{\mathbb{P}}^{n-1}$. This easily implies the required assertion. ∎

######  Remark 1.2.

Let $D$ be an arrangement of $1\leq s\leq n$ hyperplanes in a general position
in ${\mathbb{P}}^{n}$. Then by [DK93, Proposition 2.10] we have

$T_{{\mathbb{P}}^{n}}(-\log
D)\simeq{\mathcal{O}}_{{\mathbb{P}}^{n}}^{\oplus(s-1)}\oplus{\mathcal{O}}_{{\mathbb{P}}^{n}}(1)^{\oplus(n-s+1)}.$

This shows that the above theorem is optimal. Note also that $D$ is
projectively equivalent to the standard hyperplane arrangement given by
$x_{1}...x_{s}=0$. In particular, intersection of any subset of hyperplanes in
$D$ is connected.

The above theorem and the main result of M. Andreatta and J. Wisniewski [AW01,
Theorem] immediately imply the following corollary:

######  Corollary 1.3.

Let $(X,D)$ be a projective snc pair with $n=\dim X\geq 1$ and let
${\mathcal{E}}$ be an ample vector bundle contained in $T_{X}(-\log D)$. Then
we have the following possibilities for $(X,D)$ and ${\mathcal{E}}$:

  1. 1.

$(X,D)\simeq({\mathbb{P}}^{n},0)$ and
${\mathcal{E}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{n}}(1)^{\oplus m}$ for some
$m\leq n$,

  2. 2.

$(X,D)\simeq({\mathbb{P}}^{n},0)$ and ${\mathcal{E}}\simeq
T_{{\mathbb{P}}^{n}}$,

  3. 3.

$X\simeq{\mathbb{P}}^{n}$, $D$ is an arrangement of $1\leq s\leq n$
hyperplanes and
${\mathcal{E}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{n}}(1)^{\oplus m}$ for some
$m\leq{(n-s+1)}$.

###  1.2 Logarithmic cone theorem

Let us recall that a _contraction_ is a proper surjective map $\varphi:X\to Y$
of normal varieties such that $\varphi_{*}{\mathcal{O}}_{X}={\mathcal{O}}_{Y}$
(or equivalently $\varphi$ has connected fibers). We need the following
version of the cone theorem that follows from the cone theorem for dlt pairs
(see [KM98, Theorem 3.35] and [Kol96, Chapter III, Corollary 1.4]).

######  Theorem 1.4.

Let $(X,D)$ be a projective snc pair. Then:

  1. 1.

There are countably many rational curves $C_{j}\subset X$ such that
$(K_{X}+D)\cdot C_{j}<0$ and

$\overline{NE}(X)=\overline{NE}(X)_{K_{X}+D\geq
0}+\sum_{j}{\mathbb{R}}^{+}[C_{j}].$

The rays ${\mathbb{R}}^{+}[C_{j}]\subset N_{1}(X)$ are locally discrete in
$N_{1}(X)_{K_{X}+D<0}$ and every extremal ray $R$ of $\overline{NE}(X)$
contained in $N_{1}(X)_{K_{X}+D<0}$ is of the form $R={\mathbb{R}}^{+}[C_{j}]$
for some $j$.

  2. 2.

For any $(K_{X}+D)$-negative extremal face $F$ of the cone $\overline{NE}(X)$
there is a unique contraction $\varphi_{F}:X\to Y$ to projective $Y$ such that
an irreducible curve $C\subset X$ is mapped by $\varphi_{F}$ to a point if and
only if $[C]\in F$. Such $F$ is called a _$(K_{X}+D)$ -extremal face_.

  3. 3.

For any $F$ as above there exists a nef Cartier divisor $H_{F}$ such that
$F=\\{z\in\overline{NE}(X):\,H_{F}\cdot z=0\\}$ and the divisor
$mH_{F}-(K_{X}+D)$ is ample for $m\gg 0$. Such $H_{F}$ is called a _supporting
divisor_ of $F$.

A contraction $\varphi:X\to Y$ is called a _log contraction_ of $(X,D)$ if
$-(K_{X}+D)$ is $\varphi$-ample. Any log contraction is of the form
$\varphi_{F}$ for some extremal face $F$ (see [KMM87, Lemma 3.2.5]). A
$1$-dimensional $(K_{X}+D)$-extremal face is called a _ $(K_{X}+D)$-extremal
ray_. If $R$ is a $(K_{X}+D)$-extremal ray then the log contraction
$\varphi_{R}$ is called _elementary_.

Let $(X,D)$ be a projective snc pair such that $(K_{X}+D)$ is not nef. Then
either $b_{2}(X)=1$ and $-(K_{X}+D)$ is ample (the log Fano case) or
$b_{2}(X)\geq 2$ and there exists an elementary log contraction $\varphi:X\to
Y$ with $\dim Y>0$.

If $D=\sum_{i\in I}D_{i}$ for some smooth irreducible $D_{i}$ then we write
$D_{J}:=\bigcap_{j\in J}D_{j}$ for $J\subset I$. In particular, we set
$D_{\emptyset}=X$. For a subset $J\subset I$ we denote by $D^{J}$ the
restriction of the divisor $\sum_{i\in I\backslash J}D_{i}$ to $D_{J}$. Let us
note the following complement to the cone theorem:

######  Proposition 1.5.

Let $D=\sum_{i\in I}D_{i}$ and let us fix some $J\subset I$ (possibly
$\emptyset$). If $F$ is a $(K_{X}+D)$-extremal face and $C\subset D_{J}$ is a
curve such that $[C]\in F$ then we can find a rational curve
$C^{\prime}\subset D_{J}$ such that $[C^{\prime}]\in F$ and

$0<-(K_{X}+D)\cdot C^{\prime}\leq\dim D_{J}+1.$

Moreover, if $C\not\subset D_{i}$ for $i\in I\backslash J$ then we can assume
that $C\cap C^{\prime}\neq\emptyset$ and $C^{\prime}\not\subset D_{i}$ for
$i\in I\backslash J$.

###### Proof.

Let $J\subset I$ be the set of all $j\in I$ for which $C\subset D_{j}$. Then
we have

$0>(K_{X}+D)\cdot C=(K_{D_{J}}+D^{J})\cdot C\geq K_{D_{J}}\cdot C.$

Let $H_{F}$ be a supporting divisor for $F$. By [Kol96, Chapter II, Theorem
5.8] for any closed point $x\in C$ we can find a rational curve $C^{\prime}$
passing through $x$ and such that

$H_{F}\cdot C^{\prime}\leq 2\dim D_{I}\frac{H_{F}\cdot C}{-K_{D_{I}}\cdot
C}=0.$

Then $H_{F}\cdot C^{\prime}=0$ and hence $C^{\prime}\in[F]$. This shows that
we can assume that $C$ is rational. By [Kol96, Chapter II, Theorem 5.7] for
any $x\in C\subset D_{J}$ there is an effective $1$-cycle $\sum a_{i}C_{i}$ in
$D_{J}$ whose irreducible components are rational curves such that $[C]$ is
effectively algebraically equivalent to $\sum a_{i}C_{i}$, and
$-K_{D_{J}}\cdot C_{i}\leq\dim D_{J}+1$ for all $i$. Since $0=H_{F}\cdot
C=\sum a_{i}H_{F}\cdot C_{i}\geq a_{i}H_{F}\cdot C_{i}\geq 0$, we see that
$H_{F}\cdot C_{i}=0$ and hence $[C_{i}]\in F$. Since $x\in{\mathop{{\rm
Supp\,}}}\sum a_{i}C_{i}$ we can also take $i$ so that $x\in C_{i}$. Choosing
$x$ so that $x\not\in D_{j}$ for $j\in I\backslash J$, we obtain

$0<-(K_{X}+D)\cdot C_{i}=-(K_{D_{J}}+D^{J})\cdot C_{i}\leq-K_{D_{J}}\cdot
C_{i}\leq\dim D_{J}+1.$

∎

For a log contraction $\varphi$ we define its _length_ $l_{D}(\varphi)$ as the
minimum of $-(K_{X}+D)\cdot C$ taken among all rational curves $C$ contracted
by $\varphi$. If the minimum is attained for $C$ then we call $C$ an _extremal
rational curve_. By Proposition 1.5 we have $1\leq l_{D}(\varphi)\leq\dim
X+1.$

######  Lemma 1.6.

If $l_{D}(\varphi)=\dim X+1$ then $X\simeq{\mathbb{P}}^{n}$ and $D=0$. In
particular, if $D\neq 0$ then $l_{D}(\varphi)\leq\dim X$.

###### Proof.

Assume that $l_{D}(\varphi)=\dim X+1$. Then for any extremal rational curve
$C$ contracted by $\varphi$ we have $-(K_{X}+D)\cdot C=\dim X+1$. So
$C\not\subset{\mathop{{\rm Supp\,}}}D$ by Proposition 1.5. In particular,
$-K_{X}\cdot C\geq-(K_{X}+D)\cdot C=\dim X+1$. Let $f:{\mathbb{P}}^{1}\to C$
be the normalization and let $V$ be an irreducible component of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X)$ containing $[f]$. By Ionescu–Wiśniewski’s
inequality (see, e.g., [Kol96, Chapter IV, Corollary 2.6]), $-K_{X}\cdot
C=\dim X+1$ and $\overline{{\rm locus}(V)}=X$. If $D=0$ then $\varphi$
contracts $X$ to a point and hence $X$ is Fano with $b_{2}(X)=1$. Then
$X\simeq{\mathbb{P}}^{n}$ by [CMSB02] and [Keb02, Theorem 1.1]. If $D\neq 0$
then $D\cdot C=0$ and there exists $[g]\in V$ such that the intersection of
$C^{\prime}=g({\mathbb{P}}^{1})$ with $D$ is non-empty. Since $D\cdot
C^{\prime}=D\cdot C=0$, $C^{\prime}$ is contained in ${\mathop{{\rm
Supp\,}}}D$. But ${\mathbb{R}}^{+}[C^{\prime}]={\mathbb{R}}^{+}[C]=R$, so we
get a contradiction with Proposition 1.5. ∎

##  2 Rational curves on snc pairs

In this section we generalize several results on rational curves on smooth
projective varieties to projective snc pairs.

###  2.1 Log Bend-and-Break

Let $X$ be a smooth projective variety and let ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X)$ be the scheme parametrizing morphisms from
${\mathbb{P}}^{1}$ to $X$. This scheme comes with the evaluation morphism
$F:{\mathbb{P}}^{1}\times{\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X)\to X$. For
any family $V\subset{\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X)$ we write $F_{V}$
for the restriction of $F$ to ${\mathbb{P}}^{1}\times V$. We denote by ${\rm
locus}(V)$ the image of $F_{V}$.

For any closed point $x\in X$ we consider the subscheme ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X;0\to x)$ of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X)$ parametrizing $g:{\mathbb{P}}^{1}\to X$ mapping
$0\in{\mathbb{P}}^{1}$ to $x\in X$. For a closed point $x\in{\rm locus}(V)$,
we write $V^{x}$ for the family $V\cap{\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X;0\to x)$ and ${\rm locus}(V,x)$ for the locus of
$V^{x}$.

Now let $(X,D)$ be a projective snc pair and let us set
$U:=X\backslash{\mathop{{\rm Supp\,}}}D$. Let $\\{D_{i}\\}_{i\in I}$ be the
set of irreducible components of $D$ and let $G=\\{G_{i}\\}_{i\in I}$ be a
collection of effective Cartier divisors on ${\mathbb{P}}^{1}$. Then we can
consider the closed subscheme ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X;G\subset D)$ of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X)$ parametrizing morphisms $g:{\mathbb{P}}^{1}\to X$
such that for every irreducible component $D_{i}$ of $D$ we have $G_{i}\subset
g^{*}D_{i}$ (see [KM99, 5.1]). For any $x\in U$, ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X;G\subset D,0\to x)$ denotes the scheme-theoretic
intersection of the schemes ${\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X;G\subset
D)$ and ${\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X;0\to x)$ inside
${\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X)$.

Let $f:{\mathbb{P}}^{1}\to X$ be a fixed morphism such that
$f({\mathbb{P}}^{1})\not\subset{\mathop{{\rm Supp\,}}}D$ and $G_{i}\subset
f^{*}D_{i}$ for all $i\in I$. Let $W$ be an irreducible component of
${\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X;G\subset D)$ containing $[f]$. Then
[KM99, Proposition 5.3] implies that

$\dim_{[f]}W\geq\chi({\mathbb{P}}^{1},f^{*}T_{X})-\deg G=-\deg f^{*}K_{X}-\deg
G+\dim X.$

Moreover, if $G_{i}=f^{*}D_{i}$ for all $i\in I$ then

$T_{[f]}{\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X;G\subset
D)=H^{0}({\mathbb{P}}^{1},f^{*}T_{X}(-\log D)).$

The proof of the following proposition follows that of [KM99, Corollary 5.4]
and its improvement [CZ19, Proposition 6.4].

######  Proposition 2.1.

Assume that $K_{X}+D$ is not nef and let $C$ be an extremal rational curve for
some $(K_{X}+D)$-extremal face $F$. Let $J\subset I$ be the set of all $j\in
I$ for which $C\subset D_{j}$ and let $U_{J}=D_{J}\backslash{\mathop{{\rm
Supp\,}}}D^{J}$. Then for every closed point $x\in C\cap U_{J}$ there exists a
morphism $g:{\mathbb{P}}^{1}\to D_{J}$ such that $x\in g({\mathbb{P}}^{1})$,
$[g({\mathbb{P}}^{1})]\in F$ and $g^{*}(D^{J})$ is supported in at most one
point.

###### Proof.

Let $f:{\mathbb{P}}^{1}\to C$ be the normalization of $C$ and let $V$ be the
connected component of ${\mathop{{\rm Hom}}}({\mathbb{P}}^{1},D_{J})$
containing $[f]$. The _size_ of a $0$-dimensional scheme $G$ of
${\mathbb{P}}^{1}$ is the maximum length of a subscheme contained in $G$,
which is supported at a single point. Let $g:{\mathbb{P}}^{1}\to D_{J}$ be a
morphism with $[g]\in V^{x}$. Since $x\in C^{\prime}:=g({\mathbb{P}}^{1})$, we
have $C^{\prime}\not\subset{\mathop{{\rm Supp\,}}}D^{J}$ and $g^{*}(D^{J})$ is
a well defined Cartier divisor. Let us choose $g$ for which $g^{*}(D^{J})$ has
the greatest size that we denote by $s$. We claim that $g$ satisfies the
required assertions.

Let $G\subset g^{*}(D^{J})$ be a subscheme of length $s$ supported at a single
point. Since it is clear that $[C^{\prime}]\in F$, we need only to check the
last condition. Assume that $g^{*}D^{J}$ is supported in $m\geq 2$ points. Let
$W$ be an irreducible component of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},D_{J};G\subset D^{J},0\to x)$ containing $[g]$. Since
$D^{J}\cdot C^{\prime}\geq(m-1)+s$, we have

$\displaystyle\dim W$
$\displaystyle\geq{\chi({\mathbb{P}}^{1},g^{*}T_{D_{J}})-\deg G-\dim
D_{J}=-K_{D_{J}}\cdot C^{\prime}-s}$
$\displaystyle\geq{-(K_{D_{J}}+D^{J})\cdot C^{\prime}+m-1=-(K_{X}+D)\cdot
C+m-1\geq 2}.$

Note that $C^{\prime}$ is an extremal rational curve, so for large $m\gg 0$ it
has minimal degree with respect to ample divisors of the form
$-(K_{X}+D)+mH_{F}$. Hence we get a contradiction in the same way as in the
proof of [KM99, Corollary 5.4] (or [CZ19, Proposition 6.4]). ∎

######  Remark 2.2.

Note that $D_{J}$ need not be irreducible although every connected component
of $D_{J}$ is smooth and irreducible.

###  2.2 Loci of rational curves on snc pairs

As pointed out in the proof of [KPSW00, Proposition 2.9], the following
proposition due to P. Ionescu and J. Wiśniewski follows from [Kol96, Chapter
IV, Proposition 2.5] (note that unlike claimed in [Kol96], the proof does not
give any information if we do not assume that $V$ is generically unsplit).

######  Proposition 2.3.

Let $X$ be a smooth projective variety and let $C$ be a rational curve in $X$.
Let $f:{\mathbb{P}}^{1}\to C$ be the normalization of $C$ and let $V$ be a
closed irreducible subset of ${\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X)$, which
contains $[f]$ and which is invariant under the action of
${\mathop{\operatorname{Aut}\,}}({\mathbb{P}}^{1})$. If the family $V$ is
unsplit then for any closed point $x\in{\rm locus}(V)$ we have

$\dim V\leq\dim{\rm locus}(V)+\dim{\rm locus}(V,x)+1.$

Let $(X,D)$ be a projective snc pair with $D\neq 0$ and let
$U=X\backslash{\mathop{{\rm Supp\,}}}D$ as before. For any family
$V\subset{\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X)$ we set ${\rm
locus}_{U}(V):={\rm locus}(V)\cap U$ and ${\rm locus}_{U}(V,x):={\rm
locus}(V,x)\cap U$.

If $f:{\mathbb{P}}^{1}\to X$ is a fixed morphism such that
$f({\mathbb{P}}^{1})\not\subset{\mathop{{\rm Supp\,}}}D$ then we can consider
$f^{*}D$ as a collection of effective Cartier divisors on ${\mathbb{P}}^{1}$.
If $g$ with $g({\mathbb{P}}^{1})\not\subset{\mathop{{\rm Supp\,}}}D$ is in the
same connected component of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X;f^{*}D\subset D)$ as $f$ then
$f^{*}D_{i}=g^{*}D_{i}$ for all $i\in I$. This follows from the fact that
$\deg f^{*}D=\deg g^{*}D$ and $f^{*}D_{i}\leq g^{*}D_{i}$ for $i\in I$.

The following proposition is a logarithmic analogue of the above proposition.

######  Proposition 2.4.

Let $C$ be a rational curve in $X$ and let $f:{\mathbb{P}}^{1}\to C$ be the
normalization of $C$. Assume that $f^{-1}(U)\simeq{\mathbb{A}}^{1}$ and let
$W$ be a closed irreducible subset of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X;f^{*}D\subset D)$ containing $[f]$ and invariant
under the action of the subgroup of
${\mathop{\operatorname{Aut}\,}}({\mathbb{P}}^{1})$ fixing the support of
$f^{*}D$. If the family $W$ is unsplit then for any closed point $x\in{\rm
locus}_{U}(W)$ we have

$\dim W\leq\dim{\rm locus}_{U}(W)+\dim{\rm locus}_{U}(W,x).$

###### Proof.

Let $\tilde{W}={\mathop{\operatorname{Aut}\,}}({\mathbb{P}}^{1})\cdot
W\subset{\mathop{{\rm Hom}}}({\mathbb{P}}^{1},X)$. Since $C$ is not contained
in the support of $D$, $W$ is not
${\mathop{\operatorname{Aut}\,}}({\mathbb{P}}^{1})$-invariant. Hence
$\dim\tilde{W}=\dim W+1$. Let us consider the map
$F_{\tilde{W}}:{\mathbb{P}}^{1}\times\tilde{W}\to X$. For any $x\in{\rm
locus}_{U}(W)$ there exists $p\in{\mathbb{P}}^{1}$ and $[g]\in\tilde{W}$ such
that $g(p)=x$. Taking an automorphism $h$ of ${\mathbb{P}}^{1}$ such that
$h(0)=p$, we see that $x=gh(0)$ and $[gh]\in\tilde{W}$. So already the image
of the map $\varphi:\tilde{W}\to X$, $[g]\to g(0)$ contains ${\rm
locus}_{U}(W)$. In particular,
$(\varphi^{-1}(x))_{red}=\\{[g]\in\tilde{W}:g(0)=x\\}=\tilde{W}^{x}$ so we
have

$\dim\tilde{W}^{x}\geq\dim\tilde{W}-\dim{\rm locus}_{U}(W).$

Similarly, the image of the map $\psi:\tilde{W}^{x}\to X$, $[g]\to g(1),$
contains ${\rm locus}_{U}(W,x)\backslash\\{x\\}$. So for any closed point
$y\in{\rm locus}_{U}(W,x)\backslash\\{x\\}$ we have

$\dim\\{[g]\in\tilde{W}:g(0)=x,g(1)=y\\}\geq\dim\tilde{W}^{x}-\dim{\rm
locus}_{U}(W,x).$

If $\dim\\{[g]\in\tilde{W}:g(0)=x,g(1)=y\\}\geq 2$ then $f$ moves, fixing $0$
and $1$, in a $2$-dimensional family and so its image must move. Since the
family $\tilde{W}$ is unsplit, this gives a contradiction with Mori’s Bend and
Break. Therefore

$\dim\tilde{W}=\dim W+1\leq\dim{\rm
locus}_{U}(W)+\dim\tilde{W}^{x}\leq\dim{\rm locus}_{U}(W)+\dim{\rm
locus}_{U}(W,x)+1.$

∎

Every vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^{1}$ splits into a direct
sum of the form $\bigoplus{\mathcal{O}}_{{\mathbb{P}}^{1}}(a_{i})$. We set
${\mathop{\rm rk\,}}^{+}({\mathcal{E}}):=\\#\\{i:a_{i}>0\\}$.

The following proposition is an analogue of [Kol96, Chapter II, Proposition
3.10] in the logarithmic case.

######  Proposition 2.5.

Assume that for $f:{\mathbb{P}}^{1}\to X$, the divisor $f^{*}D$ is supported
in at most one point. Let us fix a closed point $x\in U$ and let $W$ be an
irreducible component of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},X;f^{*}D\subset D)$ containing $[f]$. Then for any
closed point $p\in{\mathbb{P}}^{1}\backslash(\\{0\\}\cup{\mathop{{\rm
Supp\,}}}f^{*}D)$ and any $g:{\mathbb{P}}^{1}\to X$ such that $[g]\in W^{x}$
we have

${\mathop{\rm rk\,}}dF_{W^{x}}(p,[g])={\mathop{\rm rk\,}}^{+}(g^{*}T_{X}(-\log
D)).$

In particular, we have

$\dim{\rm locus}_{U}(W,x)\leq\max_{\\{g:\,[g]\in W^{x}\\}}{\mathop{\rm
rk\,}}^{+}\left(g^{*}T_{X}(-\log D)\right).$

###### Proof.

As in [Kol96, II.3] we have

$T_{{\mathbb{P}}^{1}\times W^{x}}\otimes k(p,[g])=T_{{\mathbb{P}}^{1}}\otimes
k(p)+H^{0}({\mathbb{P}}^{1},g^{*}T_{X}(-\log D)\otimes I_{0})$

and $dF_{W^{x}}(p,[g])=dg(p)+\phi(p,g)$, where
$dg(p):T_{{\mathbb{P}}^{1}}\otimes k(p)\to g^{*}T_{X}\otimes k(p)=T_{X}\otimes
k(g(p))$ is the differential of $g$ at $p$ and

$\phi(p,g):H^{0}({\mathbb{P}}^{1},g^{*}T_{X}(-\log D)\otimes I_{0})\to
g^{*}T_{X}(-\log D)\otimes I_{0}\otimes k(p)=T_{X}\otimes k(g(p))$

is the evaluation map. Let us set $B=(f^{*}D)_{red}$ and recall that
$g^{*}D=f^{*}D$. A standard local computation shows that the differential of
$g$ induces the map $T_{{\mathbb{P}}^{1}}(-B)=T_{{\mathbb{P}}^{1}}(-\log B)\to
g^{*}T_{X}(-\log D)$. So the first claim follows from surjectivity of the
evaluation map

$H^{0}({\mathbb{P}}^{1},T_{{\mathbb{P}}^{1}}(-B)\otimes I_{0})\to
T_{{\mathbb{P}}^{1}}(-B)\otimes I_{0}\otimes k(p)=T_{{\mathbb{P}}^{1}}\otimes
k(p).$

The second part follows from [Har77, Chapter III, Proposition 10.6]. ∎

##  3 Symplectic and contact structures on smooth Lie algebroids

In this section we revise symplectic and contact structures in the Lie
algeborid setup. We also recall and revise Atiyah extensions in the log case
and prove some auxiliary results.

###  3.1 Lie algebroids

Let $X$ be a $k$-variety and let
${\mathbb{L}}=(L,[\cdot,\cdot]_{L},\alpha:L\to T_{X})$ be a $k$-Lie algebroid
on $X$ such that $L$ is a vector bundle (in this case we say that
${\mathbb{L}}$ is _smooth_). Let us recall that ${\mathbb{L}}$ is given by a
sheaf of $k$-Lie algebras structure on $L$ and the anchor map $\alpha$, which
is a map of both ${\mathcal{O}}_{X}$-modules and sheaves of $k$-Lie algebras,
and which satisfies Leibniz’s rule

$[x,fy]=(\alpha(x))(y)+f[x,y]$

for all local sections $f\in{\mathcal{O}}_{X}$ and $x,y\in L$ (see, e.g.,
[Lan14, Definition 2.1]). In the following we usually omit $k$ in the notation
and write ${\mathbb{L}}$ not only for a Lie algebroid but also for the
underlying vector bundle $L$.

For any Lie algebroid ${\mathbb{L}}$ we can consider the canonical de Rham
complex $(\Omega_{{\mathbb{L}}}^{\bullet},d^{\bullet}_{{\mathbb{L}}})$, where
$\Omega_{{\mathbb{L}}}^{m}=\bigwedge^{m}(L^{*})$ (see [Lan14, 2.1]). The
differential in this complex is induced by

$d_{{\mathbb{L}}}\in{\rm
Der}_{k}({\mathcal{O}}_{X},\Omega_{{\mathbb{L}}}^{1})={\mathop{{\rm
Hom}}}_{X}(\Omega_{X}^{1},\Omega_{{\mathbb{L}}}^{1}),$

which is the dual of the anchor map $\alpha:L\to T_{X}$. Note that the de Rham
complex is a complex of ${\mathcal{O}}_{X}$-modules but the maps in this
complex are only $k$-linear.

Let us recall that a map of Lie algebroids is an ${\mathcal{O}}_{X}$-linear
map of underlying vector bundles that commutes with anchors. It is useful to
use the following definition that is modelled on the Lie algebra case:

######  Definition 3.1.

An _abelian extension_ of a Lie algebroid ${\mathbb{L}}$ by
${\mathbb{L}}^{\prime\prime}$ is a short exact sequence of Lie algebroids

$0\to{\mathbb{L}}^{\prime\prime}\to{\mathbb{L}}^{\prime}\to{\mathbb{L}}\to 0$

in which ${\mathbb{L}}^{\prime\prime}$ is a trivial Lie algebroid on a trivial
bundle. An abelian extension is called _central_ if for every $x\in
H^{0}(X,{\mathbb{L}}^{\prime\prime})$, the image of $1\otimes x$ under the
composition
${\mathcal{O}}_{X}\otimes_{k}H^{0}(X,{\mathbb{L}}^{\prime\prime})\stackrel{{\scriptstyle{\mathop{\rm
ev\,}}}}{{\longrightarrow}}{\mathbb{L}}^{\prime\prime}\to{\mathbb{L}}^{\prime}$
is a central section of ${\mathbb{L}}^{\prime}$.

Both abelian and central extensions appear naturally in many places in this
paper. Note that the word “abelian” (or “central”) can be somewhat misleading
as it does not mean that ${\mathbb{L}}^{\prime\prime}$ is abelian (central) in
${\mathbb{L}}^{\prime}$.

###  3.2 Symplectic structure

######  Definition 3.2.

An _${\mathcal{L}}$ -valued symplectic form_ on ${\mathbb{L}}$ is a section
$\omega\in
H^{0}(X,\Omega_{{\mathbb{L}}}^{2}\otimes_{{\mathcal{O}}_{X}}{\mathcal{L}})$
such that

  1. 1.

$\omega$ is _closed_ , i.e., the composition of the induced homomorphism
${\mathcal{L}}^{-1}\to\Omega^{2}_{{\mathbb{L}}}$ with the exterior derivative
$d_{{\mathbb{L}}}:\Omega^{2}_{{\mathbb{L}}}\to\Omega^{3}_{{\mathbb{L}}}$
vanishes,

  2. 2.

$\omega$ is _nondegenerate_ , i.e., the map
$\hat{\omega}:{{\mathbb{L}}}\to\Omega^{1}_{{\mathbb{L}}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{L}}={\mathop{{\mathcal{H}}om}}_{{\mathcal{O}}_{X}}({\mathbb{L}},{\mathcal{L}})$,
$x\to\omega(x\wedge\cdot)$ is an isomorphism.

The second condition implies that the rank of $L$ is even. If ${\mathop{\rm
rk\,}}L=2r$ then $\omega^{\wedge r}$ induces an isomorphism
${\mathcal{O}}_{X}{\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}}\Omega^{2r}_{{\mathbb{L}}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{L}}^{\otimes
r}.$

The above definition generalizes holomorphic symplectic manifolds that
correspond to varieties with ${\mathcal{O}}_{X}$-valued symplectic form on the
canonical Lie algebroid structure on the tangent sheaf. However, the above
notion is not sufficiently well-behaved in general and we need to put on
$\omega$ a slightly stronger assumption:

######  Definition 3.3.

We say that an ${\mathcal{L}}$-valued symplectic form $\omega$ on
${\mathbb{L}}$ is _strong_ if it is locally exact in the analytic topology,
i.e., if for every $p\in X$ there exists an open analytic neighbourhood $p\in
U\subset X^{{\mathop{\rm an}}}$ and
$\eta\in(\Omega^{1}_{{\mathbb{L}}})^{{\mathop{\rm an}}}$ such that
$\omega|_{U}=d_{{\mathbb{L}}}(\eta)$.

In principle, the above notion makes sense only if $k={\mathbb{C}}$. But one
can make sense of it in general by defining all the data over a finitely
generated extension of ${\mathbb{Q}}$ and extending the coefficient field by
embedding this field into ${\mathbb{C}}$. In the following, we use this notion
only to establish bijection in Proposition 3.6.

###  3.3 Contact structure

Let ${\mathcal{L}}$ be a line bundle on $X$ and let $\theta:L\to{\mathcal{L}}$
be an ${\mathcal{O}}_{X}$-linear map with kernel ${\mathcal{F}}$. Although the
Lie bracket $[\cdot,\cdot]_{L}:{\bigwedge}^{2}_{k}L\to L$ is not
${\mathcal{O}}_{X}$-linear, the composition

${\bigwedge}^{2}{\mathcal{F}}\longrightarrow{\bigwedge}^{2}_{k}L\mathop{\longrightarrow}^{[\cdot,\cdot]_{L}}L\mathop{\longrightarrow}^{\theta}{\mathcal{L}},\quad
x\wedge y\to\theta([x,y]_{L})$

is ${\mathcal{O}}_{X}$-linear (see, e.g., [Lan14, Lemma 4.4]).

######  Definition 3.4.

An _${\mathcal{L}}$ -valued contact structure_ on ${\mathbb{L}}$ is a
surjective ${\mathcal{O}}_{X}$-linear map $\theta:L\to{\mathcal{L}}$ such that
${\bigwedge}^{2}{\mathcal{F}}\to{\mathcal{L}},$ $x\wedge
y\to\theta([x,y]_{L})$ is nondegenerate, i.e., the map
$\hat{\theta}:{\mathcal{F}}\to{\mathcal{F}}^{*}\otimes_{{\mathcal{O}}_{X}}{\mathcal{L}}={\mathop{{\mathcal{H}}om}}_{{\mathcal{O}}_{X}}({\mathcal{F}},{\mathcal{L}})$,
$x\to\theta([x,\cdot]_{L})$ is an isomorphism.

Note that our definition implies that ${\mathop{\rm rk\,}}{\mathcal{F}}\geq
2$. Since $\hat{\theta}$ is a non-degenerate skew-symmetric form on
${\mathcal{F}}$ with values in ${\mathcal{L}}$ (i.e.,
$\hat{\theta}=-\hat{\theta}^{*}\otimes\mathop{\rm Id}_{{\mathcal{L}}}$), the
rank of ${\mathcal{F}}$ is even and
${\bigwedge}^{2}{\mathcal{F}}\to{\mathcal{L}}$ induces an isomorphism
$\det{\mathcal{F}}={\bigwedge}^{2r}{\mathcal{F}}{\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}}{\mathcal{L}}^{\otimes
r}$, where $2r$ is the rank of ${\mathcal{F}}$. In particular, $\det
L\simeq{\mathcal{L}}^{r+1}$. Since ${\mathcal{F}}\subset L$ is not closed
under the Lie bracket on $L$, there is no natural Lie algebroid structure on
${\mathcal{F}}$ and we cannot say that $\hat{\theta}$ is a symplectic form.
This problem will be resolved in Proposition 3.6, where we show that there
exists a natural symplectic form compatible with $\hat{\theta}$ on some other
Lie algebroid.

Using the de Rham complex of ${\mathbb{L}}$, we can dually define a contact
structure on ${\mathbb{L}}$ as an _${\mathcal{L}}$ -valued contact form_
$\omega:{\mathcal{L}}^{-1}\to\Omega_{{\mathbb{L}}}^{1}$ such that the induced
${\mathcal{O}}_{X}$-linear map $\omega\wedge(d_{{\mathbb{L}}}\omega)^{\wedge
r}:{\mathcal{L}}^{-(r+1)}\to\det\Omega_{{\mathbb{L}}}^{1}$ is an isomorphism.

The relation between the two structures is that $\omega=\theta^{*}$ and
equivalence of the two definitions follows from the fact that the dual to
${\bigwedge}^{2}{\mathcal{F}}\to{\mathcal{L}},$ $x\wedge
y\to\theta([x,y]_{L})$ is given by the composition

${\mathcal{L}}^{-1}\mathop{\longrightarrow}^{\omega}\Omega_{{\mathbb{L}}}^{1}\mathop{\longrightarrow}^{d_{{\mathbb{L}}}}\Omega_{{\mathbb{L}}}^{2}\to{\bigwedge}^{2}({\mathcal{F}}^{*})=({\bigwedge}^{2}{\mathcal{F}})^{*}.$

This follows from a local computation using the formula for the exterior
derivative

$(d_{{\mathbb{L}}}\eta)(x\wedge
y)=\alpha_{x}(\eta(y))-\alpha_{y}(\eta(x))-\eta([x,y]_{L})$

for any $x,y\in L$ and $\eta\in\Omega_{{\mathbb{L}}}^{1}$.

Sometimes it is also convenient to consider contact structures defined only at
the generic point of $X$:

######  Definition 3.5.

A _generically ${\mathcal{L}}$-valued contact structure_ on ${\mathbb{L}}$ is
a non-zero ${\mathcal{O}}_{X}$-linear map
$\theta:{\mathbb{L}}\to{\mathcal{L}}$ for which there exists a non-empty
Zariski open subset $U\subset X$ such that $\theta|_{U}$ is an
${\mathcal{L}}|_{U}$-valued contact structure on ${\mathbb{L}}|_{U}$.

If $\theta:{\mathbb{L}}\to{\mathcal{L}}$ is a generically
${\mathcal{L}}$-valued contact structure and
$\omega:{\mathcal{L}}^{-1}\to\Omega_{{\mathbb{L}}}^{1}$ is its dual then
$\omega\wedge(d\omega)^{\wedge r}$ defines a non-zero section of
$\det\Omega_{{\mathbb{L}}}^{1}\otimes{\mathcal{L}}^{r+1}$. In particular,
$\det\Omega_{{\mathbb{L}}}^{1}={\mathcal{L}}^{-(r+1)}(B)$ for some effective
divisor $B$. $\omega$ defines an ${\mathcal{L}}$-valued contact structure on
${\mathbb{L}}$ if and only if $B=0$.

###  3.4 Jet bundles, the Atiyah extension and differential operators for Lie
algebroids

Let us recall Atiyah classes defined by Lie algebroids after [CVdB10, 4.2.5
and 8.1]. If ${\mathbb{L}}$ is a Lie algebroid then we define the _first
${\mathbb{L}}$-jet $k$-algebra bundle_ $J^{1}_{{\mathbb{L}}}$ as
${\mathop{{\mathcal{H}}om}}_{{\mathcal{O}}_{X}}({L}\oplus{\mathcal{O}}_{X},{\mathcal{O}}_{X})=\Omega_{{\mathbb{L}}}^{1}\oplus{\mathcal{O}}_{X}$
with a natural commutative $k$-algebra structure given by
$(\eta_{1},f_{1})\cdot(\eta_{2},f_{2})=(f_{2}\eta_{1}+f_{1}\eta_{2},f_{1}f_{2})$.
$J^{1}_{{\mathbb{L}}}$ comes with an
${\mathcal{O}}_{X}$-${\mathcal{O}}_{X}$-bimodule structure, where the left
${\mathcal{O}}_{X}$-module structure is given by the monomorphism

$\alpha_{1}:{\mathcal{O}}_{X}\to J^{1}_{{\mathbb{L}}},\quad f\to(0,f)$

and the right ${\mathcal{O}}_{X}$-module structure is given by the
monomorphism

$\alpha_{2}:{\mathcal{O}}_{X}\to J^{1}_{{\mathbb{L}}},\quad
f\to(d_{{\mathbb{L}}}(f),f).$

Since $\Omega_{{\mathbb{L}}}^{1}$ is an ideal in $J^{1}_{{\mathbb{L}}}$, we
have the short exact sequence

$0\to\Omega_{{\mathbb{L}}}^{1}\longrightarrow
J^{1}_{{\mathbb{L}}}\longrightarrow{\mathcal{O}}_{X}\to 0$

of $J^{1}_{{\mathbb{L}}}$-modules. In general, this sequence is not split
although it is split both as a sequence of left ${\mathcal{O}}_{X}$-modules
and as a sequence of right ${\mathcal{O}}_{X}$-modules.

For a vector bundle ${\mathcal{E}}$ on $X$, we define its _first
${\mathbb{L}}$-jet bundle _ as
$J^{1}_{{\mathbb{L}}}({\mathcal{E}})=J^{1}_{{\mathbb{L}}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}$.
It has a natural structure of a left $J^{1}_{{\mathbb{L}}}$-module and the
above short exact sequence induces the short exact sequence

$0\to{\mathcal{E}}\otimes_{{\mathcal{O}}_{X}}\Omega_{{\mathbb{L}}}^{1}\simeq\Omega_{{\mathbb{L}}}^{1}\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}\longrightarrow
J^{1}_{{\mathbb{L}}}({\mathcal{E}})\longrightarrow{\mathcal{E}}\to 0$

of left $J^{1}_{{\mathbb{L}}}$-modules (hence also of left
${\mathcal{O}}_{X}$-modules). This sequence is called the _${\mathbb{L}}$
-Atiyah extension_ of ${\mathcal{E}}$ and it defines so called _${\mathbb{L}}$
-Atiyah class_ of ${\mathcal{E}}$:

$\alpha_{{\mathcal{E}}}\in{\mathop{{\rm
Ext\,}}}^{1}_{X}({\mathcal{E}},\Omega_{{\mathbb{L}}}^{1}\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}})=H^{1}(X,\Omega_{{\mathbb{L}}}^{1}\otimes_{{\mathcal{O}}_{X}}{\mathop{{{\mathcal{E}}nd\,}}}{\mathcal{E}}).$

This sequence is split as a sequence of sheaves of abelian groups (or even of
sheaves of $k$-vector spaces) with the canonical splitting $d:{\mathcal{E}}\to
J^{1}_{{\mathbb{L}}}({\mathcal{E}})$ induced by $\alpha_{2}$. Existence of an
${\mathcal{O}}_{X}$-linear splitting of this sequence (treated as a sequence
of left ${\mathcal{O}}_{X}$-modules) is equivalent to existence of an
${\mathbb{L}}$-connection
${\mathcal{E}}\to\Omega_{{\mathbb{L}}}^{1}\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}$.

If ${\mathcal{E}}={\mathcal{L}}$ is a line bundle, tensoring the
${\mathbb{L}}$-Atiyah extension of ${\mathcal{L}}$ by ${\mathcal{L}}^{-1}$ (on
the left) gives the extension

$0\to\Omega_{{\mathbb{L}}}^{1}\longrightarrow{\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}})\longrightarrow{\mathcal{O}}_{X}\to
0$

in which
${\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}})={\mathcal{L}}^{-1}\otimes_{{\mathcal{O}}_{X}}J^{1}_{{\mathbb{L}}}({\mathcal{L}})$.
Sometimes we also call this extension the ${\mathbb{L}}$-Atiyah extension of
${\mathcal{L}}$. By construction we have a canonical pushout diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{X}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{{\mathbb{L}}}}$$\textstyle{{\mathcal{A}}_{T_{X}}({\mathcal{L}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{{\mathbb{L}}}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

The $T_{X}$-Atiyah class of ${\mathcal{L}}$ can be identified with the first
Chern class $c_{1}({\mathcal{L}})\in H^{1}(X,\Omega_{X}^{1})$ (sometimes
people use a different normalization in the complex case). The
${\mathbb{L}}$-Atiyah class is therefore the image of $c_{1}({\mathcal{L}})$
under the canonical map $H^{1}(X,\Omega_{X}^{1})\to
H^{1}(X,\Omega_{{\mathbb{L}}}^{1})={\mathop{{\rm
Ext\,}}}^{1}({\mathcal{O}}_{X},\Omega_{{\mathbb{L}}}^{1})$ induced by
$d_{{\mathbb{L}}}$. We denote the corresponding class by
$c_{1}^{{\mathbb{L}}}({\mathcal{L}})$.

In the following we need also a generalization of the Atiyah extension for a
collection
$\underline{{\mathcal{L}}}=({\mathcal{L}}_{1},...,{\mathcal{L}}_{s})$ of line
bundles on $X$. Then we define the _multi- ${\mathbb{L}}$-Atiyah extension of
$\underline{{\mathcal{L}}}$_

$0\to\Omega_{{\mathbb{L}}}^{1}\to{\mathcal{A}}_{{\mathbb{L}}}({\underline{{\mathcal{L}}}})\to{\mathcal{O}}_{X}\to
0$

as the pushout of the direct sum of Atiyah extensions for line bundles
${\mathcal{L}}_{1},...,{\mathcal{L}}_{s}$

$0\to(\Omega_{{\mathbb{L}}}^{1})^{\oplus
s}\to{\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}}_{1})\oplus...\oplus{\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}}_{s})\to{\mathcal{O}}_{X}^{\oplus
s}\to 0$

by the codiagonal map $\nabla:(\Omega_{{\mathbb{L}}}^{1})^{\oplus
s}\to\Omega_{{\mathbb{L}}}^{1}$. Clearly, the extension class of the multi-
Atiyah extension in ${\mathop{{\rm Ext\,}}}^{1}({\mathcal{O}}_{X}^{\oplus
s},\Omega_{{\mathbb{L}}}^{1})=H^{1}(X,\Omega_{{\mathbb{L}}}^{1})^{\oplus s}$
is given by $\oplus_{i=1}^{s}c_{1}^{{\mathbb{L}}}({\mathcal{L}}_{i})$.

Now let ${\mathcal{D}}_{{\mathbb{L}}}$ be the universal enveloping algebra of
the Lie algebroid ${\mathbb{L}}$ and let

${\mathcal{D}}_{{\mathbb{L}}}({\mathcal{L}}):={\mathcal{L}}^{-1}\otimes_{{\mathcal{O}}_{X}}{\mathcal{D}}_{{\mathbb{L}}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{L}}$

be the sheaf of ${\mathbb{L}}$-differential operators acting on
${\mathcal{L}}^{-1}$. A better definition should be in terms of operators but
the obtained sheaf of rings is canonically isomorphic to the above one and we
do not need to use another interpretation. The natural filtration of
${\mathcal{D}}_{{\mathbb{L}}}$ given by the order of differential operators
leads to the increasing filtration
${\mathcal{D}}_{{\mathbb{L}}}^{0}({\mathcal{L}})={\mathcal{O}}_{X}\subset{\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\subset...\subset{\mathcal{D}}_{{\mathbb{L}}}({\mathcal{L}})$.
This induces a central extension of Lie algebroids

$0\to{\mathcal{O}}_{X}\longrightarrow{\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\longrightarrow{\mathbb{L}}\to 0.$

The Lie algebroid structure on ${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})$ is defined by the Lie bracket coming from the associative
algebra structure on ${\mathcal{D}}_{{\mathbb{L}}}({\mathcal{L}})$ and the
anchor map ${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\longrightarrow{\mathbb{L}}\to T_{X}$. The above extension
can be identified with the dual to the ${\mathbb{L}}$-Atiyah extension of
${\mathcal{L}}$ and its extension class in ${\mathop{{\rm
Ext\,}}}^{1}({\mathbb{L}},{\mathcal{O}}_{X})=H^{1}(X,\Omega^{1}_{{\mathbb{L}}})$
is given by $-c_{1}^{{\mathbb{L}}}({\mathcal{L}})$. In particular, we have
${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})=({\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}}))^{*}$, i.e.,
${\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}})=\Omega_{{\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})}^{1}$.

If $\underline{{\mathcal{L}}}=({\mathcal{L}}_{1},...,{\mathcal{L}}_{s})$ is a
collection of line bundles then we can construct

$0\to{\mathcal{O}}_{X}^{\oplus
s}\longrightarrow{\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}(\underline{{\mathcal{L}}})\longrightarrow{\mathbb{L}}\to 0$

as the pullback of the direct sum of

$0\to{\mathcal{O}}_{X}^{\oplus
s}\longrightarrow{\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({{\mathcal{L}}_{1}})\oplus...\oplus{\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({{\mathcal{L}}_{s}})\longrightarrow{\mathbb{L}}^{\oplus s}\to 0$

by the diagonal map $\Delta:{\mathbb{L}}\to{\mathbb{L}}^{\oplus s}$. The class
of this extension is given by
$-\oplus_{i=1}^{s}c_{1}^{{\mathbb{L}}}({\mathcal{L}}_{i})$. Note that
${\mathcal{D}}_{{\mathbb{L}}}^{\leq 1}(\underline{{\mathcal{L}}})$ has a
natural structure of a Lie algebroid for which the above extension is a
central extension of Lie algebroids. Under the identification
${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}(\underline{{\mathcal{L}}})=({\mathcal{A}}_{{\mathbb{L}}}({\underline{{\mathcal{L}}}}))^{*}$,
the anchor map corresponds to the dual of
$\Omega_{X}^{1}\to\Omega_{{\mathbb{L}}}^{1}\to{\mathcal{A}}_{{\mathbb{L}}}({\underline{{\mathcal{L}}}})$.

###  3.5 Relation between contact and symplectic structures

Below we describe generalization of the classical process of symplectification
of a contact manifold (see, e.g., [Arn89, Appendix 4 E]) to arbitrary contact
structures on Lie algebroids. We follow the approach of [LeB95, Proposition
2.4] and [Bea98, Lemma 1.4] (see also [KPSW00, 2.1 and 2.2]). Classically this
gives a relation between contact structures on a variety and
${\mathbb{G}}_{m}$-invariant symplectic structures on the associated principal
${\mathbb{G}}_{m}$-bundle . The Lie algebroid point of view makes this a
relation between contact and symplectic structures on the same variety but for
different Lie algebroids.

######  Proposition 3.6.

Let ${\mathbb{L}}$ be a $k$-Lie algebroid on projective $X$ and let
${\mathcal{L}}$ be a fixed line bundle on $X$. There exists a natural
bijection between ${\mathcal{L}}$-valued contact structures on ${\mathbb{L}}$
and strong ${\mathcal{L}}$-valued symplectic forms on the Lie algebroid
${\mathcal{D}}_{{\mathbb{L}}}^{\leq 1}({\mathcal{L}})$.

###### Proof.

In the proof we can assume that $k={\mathbb{C}}$. Let

$\pi:{{\mathcal{L}}}^{\bullet}={\mathop{{\rm
Spec\,}}}_{X}(\bigoplus_{m\in{\mathbb{Z}}}{\mathcal{L}}^{m})\to X$

be the principal ${\mathbb{G}}_{m}$-bundle associated to ${\mathcal{L}}^{-1}$
(in particular it comes with a canonical ${\mathbb{G}}_{m}$-action). By the
projection formula
$\pi_{*}(\pi^{*}{\mathcal{L}}^{-1})\simeq\bigoplus_{m\in{\mathbb{Z}}}{\mathcal{L}}^{m}$,
so $\pi^{*}{\mathcal{L}}^{-1}$ has a canonical section $s$ corresponding to
$1\in H^{0}(X,{\mathcal{L}}^{0}\simeq{\mathcal{O}}_{X})$. This section
trivializes $\pi^{*}{\mathcal{L}}^{-1}$. Then $\mu=s^{-1}{ds}$ gives a well-
defined ${\mathbb{G}}_{m}$-invariant relative $1$-form that trivializes
$\Omega^{1}_{{\mathcal{L}}^{\bullet}/X}$. The push-out of the sequence

$0\to\pi^{*}(\Omega_{X}^{1})\longrightarrow\Omega^{1}_{{\mathcal{L}}^{\bullet}}\longrightarrow\Omega^{1}_{{\mathcal{L}}^{\bullet}/X}\simeq{\mathcal{O}}_{{\mathcal{L}}^{\bullet}}\to
0$

by pullback of the dual of the anchor map gives a short exact sequence

$0\to\pi^{*}(\Omega_{{\mathbb{L}}}^{1})\longrightarrow{\mathcal{E}}\longrightarrow{\mathcal{O}}_{{\mathcal{L}}^{\bullet}}\to
0.$

Pushing down this sequence to $X$ we get

$0\to\bigoplus_{m\in{\mathbb{Z}}}{\mathcal{L}}^{m}\otimes\Omega_{{\mathbb{L}}}^{1}\longrightarrow\pi_{*}{\mathcal{E}}=\bigoplus_{m\in{\mathbb{Z}}}({\mathcal{E}})_{m}\longrightarrow\bigoplus_{m\in{\mathbb{Z}}}{\mathcal{L}}^{m}\to
0,$

where $({\mathcal{E}})_{m}$ denotes the weight $m$ subbundle of
$\pi_{*}{\mathcal{E}}$ with respect to the ${\mathbb{G}}_{m}$-action. We have
$({\mathcal{E}})_{0}={\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}})$ and the
$0$-th grading of the above sequence gives the ${\mathbb{L}}$-Atiyah extension
of ${\mathcal{L}}$. Therefore

$({\bigwedge}^{2}{\mathcal{E}})_{1}={\mathcal{L}}\otimes{\bigwedge}^{2}{\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}})={\mathcal{L}}\otimes\Omega_{{\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})}^{2}$

Let $\omega\in H^{0}(X,{\mathcal{L}}\otimes\Omega_{{\mathbb{L}}}^{1})$ be an
${\mathcal{L}}$-valued contact form on ${\mathbb{L}}$.

Let us choose an open covering $\\{U_{i}\\}$ of $X$ such that
${\mathcal{L}}^{-1}|_{U_{i}}$ is free with generator $\tau_{i}$ and write
$\tau_{i}=g_{ij}\tau_{j}$ on $U_{i}\cap U_{j}$. Then we can write
$\omega|_{U_{i}}=\tau_{i}^{-1}\omega_{i}$, where
$\omega_{i}\in\Omega^{1}_{{\mathbb{L}}}(U_{i})$ satisfy
$\omega_{i}=g_{ij}\omega_{j}$ on $U_{i}\cap U_{j}$. If we write
$s_{i}=s|_{\pi^{-1}(U_{i})}$ then we have $s_{i}=\pi^{*}(g_{ij})s_{j}$ on
$\pi^{-1}(U_{i}\cap U_{j})$ and $\eta_{i}=s_{i}^{-1}\pi^{*}(\omega_{i})$ glue
to a globally defined section $\eta$ of ${\mathcal{E}}$. Then

$d\eta_{i}=s_{i}^{-1}\left(\pi^{*}(d_{{\mathbb{L}}}\omega_{i})+\pi^{*}\omega_{i}\wedge\frac{ds_{i}}{s_{i}}\right)$

glue to a global section $d\eta$ of ${\bigwedge}^{2}{\mathcal{E}}$. To check
this note that

$\displaystyle
s_{i}^{-1}\left(d_{{\mathbb{L}}}\omega_{i}+\omega_{i}\wedge\frac{ds_{i}}{s_{i}}\right)$
$\displaystyle=s_{i}^{-1}\left(\left(g_{ij}d_{{\mathbb{L}}}\omega_{j}-\omega_{j}\wedge
dg_{ij}\right)+\left(g_{ij}\omega_{j}\wedge\left(\frac{ds_{j}}{s_{j}}+\frac{dg_{ij}}{g_{ij}}\right)\right)\right)$
$\displaystyle=s_{j}^{-1}\left(d_{{\mathbb{L}}}\omega_{j}+\omega_{j}\wedge\frac{ds_{j}}{s_{j}}\right),$

where we omitted pullbacks in the notation. Note that $d\eta$ is non-
degenerate as

$(d\eta)^{\wedge(r+1)}|_{\pi^{-1}(U_{i})}=(r+1)s_{i}^{-(r+1)}(\pi^{*}(d_{{\mathbb{L}}}\omega_{i}))^{\wedge
r}\wedge\pi^{*}\omega_{i}\wedge\frac{ds_{i}}{s_{i}}$

gives a trivialization of $\det{\mathcal{E}}$. Since $d\eta$ is a weight $1$
section of ${\bigwedge}^{2}{\mathcal{E}}$, its pushforward to $X$ gives an
${\mathcal{L}}$-valued symplectic form $\omega_{{\mathcal{L}}}$ on the Lie
algebroid ${\mathcal{D}}_{{\mathbb{L}}}^{\leq 1}({\mathcal{L}})$. The equality
$d\omega_{{\mathcal{L}}}=0$ follows easily from the construction.

To see the converse we note that an ${\mathcal{L}}$-valued symplectic form on
${\mathcal{D}}_{{\mathbb{L}}}^{\leq 1}({\mathcal{L}})$ gives a section of
$({\bigwedge}^{2}{\mathcal{E}})_{1}$. This induces a section of the
analytification $({\bigwedge}^{2}{\mathcal{E}})_{1}^{{\mathop{\rm an}}}$,
which as a section on ${{\mathcal{L}}^{\bullet}}^{{\mathop{\rm an}}}$ is of
the form $d\eta$ for some weight $1$ section $\eta$ of
${\mathcal{E}}^{{\mathop{\rm an}}}$ (here we use our assumption that the
symplectic form is locally exact in the analytic topology). Then $s\eta$ gives
rise to the required ${\mathcal{L}}$-valued contact form on
${{\mathbb{L}}}^{{\mathop{\rm an}}}$. The obtained form is only analytic but
the final contact form is defined on projective $X$, so we can use GAGA to
prove that it comes from an algebraic contact form. ∎

######  Remark 3.7.

As in the proof of [LeB95, Proposition 2.4] we have the following commutative
diagram:

$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\theta}}$$\scriptstyle{\simeq}$$\textstyle{\mathbb{L}}$$\textstyle{{\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\omega}_{{\mathcal{L}}}}$$\scriptstyle{\simeq}$$\textstyle{{\mathcal{F}}^{*}\otimes{\mathcal{L}}}$$\textstyle{\Omega^{1}_{{\mathbb{L}}}\otimes{\mathcal{L}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{{\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})}\otimes{\mathcal{L}}}$

This diagram shows that the induced map
$\Omega^{1}_{{\mathbb{L}}}\otimes{\mathcal{L}}\to{\mathbb{L}}$ (using inverses
of the isomorphisms) is compatible with both contact form on ${\mathbb{L}}$
and symplectic form on ${\mathcal{D}}_{{\mathbb{L}}}^{\leq 1}({\mathcal{L}})$.

######  Remark 3.8.

Proposition 3.6 holds also if we replace strong symplectic form by
“algebraically strong” symplectic forms defined using étale (or even Zariski)
topology. However, this would not be very natural as the Poincaré lemma does
not hold even for the usual algebraic de Rham complex.

###  3.6 Atiyah extensions on snc pairs

Let $(X,D)$ be a projective snc pair. The logarithmic tangent bundle
$T_{X}(-\log D)$ carries a canonical Lie algebroid structure with canonical
inclusion $\alpha:T_{X}(-\log D)\to T_{X}$ as the anchor map and with the Lie
bracket induced from the Lie bracket on
$T_{X}=Der_{k}({\mathcal{O}}_{X},{\mathcal{O}}_{X})$. By abuse of notation we
denote this Lie algebroid again by $T_{X}(-\log D)$ and we call it the
canonical Lie algebroid.

If ${\mathcal{L}}$ is a line bundle on $X$ then we usually write
${\mathcal{A}}_{{\mathcal{L}}}$ for the Atiyah bundle
${\mathcal{A}}_{T_{X}}({\mathcal{L}})$ and ${\mathcal{B}}_{{\mathcal{L}}}$ for
the logarithmic Atiyah bundle ${\mathcal{A}}_{T_{X}(-\log D)}({\mathcal{L}})$.
In the following we write $c_{1}^{\log}({\mathcal{L}})$ for
$c_{1}^{T_{X}(-\log D)}({\mathcal{L}})$, i.e., for the image of
$c_{1}({\mathcal{L}})$ under the canonical map $H^{1}(X,\Omega_{X}^{1})\to
H^{1}(X,\Omega_{X}^{1}(\log D))$. Similarly, if $\underline{{\mathcal{L}}}$ is
a collection of line bundles then we write
${\mathcal{A}}_{\underline{{\mathcal{L}}}}$ for the multi-Atiyah bundle of
$\underline{{\mathcal{L}}}$ and ${\mathcal{B}}_{\underline{{\mathcal{L}}}}$
for the logarithmic multi-Atiyah bundle.

Let $(Y,B)$ be another projective snc pair and let $\varphi:X\to Y$ be a
morphism. The canonical surjection $\Omega_{X}^{1}\to\Omega^{1}_{X/Y}$ allows
us to consider the relative Atiyah extension

$0\to\Omega_{X/Y}^{1}\to{\mathcal{A}}_{{\mathcal{L}}/Y}\to{\mathcal{O}}_{X}\to
0$

defined by the pushout of the standard Atiyah extension. The class of this
extension in $H^{1}(X,\Omega_{X/Y}^{1})={\mathop{{\rm
Ext\,}}}^{1}({\mathcal{O}}_{X},\Omega_{X/Y}^{1})$ is the image of
$c_{1}({\mathcal{L}})$ under the induced map $H^{1}(X,\Omega_{X}^{1})\to
H^{1}(X,\Omega_{X/Y}^{1})$.

Let us write $D=\sum_{i\in I}D_{i}$ as a sum of smooth irreducible $D_{i}$.
Then for any $J\subset I$ we write $D_{J}$ for $\bigcap_{j\in J}D_{j}$. Let us
assume that $\varphi$ is smooth and the restriction of $\varphi$ to each
$D_{J}$ is also smooth. Then we can decompose $D$ into the horizontal part
$D^{h}$ corresponding to components that dominate $Y$ and the vertical part
$D^{v}$ that consists of the remaining components. Let us assume that
$D^{v}=\varphi^{-1}(B)$.

Then $\Omega^{1}_{X/Y}(\log D^{h})=\Omega^{1}_{X}(\log
D^{h})/\varphi^{*}\Omega^{1}_{Y}$ is locally free and it fits into the short
exact sequence

$0\to\Omega^{1}_{X/Y}\to\Omega^{1}_{X/Y}(\log
D^{h})\to{\bigoplus}{\mathcal{O}}_{D^{h}_{i}}\to 0$

in which $D^{h}_{i}$ are the irreducible components of $D^{h}$. Let us also
note that $\Omega^{1}_{X/Y}(\log D^{h})$ is canonically isomorphic to
$\Omega^{1}_{X}(\log D)/\varphi^{*}\Omega^{1}_{Y}(\log B)$. So as above we can
define the relative logarithmic Atiyah extension of $(X,D)/(Y,B)$ as the
extension

$0\to\Omega^{1}_{X/Y}(\log
D^{h})\to{\mathcal{B}}_{{\mathcal{L}}/Y}\to{\mathcal{O}}_{X}\to 0,$

whose class in $H^{1}(X,\Omega^{1}_{X/Y}(\log D^{h}))$ is the image of
$c_{1}^{\log}({\mathcal{L}})$ under the canonical map

$H^{1}(X,\Omega^{1}_{X}(\log D))\to H^{1}(X,\Omega^{1}_{X/Y}(\log D^{h})).$

This extension can be identified with the relative logarithmic Atiyah
extension of ${\mathcal{L}}$ for $(X,D^{h})/Y$. By construction we have the
following commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{X/Y}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{A}}_{{\mathcal{L}}/Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{X/Y}(\log
D^{h})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{B}}_{{\mathcal{L}}/Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bigoplus}{\mathcal{O}}_{D^{h}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\mathop{\rm
coker\,}}\alpha\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

Note also that our definitions imply that the following diagram is
commutative:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varphi^{*}\Omega^{1}_{Y}(\log
B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varphi^{*}\Omega^{1}_{Y}(\log
B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{X}(\log
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{B}}_{{\mathcal{L}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{X/Y}(\log
D^{h})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{B}}_{{\mathcal{L}}/Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

Therefore we get the following lemma:

######  Lemma 3.9.

Under the above assumptions there exist canonical short exact sequences:

$0\to{\mathcal{A}}_{{\mathcal{L}}/Y}\to{\mathcal{B}}_{{\mathcal{L}}/Y}\to{\bigoplus}{\mathcal{O}}_{D^{h}_{i}}\to
0$

and

$0\to\varphi^{*}\Omega^{1}_{Y}(\log
B)\to{\mathcal{B}}_{{\mathcal{L}}}\to{\mathcal{B}}_{{\mathcal{L}}/Y}\to 0.$

For future reference we also note the following easy lemma:

######  Lemma 3.10.

Let

$0\to{\mathcal{O}}_{X}^{\oplus m}\to{\mathbb{L}}\to T_{X}(-\log D)\to 0$

be a short exact sequence of Lie algebroids and let ${\mathcal{L}}$ be a line
bundle on $X$. Then we have an induced short exact sequence

$0\to{\mathcal{B}}_{{\mathcal{L}}}\to{\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}})\to{\mathcal{O}}_{X}^{\oplus
m}\to 0.$

###### Proof.

The assertion follows from the following commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{X}^{1}(\log
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{{\mathbb{L}}}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}^{\oplus
m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{B}}_{{\mathcal{L}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{{\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathop{\rm
coker\,}}\beta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

(cf. Subsection 3.4). ∎

###  3.7 Hyperplane subbundles in projective bundles

Let us consider the following situation. Let $Y$ be a smooth projective
variety and let $\pi:X={\mathbb{P}}({\mathcal{E}})\to Y$ be a projectivization
of a vector bundle ${\mathcal{E}}$ of rank $m+1$. Let us set
${\mathcal{L}}:={\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}})}(1)$. In $X$ we
consider a smooth divisor $D$, which is also a projective bundle over $Y$.
Assume that over some point $y\in Y$ we have
$(X_{y},D_{y})\simeq({\mathbb{P}}^{m},H)$, where $H$ is a hyperplane. Then
${\mathcal{L}}(-D)|_{X_{y}}\simeq{\mathcal{O}}_{X_{y}}$ and since
${\mathop{\rm Pic\,}}{\mathbb{P}}({\mathcal{E}})\simeq{\mathop{\rm
Pic\,}}Y\times{\mathbb{Z}}$, there exists some line bundle ${\mathcal{M}}$ on
$Y$ such that

${\mathcal{L}}(-D)\simeq\pi^{*}{\mathcal{M}}.$

Let us set ${\mathcal{G}}:=\pi_{*}({\mathcal{L}}|_{D})$. This is a vector
bundle and
$(D,{\mathcal{L}}|_{D})\simeq({\mathbb{P}}({\mathcal{G}}),{\mathcal{O}}_{{\mathbb{P}}({\mathcal{G}})}(1))$.
By the projection formula we have $\pi_{*}({\mathcal{L}}(-D))={\mathcal{M}}$
and $R^{1}\pi_{*}({\mathcal{L}}(-D))=0$. Pushing down the standard short exact
sequence

$0\to{\mathcal{L}}(-D)\to{\mathcal{L}}\to{\mathcal{L}}|_{D}\to 0,$

we get the short exact sequence

$0\to{\mathcal{M}}\to{\mathcal{E}}\to{\mathcal{G}}\to 0.$

In particular, we have
$\det{\mathcal{G}}\simeq\det{\mathcal{E}}\otimes{\mathcal{M}}^{-1}$.

######  Lemma 3.11.

$\Omega_{X/Y}^{1}(\log D)$ is canonically isomorphic to
$\pi^{*}{\mathcal{G}}(-1)$.

###### Proof.

Since $\pi$ is a projective bundle, we have $\pi_{*}(\Omega_{X/Y}^{1}(1))=0$
and $R^{1}\pi_{*}(\Omega_{Z/T}^{1}(1))=0$. We have a short exact sequence

$0\to\Omega_{X/Y}^{1}(1)\to(\Omega_{X/Y}^{1}(\log
D))(1)\to{\mathcal{O}}_{D}(1)\to 0,$

where the first non-zero map is the canonical inclusion and the second one is
the relative Poincaré residue. Pushing the above sequence down to $Y$, we get
a canonical isomorphism

$\pi_{*}((\Omega_{X/Y}^{1}(\log{D}))(1))\mathop{\longrightarrow}^{\simeq}\pi_{*}({\mathcal{O}}_{D}(1))={\mathcal{G}}.$

Let us recall that if $H$ is a hyperplane in ${\mathbb{P}}^{m}$ then
$(\Omega_{{\mathbb{P}}^{m}}^{1}(\log
H))(1)\simeq{\mathcal{O}}_{{\mathbb{P}}^{m}}^{\oplus m}$. So the relative
evaluation map

$\pi^{*}\pi_{*}((\Omega_{X/Y}^{1}(\log{D}))(1))\longrightarrow(\Omega_{X/Y}^{1}(\log{D}))(1)$

is an isomorphism on the fibers of $\pi$ and hence it is an isomorphism. This
shows that the relative logarithmic cotangent bundle
$\Omega_{X/Y}^{1}(\log{D})$ is canonically isomorphic to
$\pi^{*}{\mathcal{G}}(-1)$. ∎

In the following lemma we use the notation from Subsection 3.6.

######  Lemma 3.12.

We have a canonical short exact sequence

$0\to\pi^{*}{\mathcal{A}}_{{{\mathcal{M}}}}\to{\mathcal{B}}_{{\mathcal{L}}}\to\pi^{*}{\mathcal{G}}(-1)\to
0.$

###### Proof.

In the short exact sequence

$0\to\Omega_{X}^{1}\to\Omega_{X}^{1}(\log{D})\to{\mathcal{O}}_{D}\to 0,$

the connecting map $H^{0}(X,{\mathcal{O}}_{D})\to H^{1}(X,\Omega_{X}^{1})$
maps $1$ to $c_{1}({\mathcal{O}}_{X}({D}))$. So the image of
$c_{1}({\mathcal{O}}_{X}({D}))=c_{1}({\mathcal{L}})-c_{1}(\pi^{*}{\mathcal{M}})$
in $H^{1}(X,\Omega_{X}^{1}(\log{D}))$ vanishes. This shows that
$c_{1}^{\log}({\mathcal{L}})=c_{1}^{\log}(\pi^{*}{\mathcal{M}})$ and hence the
logarithmic Atiyah extensions of ${\mathcal{L}}$ and $\pi^{*}{\mathcal{M}}$
for the pair $(X,D)$ are isomorphic. By naturality of the first Chern class
$\pi^{*}c_{1}({\mathcal{M}})\in H^{1}(X,\pi^{*}\Omega_{Y}^{1})$ is mapped to
$c_{1}(\pi^{*}{\mathcal{M}})\in H^{1}(X,\Omega_{X}^{1})$. Thus we get the
following commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{*}\Omega_{Y}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{*}{\mathcal{A}}_{{\mathcal{M}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{*}{\mathcal{O}}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{X}^{1}(\log
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{B}}_{{\mathcal{L}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Hence by Lemma 3.11 we have the short exact sequence

$0\to\pi^{*}{\mathcal{A}}_{{{\mathcal{M}}}}\to{\mathcal{B}}_{{\mathcal{L}}}\to\Omega_{X/Y}^{1}(\log{D})\simeq\pi^{*}{\mathcal{G}}(-1)\to
0.$

∎

######  Remark 3.13.

Let us point out that the image of
$\pi^{*}{\mathcal{A}}_{{\mathcal{M}}}\to{\mathcal{B}}_{{\mathcal{L}}}\to{\mathcal{B}}_{{\mathcal{L}}/Y}$
is isomorphic to ${\mathcal{O}}_{X}$, so it gives a canonical splitting of the
relative logarithmic Atiyah extension

$0\to\Omega_{X/Y}^{1}(\log{D})\to{\mathcal{B}}_{{\mathcal{L}}/Y}\to{\mathcal{O}}_{X}\to
0.$

Since $\Omega_{X/Y}^{1}(\log{D})\simeq\pi^{*}{\mathcal{G}}(-1)$, this
splitting can be also given by the relative evaluation map
${\mathcal{O}}_{X}\simeq\pi^{*}\pi_{*}{\mathcal{B}}_{{\mathcal{L}}/Y}\to{\mathcal{B}}_{{\mathcal{L}}/Y}.$
This follows from the fact that
$\pi^{*}{\mathcal{A}}_{{\mathcal{M}}}\to{\mathcal{B}}_{{\mathcal{L}}/Y}$
factors through
$\pi^{*}{\mathcal{A}}_{{\mathcal{M}}}\to\pi^{*}\pi_{*}{\mathcal{B}}_{{\mathcal{L}}/Y}$.

##  4 Contact structures on special Lie algebroids

In this section we define a semi-contact structure on an snc pair and classify
such structures in case $b_{2}=1$. Then we study the Kodaira dimension of
semi-contact snc pairs and prove some result on existence of lines.

###  4.1 Contact snc pairs and special Lie algebroids

Let $(X,D)$ be an snc pair.

######  Definition 4.1.

Let ${\mathcal{L}}$ be a line bundle on $X$. A _(generically)
${\mathcal{L}}$-valued contact structure_ on $(X,D)$ is a (generically)
${\mathcal{L}}$-valued contact structure on the canonical Lie algebroid
$T_{X}(-\log D)$. We say that $(X,D,{\mathcal{L}})$ is _(generically) contact_
if $(X,D)$ is an snc pair, and there exists a line bundle ${\mathcal{L}}$ on
$X$ and a (generically) ${\mathcal{L}}$-valued contact structure on $(X,D)$.
Sometimes we say that $(X,D)$ is _(generically) contact_ if there exists a
line bundle ${\mathcal{L}}$ on $X$ such that $(X,D,{\mathcal{L}})$ is
(generically) contact.

Note that if $\theta$ is a generically ${\mathcal{L}}$-valued contact
structure on $(X,D)$ then there exists a non-empty Zariski open subset
$V\subset X$ such that $\theta|_{V}$ is an ${\mathcal{L}}|_{V}$-valued contact
structure on $V$ (it is sufficient to take $V=U\backslash{\mathop{{\rm
Supp\,}}}D$ for $U$ in the definition). So $(X,D)$ is generically contact if
and only if $X$ is generically contact. Clearly, a smooth projective variety
which is birational to a generically contact variety is also generically
contact.

Note that [BKK22, Section 1] uses a different definition of a generically
contact variety in which the map $T_{X}\to{\mathcal{L}}$ is surjective. This
notion strongly restricts possible varieties and it depends on the choice of a
birational model.

Let $(X,D)$ be a projective snc pair. We have on $X$ a stratification induced
by the components of the divisor $D$. To describe a stratum, we write
$D=\sum_{i\in I}D_{i}$ as a sum of irreducible divisors and we fix $J\subset
I$. Then a stratum $Z$ is an irreducible component of $D_{J}$. In the
following we write $D^{Z}$ for the divisor $D^{J}|_{Z}$. To make things more
precise we write $\imath$ for the closed embedding $Z\hookrightarrow X$. An
important role in this paper is played by the following Lie algebroid
structure induced on $Z$.

######  Lemma 4.2.

Let $m=\dim X-\dim Z$. Then we have a canonical Lie algebroid structure on
$\imath^{*}T_{X}(-\log D)$ and a canonical central extension

$0\to{\mathcal{O}}_{Z}^{\oplus m}\to\imath^{*}T_{X}(-\log D)\to T_{Z}(-\log
D^{Z})\to 0$

of Lie algebroids.

###### Proof.

Exactness of the above sequence as a sequence of ${\mathcal{O}}_{Z}$-modules
follows from standard short exact sequences for logarithmic bundles by taking
duals. For example, the first non-zero map in the sequence is dual to the
direct sum of restrictions to $Z$ of Poincaré residues $\Omega^{1}_{X}(\log
D)\to{\mathcal{O}}_{D_{i}}$ for $i\in J$. The anchor map for
$\imath^{*}T_{X}(-\log D)$ is given by the composition $\imath^{*}T_{X}(-\log
D)\to T_{Z}(-\log D^{Z})\hookrightarrow T_{Z}$. The Lie bracket for
$\imath^{*}T_{X}(-\log
D)={\mathcal{O}}_{Z}\otimes_{\imath^{-1}{\mathcal{O}}_{X}}\imath^{-1}T_{X}(-\log
D)$ is given by the formula

$[f_{1}\otimes\delta_{1},f_{2}\otimes\delta_{2}]=1\otimes[\tilde{f}_{1}\delta_{1},\tilde{f}_{2}\delta_{2}],$

where $f_{1},f_{2}\in{\mathcal{O}}_{Z}$, $\delta_{1},\delta_{2}\in T_{X}(-\log
D)$ and $\tilde{f}_{i}\in{\mathcal{O}}_{X}$ is a lift of $f_{i}$ for $i=1,2$.
It is easy to check that the formula does not depend on the choice of lifts
(here it is convenient to interpret the Lie bracket as the standard bracket in
the ring of differential operators and use the fact that if $f\in I_{Z}$ then
$\delta(f)\in I_{Z}$ for $\delta\in T_{X}(-\log D)$). This gives rise to the
required Lie algebroid structure. It is also clear from the formula, that the
map $\imath^{*}T_{X}(-\log D)\to T_{Z}(-\log D^{Z})$ is a map of Lie
algebroids. So the anchor map on the kernel is trivial and hence the
corresponding Lie algebra structure is ${\mathcal{O}}_{Z}$-linear. Now let us
choose local (algebraic) coordinates $(x_{1},...,x_{n})$ so that $Z$ is given
by $x_{1}=...=x_{m}=0$. Then $T_{X}(-\log D)$ has a local basis
$x_{1}\frac{\partial}{\partial x_{1}},...,x_{m}\frac{\partial}{\partial
x_{m}},\frac{\partial}{\partial x_{m+1}},...,\frac{\partial}{\partial x_{n}}$
and the kernel of $\imath^{*}T_{X}(-\log D)\to T_{Z}(-\log D^{Z})$ is
generated as an ${\mathcal{O}}_{Z}$-module by
$1\otimes\left(x_{i}\frac{\partial}{\partial x_{i}}\right)$ for $i=1...,m$.
Since $\left[\frac{\partial}{\partial x_{i}},\frac{\partial}{\partial
x_{j}}\right]=0$, we have

$1\otimes\left[x_{i}\frac{\partial}{\partial x_{i}},\sum_{j\leq
m}f_{j}x_{j}\frac{\partial}{\partial
x_{j}}+\sum_{j>m}f_{j}\frac{\partial}{\partial x_{j}}\right]=\sum_{j\leq
m}x_{i}\frac{\partial f_{j}}{\partial x_{i}}\otimes
x_{j}\frac{\partial}{\partial x_{j}}+\sum_{j>m}x_{i}\frac{\partial
f_{j}}{\partial x_{i}}\otimes\frac{\partial}{\partial x_{j}}=0$

for all $1\leq i\leq m$. So the Lie bracket on the kernel is trivial and
$\left\\{1\otimes\left(x_{i}\frac{\partial}{\partial
x_{i}}\right)\right\\}_{i=1,..,m}$ define central sections of
$\imath^{*}T_{X}(-\log D)$. ∎

######  Remark 4.3.

The above lemma allows us to upgrade $\imath:Z\hookrightarrow X$ to a morphism
of D-schemes

$(Z,{\mathcal{D}}_{T_{Z}(-\log D^{Z})})\to(X,{\mathcal{D}}_{T_{X}(-\log D)})$

in the sense of [BB93, 1.4.2].

######  Remark 4.4.

One can easily check that the class of the extension from Lemma 4.2 is given
by

$\bigoplus_{i\in J}c_{1}^{\log}({\mathcal{O}}_{Z}(D_{i}))\in{\mathop{{\rm
Ext\,}}}^{1}(T_{Z}(-\log D^{Z}),{\mathcal{O}}_{Z}^{\oplus
m})=H^{1}(Z,\Omega_{Z}^{1}(\log D^{Z}))^{\oplus m}.$

Note that if $(X,D,{\mathcal{L}})$ is a contact projective log variety then
for any $Z$ as in the lemma, the restriction $\imath^{*}T_{X}(-\log
D)\to\imath^{*}{\mathcal{L}}$ is a contact structure on the above Lie
algebroid $\imath^{*}T_{X}(-\log D)$. This allows us to study logarithmic
contact structures by induction on the dimension of the underlying variety. It
also motivates the following definition of contact structures on some Lie
algebroids generalizing Picard Lie algebroids (see [BB93, Definition 2.1.3]).

######  Definition 4.5.

Let $(X,D)$ be a projective snc pair.

  1. 1.

A _special Lie algebroid_ on $(X,D)$ is a Lie algebroid ${\mathbb{L}}$
together with an abelian extension

$0\to{\mathbb{L}}^{\prime}\to{\mathbb{L}}\to T_{X}(-\log D)\to 0$

of Lie algebroids. In the following we often omit the extension in the
notation.

  2. 2.

A special Lie algebroid ${\mathbb{L}}$ on $(X,D)$ is called _super-special_ if
the corresponding extension is central.

  3. 3.

A _special (generically) contact Lie algebroid_ on $(X,D)$ is a triple
$({\mathbb{L}},{\mathcal{L}},{\mathbb{L}}\to{\mathcal{L}})$ consisting of a
special Lie algebroid ${\mathbb{L}}$, a line bundle ${\mathcal{L}}$ and a
(generically) ${\mathcal{L}}$-valued contact structure
${\mathbb{L}}\to{\mathcal{L}}$. In the following we omit the contact structure
in the notation.

  4. 4.

We say that $(X,D)$ is _(generically) semi-contact of rank $(2r+1)$_ if there
exists a special contact Lie algebroid $({\mathbb{L}},{\mathcal{L}})$ on
$(X,D)$ with ${\mathop{\rm rk\,}}{\mathbb{L}}=2r+1$.

The proof of the following lemma is similar to that of Lemma 4.2 and we leave
it to the reader.

######  Lemma 4.6.

Let $({\mathbb{L}},{\mathcal{L}})$ be a special contact Lie algebroid on
$(X,D)$. Then for any irreducible stratum $Z$ of the stratification induced by
$D$, the restriction $({\mathbb{L}}|_{Z},{\mathcal{L}}|_{Z})$ is a special
contact Lie algebroid on $(Z,D^{Z})$.

###  4.2 Bounds on existence of a semi-contact structure

The following result shows that varieties admitting a semi-contact structure
are quite rare.

######  Proposition 4.7.

Let $({\mathbb{L}},{\mathcal{L}})$ be a special contact Lie algebroid of rank
$(2r+1)$ on $(X,D)$. If $K_{X}+D$ is not nef then $\dim X\geq r$ and if
equality holds then $D=0$. If moreover $b_{2}(X)=1$ then one of the following
cases occurs:

  1. 1.

$\dim X=2r+1$ and $(X,D)$ is a contact log Fano variety.

  2. 2.

For some $0\leq s\leq r$ we have
$(X,{\mathcal{L}})\simeq({\mathbb{P}}^{r+s},{\mathcal{O}}_{{\mathbb{P}}^{r+s}}(1))$,
$D$ is an arrangement of $s$ hyperplanes in a general position and
${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\simeq{\mathcal{O}}_{{\mathbb{P}}^{r+s}}^{\oplus(r+1)}\oplus{\mathcal{O}}_{{\mathbb{P}}^{r+s}}(1)^{\oplus(r+1)}$.
Moreover, we have either
${\mathbb{L}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{r+s}}^{\oplus
r}\oplus{\mathcal{O}}_{{\mathbb{P}}^{r+s}}(1)^{\oplus(r+1)}$ or $s=0$ and
${\mathbb{L}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{r}}^{\oplus(r+1)}\oplus
T_{{\mathbb{P}}^{r}}$.

###### Proof.

Let us write $\dim X=r+s$ for some $s\leq r+1$. By assumption there exists a
short exact sequence of Lie algebroids

$0\to{\mathcal{O}}_{X}^{\oplus(r+1-s)}\to{\mathbb{L}}\to T_{X}(-\log D)\to 0$
(1)

So existence of an ${\mathcal{L}}$-valued contact structure on ${\mathbb{L}}$
implies that $-(K_{X}+D)=(r+1)c_{1}({\mathcal{L}})$. If $K_{X}+D$ is not nef
then there exists a log contraction $\varphi$ of $(X,D)$. So by Lemma 1.6 we
can find a rational curve $C$ such that

$0<-(K_{X}+D)\cdot C=(r+1){\mathcal{L}}\cdot C\leq\dim X+1=r+s+1.$

This shows that $s\geq 0$. If $s=0$ then the same lemma implies that $D=0$.

Now assume that $b_{2}(X)=1$. Then $-(K_{X}+D)$ is ample and hence
${\mathcal{L}}$ is ample. For small $\epsilon>0$ the divisor
$-(K_{X}+D)+\epsilon D$ is also ample. So by the Kawamata–Viehweg vanishing
theorem we have

$H^{j}(X,{\mathcal{O}}_{X})=H^{j}(X,K_{X}+\lceil-(K_{X}+D)+\epsilon
D\rceil)=0$

for all $j>0$. For any irreducible component $D_{i}$ of $D$,
$-(K_{D_{i}}+D^{i})=-(K_{X}+D)|_{D_{i}}$ is ample, so we also have
$H^{j}(D_{i},{\mathcal{O}}_{D_{i}})=0$ for $j>0$ (note that the above argument
for vanishing did not use $b_{2}=1$). Since $b_{2}(X)=1$, we have
$h^{1}(X,\Omega_{X}^{1})=1$. Let us consider the Atiyah extension of
${\mathcal{L}}$

$0\to\Omega_{X}^{1}\longrightarrow{\mathcal{A}}_{{\mathcal{L}}}\longrightarrow{\mathcal{O}}_{X}\to
0.$

Since $c_{1}({\mathcal{L}})\in H^{1}(X,\Omega_{X}^{1})$ is non-zero, the
sequence is non-split. Hence the connecting homomorphism
$H^{0}(X,{\mathcal{O}}_{X})\to H^{1}(X,\Omega_{X}^{1})$ is non-zero. Since
$H^{1}(X,{\mathcal{O}}_{X})=0$, this gives
$H^{1}(X,{\mathcal{A}}_{{\mathcal{L}}})=0$. Let

$0\to\Omega_{X}^{1}(\log
D)\longrightarrow{\mathcal{B}}_{{\mathcal{L}}}\longrightarrow{\mathcal{O}}_{X}\to
0$

be the logarithmic Atiyah extension of ${\mathcal{L}}$. The short exact
sequence

$0\to\Omega_{X}^{1}\longrightarrow\Omega_{X}^{1}(\log
D)\longrightarrow\bigoplus{\mathcal{O}}_{D_{i}}\to 0$

induces the short exact sequence

$0\to{\mathcal{A}}_{{\mathcal{L}}}\longrightarrow{\mathcal{B}}_{{\mathcal{L}}}\longrightarrow\bigoplus{\mathcal{O}}_{D_{i}}\to
0.$

The long exact cohomology sequence for this sequence shows that
$H^{1}(X,{\mathcal{B}}_{{\mathcal{L}}})=0$.

If the extension (1) does not split then $s\leq r$ and there exists a
commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}^{\oplus(r+1-s)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{L}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{X}(-\log
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{B}}_{{\mathcal{L}}}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{X}(-\log
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

The middle column of this diagram induces a short exact sequence

$0\to{\mathcal{O}}_{X}^{\oplus(r-s)}\to{\mathbb{L}}\to{\mathcal{B}}_{{\mathcal{L}}}^{*}\to
0.$

Since the corresponding Ext group vanishes, this sequence splits and
${\mathbb{L}}\simeq{\mathcal{O}}_{X}^{\oplus(r-s)}\oplus{\mathcal{B}}_{{\mathcal{L}}}^{*}$.
Let us consider the short exact sequence

$0\to{\mathcal{O}}_{X}\longrightarrow{\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\longrightarrow{\mathbb{L}}\to 0.$ (2)

This sequence splits as ${\mathop{{\rm
Ext\,}}}^{1}({\mathbb{L}},{\mathcal{O}}_{X})=H^{1}(X,{\mathcal{O}}_{X}^{\oplus(r-s)}\oplus{\mathcal{B}}_{{\mathcal{L}}})=0$.
Therefore ${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\simeq{\mathcal{O}}_{X}^{\oplus(r-s+1)}\oplus{\mathcal{B}}_{{\mathcal{L}}}^{*}.$
By Proposition 3.6 there exists an isomorphism
${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\simeq{\mathcal{L}}\otimes({\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}}))^{*}$. So

${\mathcal{O}}_{X}^{\oplus(r-s+1)}\oplus{\mathcal{B}}_{{\mathcal{L}}}^{*}\simeq{\mathcal{L}}^{\oplus(r-s+1)}\oplus({\mathcal{L}}\otimes{\mathcal{B}}_{{\mathcal{L}}}).$

By the Krull–Schmidt theorem, a decomposition of a vector bundle into a direct
sum of indecomposable bundles is unique up to reordering of the factors (see
[Ati56]). In particular, ${\mathcal{L}}^{\oplus(r-s+1)}$ is a direct factor of
${\mathcal{B}}_{{\mathcal{L}}}^{*}$. But we have a short exact sequence

$0\to{\mathcal{O}}_{X}\longrightarrow{\mathcal{B}}^{*}_{{\mathcal{L}}}\longrightarrow
T_{X}(-\log D)\to 0.$

So if $s<r+1$ then we get a non-zero map ${\mathcal{L}}\to T_{X}(-\log D)$.
Since ${\mathcal{L}}$ is ample and $-(K_{X}+D)=(r+1)c_{1}({\mathcal{L}})$,
Theorem 1.1 shows that $X\simeq{\mathbb{P}}^{r+s}$ and $D$ is an arrangement
of $s$ general hyperplanes. If $s=0$ then
${\mathcal{B}}^{*}_{{\mathcal{L}}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{r}}(1)^{\oplus(r+1)}$.
If $s>0$ then $T_{X}(-\log
D)\simeq{\mathcal{O}}_{{\mathbb{P}}^{r+s}}^{\oplus(s-1)}\oplus{\mathcal{O}}_{{\mathbb{P}}^{r+s}}(1)^{\oplus(r+1)}$
by [DK93, Proposition 2.10]. In both cases we have

${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\simeq{\mathcal{O}}_{{\mathbb{P}}^{r+s}}^{\oplus(r+1)}\oplus{\mathcal{O}}_{{\mathbb{P}}^{r+s}}(1)^{\oplus(r+1)}.$

Now assume that the extension (1) splits. Then
${\mathbb{L}}\simeq{\mathcal{O}}_{X}^{\oplus(r-s+1)}\oplus T_{X}(-\log D)$. If
the sequence (2) splits then ${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\simeq{\mathcal{O}}_{X}^{\oplus(r-s+2)}\oplus T_{X}(-\log
D).$ So we have an isomorphism

${\mathcal{O}}_{X}^{\oplus(r-s+2)}\oplus T_{X}(-\log
D)\simeq{\mathcal{L}}^{\oplus(r-s+2)}\oplus({\mathcal{L}}\otimes\Omega_{X}^{1}(\log
D)).$

Then ${\mathcal{L}}^{\oplus(r-s+2)}$ is a direct factor of $T_{X}(-\log D)$.
As above this implies that $X\simeq{\mathbb{P}}^{r+s}$ and $D$ is an
arrangement of $s$ general hyperplanes. But in this case $T_{X}(-\log D)$
contains only $(r-s+1)$ copies of ${\mathcal{L}}$, a contradiction. It follows
that the sequence (2) does not split, which shows that
${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\simeq{\mathcal{O}}_{X}^{\oplus(r-s+1)}\oplus{\mathcal{B}}_{{\mathcal{L}}}^{*}.$
As above this implies that if $s<r+1$ then $X\simeq{\mathbb{P}}^{r+s}$, $D$ is
an arrangement of $s$ general hyperplanes and
${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})\simeq{\mathcal{O}}_{{\mathbb{P}}^{r+s}}^{\oplus(r+1)}\oplus{\mathcal{O}}_{{\mathbb{P}}^{r+s}}(1)^{\oplus(r+1)}.$
∎

Existence of semi-contact structure in case (2) is described in the following
example:

######  Example 4.8.

Let us first consider $X={\mathbb{P}}^{2r+1}$ with an arrangement $D$ of
$(r+1)$ hyperplanes $H_{1},...,H_{r+1}$ in a general position. Then $(X,D)$
has a contact structure given by projection $T_{X}(-\log
D)\simeq{\mathcal{O}}_{X}^{\oplus
r}\oplus{\mathcal{O}}_{X}(1)^{\oplus(r+1)}\to{\mathcal{O}}_{X}(1)$ onto one of
the factors. For every $0\leq s\leq r$, restricting to $H_{1}\cap...\cap
H_{r+1-s}\simeq{\mathbb{P}}^{r+s}$ and using Lemma 4.2, we get a semi-contact
structure on ${\mathbb{P}}^{r+s}$ with an arrangement of $s$ hyperplanes in a
general position.

Note that in the proof of Proposition 4.7 we did not use the Lie algebra
structure on ${\mathbb{L}}$. But we use it to show the following slightly
stronger form of the first assertion.

######  Lemma 4.9.

If $(X,D)$ is a semi-contact snc pair of rank $(2r+1)$ then $r\leq\dim X$.

###### Proof.

Let $({\mathbb{L}},{\mathcal{L}})$ be a special contact Lie algebroid of rank
$(2r+1)$ on $(X,D)$. Let us consider a closed point $x\in X$ and the sequence
of Lie $k$-algebras

$0\to\mathfrak{a}\to\mathfrak{g}={\mathbb{L}}(x)\to(T_{X}(-\log D))(x)\to 0$

of fibers at the point $x$, where $\mathfrak{a}$ is an abelian Lie
$k$-algebra. So the map $\mathfrak{a}\to\mathfrak{a}^{*}$, $x\to[x,\cdot]$,
vanishes and the rank of $\mathfrak{a}\to\mathfrak{g}^{*}$, $x\to[x,\cdot]$,
is at most $\dim X$. We also have an exact sequence of $k$-vector spaces

$0\to\mathfrak{b}={\mathcal{F}}(x)\to\mathfrak{g}\mathop{\longrightarrow}^{\alpha}k={\mathcal{L}}(x)\to
0.$

By definition of a contact structure, the map
$\mathfrak{b}\to\mathfrak{b}^{*}$ defined by $x\to\alpha([x,\cdot])$, is an
isomorphism. So the map $\mathfrak{b}\to\mathfrak{g}^{*}$, $x\to[x,\cdot]$ is
injective. Therefore the intersection $\mathfrak{a}\cap\mathfrak{b}$ has
dimension $\leq\dim X$. But $\mathfrak{a}\cap\mathfrak{b}$ is the kernel of
the map $\mathfrak{g}\to(T_{X}(-\log D))(x)\oplus{\mathcal{L}}(x)$, so it has
dimension $\geq(2r+1)-(\dim X+1)=2r-\dim X$. This shows that $r\leq\dim X$. ∎

###  4.3 Kodaira dimension of semi-contact snc pairs

The proof of the following lemma follows that of [Dru98, Proposition 2].

######  Lemma 4.10.

If $(X,D,{\mathcal{L}})$ is a generically contact projective snc pair then
$\kappa({\mathcal{L}}^{-1})=-\infty$. In particular, if $(X,D)$ is contact
then $\kappa(K_{X}+D)=-\infty$.

###### Proof.

Assume that $\kappa({\mathcal{L}}^{-1})\geq 0$. Then there exists a smooth
projective variety $\bar{X}$ and a generically finite morphism $\pi:\bar{X}\to
X$ such that $h^{0}(\bar{X},\bar{\mathcal{L}}^{-1})>0$, where
$\bar{\mathcal{L}}=\pi^{*}{\mathcal{L}}$. Passing to a log resolution, we can
also assume that $\bar{D}:=(\pi^{*}D)_{red}$ is a simple normal crossing
divisor. By assumption there exists a non-empty Zariski open subset $U\subset
X$ and $\omega\in H^{0}(X,\Omega_{X}^{1}(\log
D)\otimes_{{\mathcal{O}}_{X}}{\mathcal{L}})$ such that $\omega|_{U}$ is an
${\mathcal{L}}|_{U}$-valued contact form on $U$. There also exists a non-empty
Zariski open subset $V\subset U\backslash{\mathop{{\rm Supp\,}}}D$ such that
$\bar{V}=\pi^{-1}(V)\to V$ is finite étale. Taking any non-zero section $s\in
H^{0}(\bar{X},\bar{\mathcal{L}}^{-1})$ we can find a non-empty Zariski open
subset $W\subset\bar{V}$ such that $s$ has no zeroes on $W$. Note that by the
logarithmic ramification formula, we have a natural injective
${\mathcal{O}}_{\bar{X}}$-linear map $\pi^{*}\Omega_{X}^{1}(\log
D)\hookrightarrow\Omega^{1}_{\bar{X}}(\log\bar{D})$ that allows us to consider
$\pi^{*}\omega:\bar{\mathcal{L}}^{-1}\to\Omega^{1}_{\bar{X}}(\log\bar{D})$.
Since $\pi^{*}(\omega|_{V})|_{W}$ gives an $\bar{\mathcal{L}}|_{W}$-valued
contact form, we have $d(s|_{W})\neq 0$ in
$H^{0}(W,\Omega^{2}_{\bar{X}}|_{W})=H^{0}(W,\Omega^{2}_{\bar{X}}(\log\bar{D})|_{W})$.
But by degeneracy of the logarithmic Hodge to de Rham spectral sequence on
$(\bar{X},\bar{D})$, all global logarithmic $1$-forms on $(\bar{X},\bar{D})$
are closed. Hence $ds=0\in H^{0}(\bar{X},\Omega^{2}_{\bar{X}}(\log\bar{D}))$.
So $d(s|_{W})=(ds)|_{W}$ also vanishes, a contradiction. ∎

Let us recall that a line bundle ${\mathcal{M}}$ on a smooth projective
variety $X$ is _pseudoeffective_ if and only if $c_{1}({\mathcal{M}})\in
N^{1}(X)_{{\mathbb{R}}}$ lies in the closure of the cone generated by classes
of effective divisors. The following theorem was proven by J.-P. Demailly in
[Dem02, Corollary 1] in the non-logarithmic case and then generalized by F.
Touzet in [Tou16, Theorem 5 and Corollary 7.6] to the logarithmic case.

######  Theorem 4.11.

Let $(X,D)$ be a projective snc pair and let ${\mathcal{L}}$ be a line bundle
on $X$. If ${\mathcal{L}}^{-1}$ is pseudo-effective and $\omega\in
H^{0}(X,\Omega_{X}^{1}(\log D)\otimes{\mathcal{L}})$ is a logarithmic $1$-form
with values in ${\mathcal{L}}$ then $\omega\wedge d\omega=0$.

This theorem implies a much stronger form of Lemma 4.10:

######  Corollary 4.12.

Let $(X,D)$ be a projective snc pair. If $(X,D,{\mathcal{L}})$ is generically
contact then ${\mathcal{L}}^{-1}$ is not pseudo-effective. In particular,
$\kappa({\mathcal{L}}^{-1})=-\infty$. If $(X,D)$ is a contact snc pair then
$K_{X}+D$ is not pseudoeffective. In particular, $X$ is uniruled.

The following lemma deals with semi-contact, non-contact snc pairs.

######  Lemma 4.13.

Let $({\mathbb{L}},{\mathcal{L}})$ be a special generically contact Lie
algebroid of rank $(2r+1)$ on a projective snc pair $(X,D)$. If $\dim X<2r+1$
then $H^{0}(X,{\mathcal{L}})\neq 0$.

###### Proof.

By assumption, we have a short exact sequence

$0\to{\mathcal{O}}_{X}^{\oplus s}\to{\mathbb{L}}\to T_{X}(-\log D)\to 0,$

in which $s\geq 1$. If $H^{0}(X,{\mathcal{L}})=0$ then the composition
${\mathcal{O}}_{X}^{\oplus s}\to{\mathbb{L}}\to{\mathcal{L}}$ vanishes. So
${\mathbb{L}}\to{\mathcal{L}}$ induces a generically surjective map
$T_{X}(-\log D)\to{\mathcal{L}}$. Passing to the dual we see that the contact
form $\omega:{\mathcal{L}}^{-1}\to\Omega_{{\mathbb{L}}}^{1}$ factors through
${\mathcal{L}}^{-1}\to\Omega_{X}^{1}(\log D)$. But then
$\omega\wedge(d_{{\mathbb{L}}}\omega)^{\wedge
r}:{\mathcal{L}}^{-(r+1)}\to\det\Omega_{{\mathbb{L}}}^{1}$ factors through
${\mathcal{L}}^{-(r+1)}\to\Omega_{X}^{2r+1}(\log D)=0$ and we get a
contradicton. ∎

######  Corollary 4.14.

Let $(X,D)$ be a projective snc pair. If $(X,D,{\mathbb{L}},{\mathcal{L}})$ is
generically semi-contact and $K_{X}+D$ is pseudoeffective then $\dim
X<{\mathop{\rm rk\,}}{\mathbb{L}}$ and ${\mathcal{L}}\simeq{\mathcal{O}}_{X}$
(in particular, $K_{X}+D\sim 0$).

###### Proof.

By Corollary 4.12, $\dim X<{\mathop{\rm rk\,}}{\mathbb{L}}$. So Lemma 4.13
implies that $H^{0}(X,{\mathcal{L}})\neq 0$. Since the cone of pseudo-
effective divisors is convex, the class of ${\mathcal{L}}$ in $N^{1}(X)$
vanishes. Since ${\mathcal{L}}$ has a non-zero section this gives
${\mathcal{L}}\simeq{\mathcal{O}}_{X}$. ∎

The following example shows that semi-contact snc pairs with $K_{X}+D\sim 0$
really occur.

######  Example 4.15.

Let $(X,D)$ be a projective snc pair of dimension $n$ and assume that a semi-
abelian variety $G$ acts on $X$ with $X\backslash{\mathop{{\rm Supp\,}}}D$ as
an open orbit. Since $G$ is a commutative $k$-group, the Lie algebra
$\mathfrak{g}$ of $G$ is abelian. By assumption $\mathfrak{g}$ acts on $X$ and
${\mathcal{O}}_{X}\otimes_{k}\mathfrak{g}$ has a natural structure of Lie
algebroid described in [BB93, 1.2.2]. The image of the anchor map
$\alpha:{\mathcal{O}}_{X}\otimes_{k}\mathfrak{g}\to T_{X}$ coincides with
$T_{X}(-\log D)$ (see, e.g., [Win04, p. 478]). The Lie bracket on
${\mathcal{O}}_{X}\otimes_{k}\mathfrak{g}$ is trivial on the fibers at all
points of $X$ as

$[1\otimes\delta,1\otimes\delta^{\prime}]=1\otimes[\delta,\delta^{\prime}]=0$

but it is non-zero on the corresponding sheaf. Note that the above formula,
together with the requirement that the anchor map is a map of Lie
$k$-algebras, uniquely determines the Lie bracket on
${\mathcal{O}}_{X}\otimes_{k}\mathfrak{g}$.

Let $\sigma$ be a non-degenerate alternating form on a $2r$-dimensional
$k$-vector space $V$ and let $\mathfrak{h}=V\oplus k\cdot z$ be a Lie
$k$-algebra of dimension $(2r+1)$ with the Lie bracket determined by

$[v,w]=\left\\{\begin{array}[]{cl}\sigma(v,w)\cdot z&\hbox{if $v,w\in V$,}\\\
0&\hbox{if $v\in V$ and $w=z$.}\\\ \end{array}\right.$

Assume that $r\leq n\leq 2r$ and let $W\subset V$ be an isotropic $k$-linear
subspace of dimension $(2r-n)$. Then $I=W\oplus k\cdot z\subset\mathfrak{h}$
is an ideal and the quotient $\mathfrak{h}/I$ is an abelian Lie algebra of
dimension $n$, so it is isomorphic to $\mathfrak{g}$. Let us fix a surjective
map of Lie algebras $\mathfrak{h}\to\mathfrak{g}$. Then
${\mathbb{L}}={\mathcal{O}}_{X}\otimes_{k}\mathfrak{h}$ has a canonical
structure of a Lie algebroid for which the extension

$0\to{\mathcal{O}}_{X}\otimes_{k}I\to{\mathbb{L}}\to{\mathcal{O}}_{X}\otimes_{k}\mathfrak{g}\simeq
T_{X}(-\log D)\to 0$

is abelian (one can again use [BB93, 1.2.2]). A contact structure on
${\mathbb{L}}$ is defined by the map
${\mathbb{L}}\to{\mathcal{L}}={\mathcal{O}}_{X}$ induced by the projection
$\mathfrak{g}\to k\cdot z$. In this way for every $n/2\leq r\leq n$ we obtain
a rank $(2r+1)$ semi-contact structure on $(X,D)$. Note that the above
extension is central only for $n=2r$.

###  4.4 Lines on semi-contact snc pairs

Let us first note the following easy lemma:

######  Lemma 4.16.

Let

$0\to{\mathcal{O}}_{{\mathbb{P}}^{1}}\to{\mathcal{E}}_{1}\to{\mathcal{E}}_{2}\to
0$

be an exact sequence of vector bundles on ${\mathbb{P}}^{1}$. Then one of the
following holds:

  1. 1.

the sequence does not split and ${\mathop{\rm
rk\,}}^{+}{\mathcal{E}}_{2}={\mathop{\rm rk\,}}^{+}{\mathcal{E}}_{1}-1$, or

  2. 2.

the sequence splits and ${\mathop{\rm rk\,}}^{+}{\mathcal{E}}_{2}={\mathop{\rm
rk\,}}^{+}{\mathcal{E}}_{1}$.

###### Proof.

If the sequence splits then the assertion is clear. If it does not split then
${\mathop{\rm rk\,}}^{+}{\mathcal{E}}>0$. In this case, after dualizing the
above sequence, the required equality follows from [KPSW00, Lemma 2.7]. ∎

By passing to the dual sequence and induction on $m$ we get the following
corollary:

######  Corollary 4.17.

Let

$0\to{\mathcal{E}}_{2}\to{\mathcal{E}}_{1}\to{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus
m}\to 0$

be an exact sequence of vector bundles on ${\mathbb{P}}^{1}$. Then
${\mathop{\rm rk\,}}^{+}{\mathcal{E}}_{2}^{*}\leq{\mathop{\rm
rk\,}}^{+}{\mathcal{E}}_{1}^{*}$ and equality holds if and only if the
sequence splits.

######  Lemma 4.18.

Let $({\mathbb{L}},{\mathcal{L}})$ be a rank $(2r+1)$ special contact Lie
algebroid on a projective snc pair $(X,D)$. Let $f:{\mathbb{P}}^{1}\to
C\subset X$ be a normalization of a rational curve with $\deg
f^{*}{\mathcal{L}}>0$. Then we have ${\mathop{\rm rk\,}}^{+}f^{*}T_{X}(-\log
D)\leq r+1$. Moreover, if $C\cap{\mathop{{\rm Supp\,}}}D=\emptyset$ then
${\mathop{\rm rk\,}}^{+}f^{*}T_{X}\leq r$.

###### Proof.

Let us set ${\mathcal{B}}={\mathcal{A}}_{{\mathbb{L}}}({\mathcal{L}})$ and
consider the ${\mathbb{L}}$-Atiyah extension of ${\mathcal{L}}$

$0\to\Omega_{{\mathbb{L}}}^{1}\longrightarrow{\mathcal{B}}\longrightarrow{\mathcal{O}}_{X}\to
0.$

By Proposition 3.6, ${\mathcal{D}}_{{\mathbb{L}}}^{\leq
1}({\mathcal{L}})={\mathcal{B}}^{*}$ carries an ${\mathcal{L}}$-valued
symplectic structure and hence we have an isomorphism
${\mathcal{B}}\simeq{\mathcal{L}}^{-1}\otimes{\mathcal{B}}^{*}$. So for some
integers $a_{1}\geq...\geq a_{r+1}\geq 0$ we have

$f^{*}{\mathcal{B}}\simeq\bigoplus_{i=1}^{r+1}({\mathcal{O}}_{{\mathbb{P}}^{1}}(a_{i})\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}(-a_{i}-d)),$

where $d=\deg f^{*}{\mathcal{L}}>0$. In particular, ${\mathop{\rm
rk\,}}^{+}f^{*}({\mathcal{B}}^{*})=r+1$. If the sequence

$0\to f^{*}\Omega_{{\mathbb{L}}}^{1}\to f^{*}{\mathcal{B}}\to
f^{*}{\mathcal{O}}_{X}={\mathcal{O}}_{{\mathbb{P}}^{1}}\to 0$

does not split then by Lemma 4.16 we have ${\mathop{\rm
rk\,}}^{+}f^{*}{\mathbb{L}}=r$. If the above sequence splits then
${\mathop{\rm rk\,}}^{+}f^{*}{\mathbb{L}}=r+1$.

We also have a short exact sequence

$0\to f^{*}\Omega_{X}^{1}(\log D)\to
f^{*}\Omega_{{\mathbb{L}}}^{1}\to{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus(2r+1-\dim
X)}\to 0$

obtained by pulling back the dual to the sequence defining a special contact
Lie algebroid. So using Corollary 4.17 we get ${\mathop{\rm
rk\,}}^{+}f^{*}T_{X}(-\log D)\leq r+1$ with equality if and only if both the
above sequences split.

Now assume that $C\cap{\mathop{{\rm Supp\,}}}D=\emptyset$ so that
$f^{*}\Omega_{X}^{1}\simeq f^{*}\Omega_{X}^{1}(\log D)$. By construction of
the Atiyah extensions, we have a commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f^{*}\Omega_{X}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f^{*}({\mathcal{A}}_{{\mathcal{L}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{{\mathbb{P}}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f^{*}\Omega_{{\mathbb{L}}}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f^{*}{\mathcal{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{{\mathbb{P}}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus
m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus
m}}$

If ${\mathop{\rm rk\,}}^{+}f^{*}T_{X}={\mathop{\rm rk\,}}^{+}f^{*}T_{X}(-\log
D)=r+1$ then we have an induced splitting

$f^{*}({\mathcal{A}}_{{\mathcal{L}}})\to f^{*}{\mathcal{B}}\to
f^{*}\Omega_{{\mathbb{L}}}^{1}\to f^{*}\Omega_{Z}^{1}$

of the first row in the above diagram. But $\deg f^{*}{\mathcal{L}}\neq 0$, so
the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f^{*}\Omega_{X}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f^{*}({\mathcal{A}}_{{\mathcal{L}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{{\mathbb{P}}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{{\mathbb{P}}^{1}}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{A}}_{f^{*}{\mathcal{L}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{{\mathbb{P}}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

shows that this sequence does not split. Therefore ${\mathop{\rm
rk\,}}^{+}f^{*}T_{X}\leq r$. ∎

######  Lemma 4.19.

Let $({\mathbb{L}},{\mathcal{L}})$ be a rank $(2r+1)$ special contact Lie
algebroid on a projective snc pair $(X,D)$. Assume that $K_{X}+D$ is not nef
and let $\varphi$ be a log contraction of $(X,D)$. Then either
$(X,D,{\mathbb{L}},{\mathcal{L}})\simeq({\mathbb{P}}^{2r+1},0,T_{{\mathbb{P}}^{2r+1}},{\mathcal{O}}_{{\mathbb{P}}^{2r+1}}(2))$
or there exists a $\varphi$-exceptional rational curve $C$ with
${\mathcal{L}}\cdot C=1$.

###### Proof.

Without loss of generality we can assume that $\varphi$ is elementary and it
contracts some $(K_{X}+D)$-extremal ray $R$. If
$(X,D)\simeq({\mathbb{P}}^{2r+1},0)$ then
$({\mathbb{L}},{\mathcal{L}})\simeq(T_{{\mathbb{P}}^{2r+1}},{\mathcal{O}}_{{\mathbb{P}}^{2r+1}}(2))$.
Otherwise, Lemma 1.6 implies that $l_{D}(\varphi)\leq 2r+1$. Since
$-(K_{X}+D)=(r+1)c_{1}({\mathcal{L}})$, we have $l_{D}(\varphi)=r+1$ and if
$C$ is an extremal rational curve with $[C]\in R$ then ${\mathcal{L}}\cdot
C=1$. ∎

If $(X,D)$ is a contact projective snc pair then Corollary 4.12 implies that
$K_{X}+D$ is not nef. So we have the following corollary.

######  Corollary 4.20.

If $(X,D,{\mathcal{L}})$ is a contact projective snc pair and
$(X,D,{\mathcal{L}})\not\simeq({\mathbb{P}}^{2r+1},0,{\mathcal{O}}_{{\mathbb{P}}^{2r+1}}(2))$
then $K_{X}+D$ is not nef and for every log contraction $\varphi$ there exists
an extremal rational curve $C$ with ${\mathcal{L}}\cdot C=1$.

##  5 Log contractions of semi-contact snc pairs

This section contains proof of a generalization of Theorem 0.1. First we show
that we can study log contractions of semi-contact varieties by studying maps
induced on the strata of stratification induced by $D$. This allows us to
reduce to studying contractions that are not isomorphic on the complement of
the boundary. Then we use results of Section 2 to describe the general fiber
and we prove that log contractions are equidimensional. This allows us to
conclude that our contractions are projective bundles (we also need to prove
this for every stratum of the stratification iinduced by the boundary
divisor). The main result of the section is Theorem 5.14. Its proof requires
further study of the geometry using various Atiyah extensions.

###  5.1 Restriction to strata

The proof of the following lemma follows standard arguments used, e.g., in the
proof of [Kaw97, Theorem 1.6].

######  Lemma 5.1.

Let $(X,D)$ be a semi-contact projective snc pair and let $\varphi:X\to Y$ be
a log contraction of $(X,D)$. Let $Z$ be an irreducible component of $D$ and
let $D^{Z}$ be the restriction of $(D-Z)$ to $Z$. Then $T=\varphi(Z)$ is
normal and $\varphi|_{Z}:Z\to T$ is a log contraction of $(Z,D^{Z})$.

###### Proof.

Since $-(K_{X}+D)$ is $\varphi$-ample, for small $\epsilon>0$ the divisor
$-(K_{X}+D)+\epsilon(D-Z)$ is also $\varphi$-ample. Since

$-Z=K_{X}+(-(K_{X}+D)+\epsilon(D-Z))+(1-\epsilon)(D-Z),$

the relative version of the Kawamata-Viehweg vanishing theorem implies that
$R^{i}\varphi_{*}({\mathcal{O}}_{X}(-Z))=0$ for all $i>0$. In particular, the
map
${\mathcal{O}}_{Y}=\varphi_{*}{\mathcal{O}}_{X}\to\varphi_{*}{\mathcal{O}}_{Z}$
is surjective. So if $T=\varphi(Z)$ then
${\mathcal{O}}_{T}\to\varphi_{*}{\mathcal{O}}_{Z}$ is surjective. Since it is
also injective, it is an isomorphism and hence $T$ is normal. To see that
$\varphi|_{Z}$ is a log contraction it is sufficient to note that
$K_{Z}+D^{Z}=(K_{X}+D)|_{Z}$. ∎

By induction the above lemma implies the following corollary.

######  Corollary 5.2.

Let $\varphi:X\to Y$ be a log contraction of a semi-contact projective snc
pair $(X,D)$. Let us fix $J\subset I$ and an irreducible component $Z$ of
$D_{J}$. Then $T=\varphi(Z)$ is normal and $\varphi|_{Z}:Z\to T$ is a log
contraction of $(Z,D^{J}|_{Z})$.

This allows us to reduce the study of log contractions to those, whose
exceptional locus is not contained in the boundary. From now on in this
section we fix the following notation. Let $({\mathbb{L}},{\mathcal{L}})$ be a
special contact Lie algebroid of rank $(2r+1)$ on a projective snc pair
$(Z,D^{Z})$. Let us write $n=\dim Z$ and $U_{Z}=Z\backslash{\mathop{{\rm
Supp\,}}}D^{Z}$. Assume that $(K_{Z}+D^{Z})$ is not nef so that $(Z,D^{Z})$
admits some log contraction $\varphi_{Z}:Z\to T$. Let $F$ be the
$(K_{Z}+D^{Z})$-extremal face corresponding to $\varphi_{Z}$. Till the end of
the section we assume that the exceptional locus ${\mathop{{\rm
Exc\,}}}(\varphi_{Z})$ is not contained in the support of $D^{Z}$. We also
assume that
$(Z,D^{Z},{\mathbb{L}},{\mathcal{L}})\not\simeq({\mathbb{P}}^{2r+1},0,T_{{\mathbb{P}}^{2r+1}},{\mathcal{O}}_{{\mathbb{P}}^{2r+1}}(2))$.

###  5.2 General structure of log contractions

Let $C$ be an extremal rational curve with $[C]\in F$ and let
$f:{\mathbb{P}}^{1}\to C$ be the normalization of $C$. We can choose $C$ so
that $C\cap U_{Z}\neq\emptyset$. By Proposition 2.1 we can also assume that
$f^{*}D^{Z}$ is supported in at most one point so that
$f^{-1}(U_{Z})={\mathbb{P}}^{1}$ or $f^{-1}(U_{Z})\simeq{\mathbb{A}}^{1}$.

If $H_{F}$ is a supporting divisor of $F$ then for large $m$ the divisor
$mH_{F}+c_{1}({\mathcal{L}})$ is ample and by Lemma 4.19 the degree of $C$
with respect to this divisor is $1$. In particular, any closed subset of the
connected component of ${\mathop{{\rm Hom}}}({\mathbb{P}}^{1},Z)$ containing
$[f]$ is unsplit.

Let $V$ be an irreducible component of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},Z)$, which contains $[f]$.

######  Lemma 5.3.

If $f^{-1}(U_{Z})={\mathbb{P}}^{1}$ then ${\rm locus}(V)=Z$ and $\dim{\rm
locus}(V,x)=r$ for any $x\in U_{Z}$. Moreover, we have ${\mathop{\rm
rk\,}}^{+}f^{*}T_{Z}=r$.

###### Proof.

Since the family $V$ is unsplit, we have

$-(K_{Z}+D^{Z})\cdot C+\dim Z=r+1+n\leq-K_{Z}\cdot C+\dim Z\leq\dim
V\leq\dim{\rm locus}(V)+\dim{\rm locus}(V,x)+1.$

But $\dim{\rm locus}(V)\leq n$, so $\dim{\rm locus}(V,x)\geq r$ for all $x\in
Z$. Lemma 4.18 implies that

$r\leq\dim{\rm locus}(V,x)\leq{\mathop{\rm rk\,}}^{+}f^{*}T_{Z}\leq r.$

So $\dim{\rm locus}(V,x)=r$ for all $x\in U_{Z}$. This implies that $\dim{\rm
locus}(V)=n$ and hence ${\rm locus}(V)=Z$. ∎

Let $W$ be an irreducible component of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},Z;f^{*}D^{Z}\subset D^{Z})$ containing $[f]$.

######  Lemma 5.4.

If $f^{-1}(U_{Z})\simeq{\mathbb{A}}^{1}$ then ${\rm locus}(W)=Z$ and $\dim{\rm
locus}_{U_{Z}}(W,x)=r+1$ for any $x\in U_{Z}$.

###### Proof.

By Lemma 4.18 for any $g:{\mathbb{P}}^{1}\to X$ such that $[g]\in W$, we have
${\mathop{\rm rk\,}}^{+}g^{*}T_{Z}(-\log D^{Z})\leq r+1$. So by Proposition
2.5 we have $\dim{\rm locus}_{U_{Z}}(W,x)\leq r+1.$ Since by Proposition 2.4

$-(K_{Z}+D^{Z})\cdot C+\dim Z=r+1+n\leq\dim W\leq\dim{\rm
locus}_{U_{Z}}(W)+\dim{\rm locus}_{U_{Z}}(W,x)\leq n+r+1,$

we have $\dim{\rm locus}_{U_{Z}}(W)=n$ and $\dim{\rm locus}_{U_{Z}}(W,x)=r+1$.
∎

######  Remark 5.5.

Note that if $f^{-1}(U_{Z})={\mathbb{P}}^{1}$ then $f^{*}D^{Z}=0$ and $W=V$.
So it is always sufficient to consider $W$. We keep the above notation to show
differences and similarities to the usual non-logarithmic case (and anyway we
need to consider two cases).

The following proposition is an analogue of [KPSW00, Proposition 2.11] in our
setting. The strategy of its proof is similar but we have more cases than just
one.

######  Proposition 5.6.

Assume that with $\dim T>0$. Then there exists an open subset $U\subset T$
such that for some $0\leq s\leq r$ and for any closed point $y\in U$ we have
$(Z_{y},D_{y}^{Z})\simeq({\mathbb{P}}^{r+s},H_{1}+...+H_{s})$, where
$H_{1},...,H_{s}$ are hyperplanes in a general position. Moreover, we have
${\mathcal{L}}|_{Z_{y}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{r+s}}(1)$.

###### Proof.

Lemmas 5.3 and 5.4 imply that $\varphi_{Z}$ is of fiber type, i.e., $\dim
T<\dim Z=n$. By generic smoothness in characteristic zero, there exists an
open subset $U\subset T$ such that $\varphi|_{Z_{U}}:Z_{U}\to V$ is smooth and
$D_{U}^{Z}\subset Z_{U}$ is a divisor with strict normal crossings relative to
$U$. In particular, for every $y\in U$ we have a short exact sequence

$0\to T_{Z_{y}}(-\log D^{Z}_{y})\to T_{Z}(-\log
D^{Z})|_{Z_{y}}\to\varphi^{*}T_{T}|_{Z_{y}}\simeq{\mathcal{O}}_{Z_{y}}^{\dim
T}\to 0.$

Let us consider the map $\alpha:\Omega_{Z}^{1}(\log
D^{Z})\otimes{\mathcal{L}}\to T_{Z}(-\log D^{Z})$ given by composition

$\Omega_{Z}^{1}(\log
D^{Z})\otimes{\mathcal{L}}\hookrightarrow\Omega_{{\mathbb{L}}}^{1}\otimes{\mathcal{L}}\twoheadrightarrow{\mathcal{F}}^{*}\otimes{\mathcal{L}}\mathop{\longrightarrow}^{\simeq}{\mathcal{F}}\hookrightarrow{\mathbb{L}}\twoheadrightarrow
T_{Z}(-\log D^{Z}).$

This yields the following commutative diagram:

$\textstyle{T_{Z_{y}}(-\log
D_{y}^{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\varphi^{*}\Omega_{T}\otimes{\mathcal{L}})|_{Z_{y}}\simeq({\mathcal{L}}|_{Z_{y}})^{\oplus\dim
T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{0}$$\scriptstyle{\beta}$$\textstyle{(\Omega_{Z}^{1}(\log
D^{Z})\otimes{\mathcal{L}})|_{Z_{y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha|_{Z_{y}}}$$\textstyle{T_{Z}(-\log
D^{Z})|_{Z_{y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{Z_{y}}^{\dim
T}}$

The lower arrow is zero because ${\mathcal{L}}|_{Z_{y}}$ is ample as
$-(K_{Z}+D^{Z})$ is $\varphi|_{Z}$-ample. This implies existence of $\beta$.
The short exact sequence

$0\to{\mathcal{O}}_{Z}^{\oplus(2r+1-n)}\to{\mathbb{L}}\to T_{Z}(-\log
D^{Z})\to 0$

shows that the kernel of
$\delta:({\mathcal{F}}^{*}\otimes{\mathcal{L}})|_{Z_{y}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\mathcal{F}}|_{Z_{y}}\hookrightarrow{\mathbb{L}}|_{Z_{y}}\twoheadrightarrow
T_{Z}(-\log D^{Z})|_{Z_{y}}$ is contained in
${\mathcal{O}}_{Z_{y}}^{\oplus(2r+1-n)}$. On the other hand, the short exact
sequence

$0\to{\mathcal{O}}_{Z_{y}}\to(\Omega_{{\mathbb{L}}}^{1}\otimes{\mathcal{L}})|_{Z_{y}}\to({\mathcal{F}}^{*}\otimes{\mathcal{L}})|_{Z_{y}}\to
0$

shows that the kernel of $\alpha|_{Z_{y}}$ is contained in an extension of
$\ker\delta$ by ${\mathcal{O}}_{Z_{y}}$. So $\ker\alpha|_{Z_{y}}$ is also
contained in an extension of ${\mathcal{O}}_{{Z_{y}}}^{\oplus(2r+1-n)}$ by
${\mathcal{O}}_{Z_{y}}$. Since $\gamma$ is injective this implies that the map
$\beta$ is non-zero. So there exists a non-zero map ${\mathcal{L}}|_{Z_{y}}\to
T_{Z_{y}}(-\log D_{y}^{Z})$. By Theorem 1.1 we see that for some $m$ we have
$Z_{y}\simeq{\mathbb{P}}^{m}$ and
${\mathcal{L}}|_{Z_{y}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{m}}(1)$. Moreover,
$D_{y}^{Z}$ is a sum of $0\leq s\leq m$ hyperplanes in a general position. Let
$C\subset Z_{y}$ be a line. Then

$r+1=-(K_{X}+D)\cdot C=-(K_{Z_{y}}+D_{y}^{Z})\cdot C_{y}=m+1-s,$

so $m=r+s$. ∎

######  Remark 5.7.

If ${\mathbb{L}}=T_{Z}(-\log D^{Z})$ and $s=0$, i.e.,
$(Z_{y},D_{y}^{Z})\simeq({\mathbb{P}}^{r},0)$, then the composition

$T_{Z_{y}}=T_{Z_{y}}(-\log D_{y}^{Z})\to T_{Z}(-\log
D^{Z})|_{Z_{y}}\to{\mathcal{L}}|_{Z_{y}}$

vanishes and $T_{Z_{y}}\subset{\mathcal{F}}|_{Z_{y}}$. Indeed, since
${\mathop{{\rm
Hom}}}(T_{{\mathbb{P}}^{r}},{\mathcal{O}}_{{\mathbb{P}}^{r}}(1))=H^{0}({\mathbb{P}}^{r},\Omega^{1}_{{\mathbb{P}}^{r}}(1))=0$,
there are no non-zero maps $T_{Z_{y}}\to{\mathcal{L}}|_{Z_{y}}$. This gives a
quick proof of the last part of [KPSW00, Proposition 2.11].

######  Remark 5.8.

Proposition 5.6 shows that a posteriori a general fiber $(Z_{y},D_{y}^{Z})$ of
$\varphi_{Z}$ is semi-contact with $b_{2}(Z_{y})=1$ (see Proposition 4.7 and
Example 4.8).

Let $D^{Z}=\sum_{i\in I}D_{i}^{Z}$ be the decomposition of $D^{Z}$ into
irreducible components. For a subset $J\subset I$ and a connected component
$Z^{\prime}$ of $D_{J}^{Z}=\bigcap_{i\in J}D_{i}^{Z}$ we write
$D^{Z^{\prime}}$ for the restriction of the divisor $\sum_{i\in I\backslash
J}D_{i}^{Z}$ to $Z^{\prime}$. If $i\in I\backslash J$ then we also write
$D^{Z^{\prime}}_{i}$ for the restriction of $D_{i}^{Z}$ to $Z^{\prime}$. We
also set $U_{Z^{\prime}}:=Z^{\prime}\backslash{\mathop{{\rm
Supp\,}}}D^{Z^{\prime}}$.

Assume that $\dim T>0$. Let $I^{h}\subset I$ be the subset corresponding to
the components dominating $T$ and let $I^{v}:=I\backslash I^{h}$ be its
complement. Then we set $D^{h}:=\sum_{i\in I^{h}}D_{i}^{Z}$ and
$D^{v}:=\sum_{i\in I^{v}}D_{i}^{Z}$. In the following $D^{h}$ is called the
_horizontal part_ of $D^{Z}$ and $D^{v}$ is called the _vertical part_ of
$D^{Z}$.

By Proposition 5.6 we have $0\leq s=|I^{h}|\leq r$. Let $U_{T}$ be the
complement of the image of $D^{v}$ in $T$. For a subset $J\subset I^{h}$, we
set $U_{J}^{\prime}=D^{h}_{J}\cap\varphi_{Z}^{-1}(U_{T})$ (note that
$D^{h}_{J}$ is irreducible). Clearly, $\varphi_{Z}(D^{h}_{J})=T$ and
$\varphi_{Z}|_{U_{J}^{\prime}}$ is a projective morphism with a general fiber
isomorphic to ${\mathbb{P}}^{r+s-|J|}$. We need the following result on the
dimension of its fibers.

######  Lemma 5.9.

If $\dim T>0$ then for any subset $J\subset I^{h}$ (possibly empty), all
fibers of $\varphi_{Z}|_{U_{J}^{\prime}}$ have dimension $r+|I^{h}\backslash
J|$.

###### Proof.

Since $-(K_{Z}+D^{Z})=(r+1)c_{1}({\mathcal{L}})$ is $\varphi_{Z}$-ample, for
sufficiently ample line bundle ${\mathcal{A}}$ on $T$ the line bundle
${\mathcal{L}}^{\prime}={\mathcal{L}}\otimes\varphi_{Z}^{*}{\mathcal{A}}$ is
ample. The proof is by induction on the number of elements in $I^{h}\backslash
J$.

_Step 1._ Assume that $|I^{h}\backslash J|=0$, i.e., $J=I^{h}$.

Let us consider $Z^{\prime}=D^{h}_{I^{h}}$ (if $I^{h}=\emptyset$ then we set
$Z^{\prime}=Z$). In this case Proposition 5.6 implies that for general $y\in
T$ we have $(Z_{y}^{\prime},D_{y}^{Z^{\prime}})\simeq({\mathbb{P}}^{r},0)$ and
${\mathcal{L}}|_{Z^{\prime}_{y}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{r}}(1)$.
Let us consider the Chow variety $\mathop{\rm Chow\,}_{r,1}(Z^{\prime})$ of
$r$-dimensional cycles on $Z^{\prime}$ of degree $1$ with respect to
${\mathcal{L}}^{\prime}$ and let $S$ be an irreducible component of
$\mathop{\rm Chow\,}_{r,1}(Z^{\prime})$ containing $Z^{\prime}_{y}$ for
general $y\in T$. This component is projective of dimension $\dim T$ and there
exists a universal family of cycles $\tau:{\mathcal{U}}\to S$ with the
evaluation map ${\mathop{\rm ev\,}}:{\mathcal{U}}\to Z^{\prime}$. The map
${\mathop{\rm ev\,}}$ is birational and either it is an isomorphism or it has
a positive dimensional fiber. In the first case,
$(\varphi_{Z}|_{Z^{\prime}}:Z^{\prime}\to T)\simeq(\tau:{\mathcal{U}}\to S)$,
so $\varphi_{Z}|_{Z^{\prime}}$ is equidimensional. In the second case, there
exists $x_{0}\in Z^{\prime}$ such that $\dim({\mathop{\rm
ev\,}}^{-1}(x_{0}))>0$. Then by the defining property of $\mathop{\rm
Chow\,}_{r,1}(Z^{\prime})$, $\dim({\mathop{\rm
ev\,}}(\tau^{-1}(\tau({\mathop{\rm ev\,}}^{-1}(x_{0})))))>r$.

Let $V_{{\mathcal{U}}}\subset{\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},{\mathcal{U}})$ be an irreducible component containing
some $f:{\mathbb{P}}^{1}\to{\mathcal{U}}$ for which the image is a line in a
general fiber
$Z^{\prime}_{y}=\tau^{-1}([Z^{\prime}_{y}])\simeq{\mathbb{P}}^{r}$ and $f$ is
an isomorphism onto its image. The family $V_{{\mathcal{U}}}$ is unsplit and
the natural map $\tilde{\mathop{\rm ev\,}}:{\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},{\mathcal{U}})\to{\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},Z^{\prime})$ maps $V_{{\mathcal{U}}}$ onto an
irreducible component $V$ of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},Z^{\prime})$ containing lines in a general fiber
$Z^{\prime}_{y}$ of $\varphi_{Z}|_{Z^{\prime}}$. For general
$u\in{\mathcal{U}}$, ${\rm
locus}(V_{{\mathcal{U}}},u)=\tau^{-1}(\tau(u))\simeq{\mathbb{P}}^{r}$. Since
$V_{{\mathcal{U}}}$ is unsplit, the equality ${\rm
locus}(V_{{\mathcal{U}}},u)=\tau^{-1}(\tau(u))$ holds for all
$u\in{\mathcal{U}}$. This implies that ${\rm
locus}(V,x_{0})\supset{\mathop{\rm ev\,}}(\tau^{-1}(\tau({\mathop{\rm
ev\,}}^{-1}(x_{0}))))$ has dimension $\geq(r+1)$.

Let us assume that $x_{0}\in U_{Z^{\prime}}$. If $g:{\mathbb{P}}^{1}\to
Z^{\prime}$ is a map from the family $V$ and $x_{0}\in C:=g({\mathbb{P}}^{1})$
then $C\subset U_{Z^{\prime}}$ as $C$ is not contained in the support of
$D^{Z^{\prime}}$ and $D^{Z^{\prime}}\cdot C=0$. But then Lemma 5.3 implies
that $\dim{\rm locus}(V,x_{0})=r$, a contradiction. It follows that
$\varphi_{Z}(x_{0})\not\in U_{T}$ and all fibers of
$\varphi_{Z}|_{U_{I^{h}}^{\prime}}:U_{I^{h}}^{\prime}\to U_{T}$ have dimension
$r$.

_Step 2._ Assume that $J\subsetneq I^{h}$ and the assertion holds for all
subsets properly containing $J$.

Let us set $m=|I^{h}\backslash J|$ and consider $Z^{\prime}=D^{h}_{J}$. By
Proposition 5.6 for general $y\in T$ we have
$(Z_{y},D_{y}^{Z})\simeq({\mathbb{P}}^{r+s},H_{1}+...+H_{s})$ and
${\mathcal{L}}|_{Z_{y}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{r+s}}(1)$. This
implies that for general $y\in T$ we have
$(Z_{y}^{\prime},D_{y}^{Z^{\prime}})\simeq({\mathbb{P}}^{r+m},H_{1}+...+H_{m})$
and
${\mathcal{L}}|_{Z_{y}^{\prime}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{r+m}}(1)$.
Let us consider the Chow variety $\mathop{\rm Chow\,}_{r+m,1}(Z^{\prime})$ of
$(r+m)$-dimensional cycles on $Z^{\prime}$ of degree $1$ with respect to
${\mathcal{L}}^{\prime}$. Let $S$ be an irreducible component of $\mathop{\rm
Chow\,}_{r+m,1}(Z^{\prime})$ containing general fibers $Z^{\prime}_{y}$ of
$\varphi_{Z}|_{Z^{\prime}}$. This component is projective of dimension equal
to the dimension of $T$ and there exists a universal family of pairs of cycles
$\tau:{\mathcal{U}}\to S$ with the evaluation map ${\mathop{\rm
ev\,}}:{\mathcal{U}}\to Z^{\prime}$. The map ${\mathop{\rm ev\,}}$ is
birational and either it is an isomorphism or it has a positive dimensional
fiber. In the first case, $(\varphi_{Z}|_{Z^{\prime}}:Z^{\prime}\to
T)\simeq(\tau:{\mathcal{U}}\to S)$, so $\varphi_{Z}|_{Z^{\prime}}$ is
equidimensional. In the second case, there exists $x_{0}\in Z^{\prime}$ such
that $\dim({\mathop{\rm ev\,}}^{-1}(x_{0}))>0$. Then by the defining property
of $\mathop{\rm Chow\,}_{r+m,1}(Z^{\prime})$, $\dim({\mathop{\rm
ev\,}}(\tau^{-1}(\tau({\mathop{\rm ev\,}}^{-1}(x_{0})))))>r+m$.

Let us choose some $f:{\mathbb{P}}^{1}\to{\mathcal{U}}$ such that $f$ is an
isomorphism onto a line in a general fiber
$Z^{\prime}_{y}=\tau^{-1}([Z^{\prime}_{y}])\simeq{\mathbb{P}}^{r+m}$, not
contained in the support of $D^{Z^{\prime}}_{y}$. By $\tilde{f}$ we denote the
composition of $f$ with ${\mathop{\rm ev\,}}$. Let us fix some $i_{0}\in
I^{h}\backslash J$ and let us set $D_{{\mathcal{U}}}:={\mathop{\rm
ev\,}}^{*}D^{Z^{\prime}}_{i_{0}}$. Let $W_{{\mathcal{U}}}\subset{\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},{\mathcal{U}};f^{*}D_{{\mathcal{U}}}\subset
D_{{\mathcal{U}}})$ be an irreducible component containing $[f]$. The family
$W_{{\mathcal{U}}}$ is unsplit and the natural map $\tilde{\mathop{\rm
ev\,}}:{\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},{\mathcal{U}};f^{*}D_{{\mathcal{U}}}\subset
D_{{\mathcal{U}}})\to{\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},{Z^{\prime}};\tilde{f}^{*}D^{Z^{\prime}}_{i_{0}}\subset
D^{Z^{\prime}}_{i_{0}})$ maps $W_{{\mathcal{U}}}$ onto an irreducible
component $W$ of ${\mathop{{\rm
Hom}}}({\mathbb{P}}^{1},{Z^{\prime}};\tilde{f}^{*}D^{Z^{\prime}}_{i_{0}}\subset
D^{Z^{\prime}}_{i_{0}})$ containing $[\tilde{f}]$. For general
$u\in{\mathcal{U}}$, ${\rm
locus}(W_{{\mathcal{U}}},u)=\tau^{-1}(\tau(u))\simeq{\mathbb{P}}^{r+m}$. Since
$W_{{\mathcal{U}}}$ is unsplit, the equality ${\rm
locus}(W_{{\mathcal{U}}},u)=\tau^{-1}(\tau(u))$ holds for all
$u\in{\mathcal{U}}$. This implies that ${\rm
locus}(W,x_{0})\supset{\mathop{\rm ev\,}}(\tau^{-1}(\tau({\mathop{\rm
ev\,}}^{-1}(x_{0}))))$, so it has dimension $\geq(r+m+1)$.

By Lemma 4.6, $(Z^{\prime},D^{Z^{\prime}})$ is semi-contact of rank $(2r+1)$.
So by Lemma 4.18 for any $g:{\mathbb{P}}^{1}\to{Z^{\prime}}$ with $[g]\in W$
we have

${\mathop{\rm rk\,}}^{+}g^{*}T_{Z^{\prime}}(-\log D^{Z^{\prime}})\leq r+1.$

If $g({\mathbb{P}}^{1})\cap{\mathop{{\rm Supp\,}}}D^{v}=\emptyset$ then
pulling back the short exact sequence

$0\to T_{Z^{\prime}}(-\log(\sum_{i\in I^{h}\backslash
J}D^{Z^{\prime}}_{i}))\longrightarrow T_{Z^{\prime}}(-\log
D^{Z^{\prime}}_{i_{0}})\longrightarrow\bigoplus_{i\in
I^{h}\backslash(J\cup\\{i_{0}\\})}{\mathcal{O}}_{D^{Z^{\prime}}_{i}}(D^{Z^{\prime}}_{i})\to
0$

by $g$ we get an exact sequence

$g^{*}T_{Z^{\prime}}(-\log D^{Z^{\prime}})\longrightarrow
g^{*}T_{Z^{\prime}}(-\log
D^{Z^{\prime}}_{i_{0}})\longrightarrow{\mathcal{O}}_{{\mathbb{P}}^{1}}(1)^{\oplus(m-1)}\to
0.$

Then

${\mathop{\rm rk\,}}^{+}g^{*}T_{Z^{\prime}}(-\log
D^{Z^{\prime}}_{i_{0}})\leq(r+1)+(m-1)=r+m.$

Since the family $W$ is unsplit, Proposition 2.5 implies that if $x_{0}\not\in
D^{{Z^{\prime}}}_{i_{0}}\cup{\mathop{{\rm Supp\,}}}D^{h}$ then $\dim{\rm
locus}(W,x_{0})\leq r+m$, a contradiction. If $x_{0}\in
D^{Z^{\prime}}_{i_{0}}\backslash{\mathop{{\rm Supp\,}}}D^{h}$ then ${\rm
locus}(W,x_{0})\cap D^{Z^{\prime}}_{i_{0}}$ has dimension $\geq(r+m)$. But
${\rm locus}(W,x_{0})$ is contained in a fiber of $\varphi_{Z}$ as it is
covered by extremal curves passing through one point. So ${\rm
locus}(W,x_{0})\cap D^{h}$ is contained in $(\varphi_{Z}|_{{Z^{\prime}}\cap