# GW Invariants Relative Normal Crossings Divisors

Eleny-Nicoleta Ionel
Stanford University
Stanford, CA 94305 Research supported in part by the NSF grants DMS-0605003
and DMS-0905738.

###### Abstract

In this paper we introduce a notion of symplectic normal crossings divisor $V$
and define the GW invariant of a symplectic manifold $X$ relative such a
divisor. Our definition includes normal crossings divisors from algebraic
geometry. The invariants we define in this paper are key ingredients in
symplectic sum type formulas for GW invariants, and extend those defined in
our previous joint work with T.H. Parker [IP1], which covered the case $V$ was
smooth. The main step is the construction of a compact moduli space of
relatively stable maps into the pair $(X,V)$ in the case $V$ is a symplectic
normal crossings divisor in $X$.

##  0 Introduction

In previous work with Thomas H. Parker [IP1] we constructed the relative
Gromov-Witten invariant $GW(X,V)$ of a closed symplectic manifold $X$ relative
a smooth “divisor” $V$, that is, a (real) codimension 2 symplectic
submanifold. These relative invariants are defined by choosing an almost
complex structure $J$ on $X$ that is compatible with both $V$ and the
symplectic form, and counting $J$-holomorphic maps that intersect $V$ with
specified multiplicities. An important application is the symplectic sum
formula that relates the GW invariant of a symplectic sum $X\\#_{V}Y$ to the
relative GW invariants of $(X,V)$ and $(Y,V)$ (see [IP2] and the independent
approaches [LR], [Li] and [EGH]).

In this paper we introduce a notion of symplectic normal crossings divisor $V$
and define the GW invariant of a symplectic manifold $X$ relative such a
divisor. Roughly speaking, a set $V\subset X$ is a symplectic normal crossings
divisor if it is locally the transverse intersection of codimension 2
symplectic submanifolds compatible with $J$ (the precise definition is given
in Section 1).

There are many reasons why one would want to extend the definition of relative
GW invariants to include normal crossings divisors, and we already have
several interesting applications in mind. One is a Mayer-Vietoris type formula
for the GW invariants: a formula describing how the GW invariants behave when
$X$ degenerates into several components and that allows one to recover the
invariants of $X$ from those of the components of the limit. The simplest such
degenerations come from the symplectic sum along a smooth divisor. But if one
wants to iterate this degeneration, one is immediately confronted with several
pieces whose intersection is no longer smooth, but instead are normal
crossings divisors. Normal crossings divisors appear frequently in algebraic
geometry, not only as the central fiber of a stable degeneration but also for
example as the toric divisor in a toric manifold which then appears in the
context of mirror symmetry. We also have some purely symplectic applications
in mind in which normal crossings divisors arise from Donaldson’s theorem;
these will appear in a subsequent paper.

The general approach in this paper is to appropriately adapt the ideas in
[IP1] but now allow the divisor to have a simple type of singularity, which we
call symplectic normal crossings. This is defined in Section 1, where we also
present many of the motivating examples. The notion of simple singularity is
of course relative: the main issue here is to be able to control the analysis
of the problem; the topology of the problem, though perhaps much more
complicated is essentially of a combinatorial nature so it is much easier
controlled.

There are several new features and problems that appear when the divisor $V$
has such singular locus. First, one must include in the moduli space
holomorphic curves that intersect the singular locus, and one must properly
record the contact information about such intersections. In Section 2 we
describe how to do this and construct the corresponding moduli space ${\cal
M}_{s}(X,V)$ of stable maps into $X$ whose contract intersection with $V$ is
described by the sequence $s$. There is a lot of combinatorics lurking in the
background that keeps track of the necessary topological information along the
singular locus, which could make the paper unnecessarily longer. We have
decided to keep the notation throughout the paper to a minimum, and expand its
layers only as needed for accuracy in each section. We give simple examples of
why certain situations have to be considered, explain in that simple example
what needs to be done, and only after that proceed to describe how such
situations can be handled in general. In the Appendix we describe various
needed stratifications associated to a normal crossings divisor, and
topological data associated to it.

The other more serious problem concerns the construction of a compactification
$\overline{{\cal M}}_{s}(X,V)$ of the relative moduli space. In the usual
Gromov compactification of stable maps into $X$, a sequence of holomorphic
maps that have a prescribed contact to $V$ may limit to a map that has
components in $V$ or even worse in the singular locus of $V$; then not only
the contact information is lost in the limit, but the formal dimension of the
corresponding boundary stratum of the stable map compactification is greater
than the dimension of the moduli space. This problem already appeared for the
moduli space relative a smooth divisor, where the solution was to rescale the
target normal to $V$ to prevent components from sinking into $V$; but now the
problem is further compounded by the presence of the singular locus of $V$. So
the main issue now is to how to precisely refine the Gromov compactness and
construct an appropriate relatively stable map compactification
$\overline{{\cal M}}_{s}(X,V)$ in such a way that its boundary strata are not
larger dimensional than the interior.

In his unpublished Ph. D. thesis, Joshua Davis [Da] described how one can
construct a relatively stable map compactification for the space of genus zero
maps relative a normal crossings divisor, by recursively blowing up the
singular locus of the divisor. As components sunk into this singular locus, he
recursively blew it up to prevent this from happening. This works for genus
zero, but unfortunately not in higher genus. The main reason for this is that
in genus zero a dimension count shows that components sinking into $V$ cause
no problem, only those sinking into the singular locus of $V$ do. However,
that is not the case in higher genus, so then one would also need to rescale
around $V$ to prevent this type of behavior. But then the process never
terminates: Josh had a simple example in higher genus where a component would
sink into the singular locus. Blowing up the singular locus forced the
component to fall into the exceptional divisor. Rescaling around the
exceptional divisor then forced the component to fall back into the next
singular locus, etc.

In this paper we present a different way to construct the a relatively stable
map compactification $\overline{{\cal M}}_{s}(X,V)$, by instead rescaling $X$
simultaneously normal to all the branches of $V$, a procedure we describe in
Section 3. When done carefully, this is essentially a souped up version of the
rescaling procedure described in [IP1] in the case $V$ was smooth.
Unfortunately, the naive compactification that one would get by simply
importing the description of that in [IP1] simply does not work when the
singular locus of $V$ is nonempty! There are two main reasons for its failure:
the first problem is that the ”boundary stratum” containing curves with
components over the singular locus is again larger dimensional than the
”interior” so it is in some sense too big; the second problem is that it still
does not capture all the limits of curves sinking into the singular locus, so
it is too small! This seems to lead into a dead end, but upon further analysis
in Sections 5 and 6 of the limiting process near the singular locus two new
features appear that allows us to still proceed.

The first new feature is the enhanced matching condition that the limit curves
must satisfy along the singular locus of $V$. It turns out that not all the
curves which satisfy the naive matching conditions can appear as limits of
maps in ${\cal M}_{s}(X,V)$. The naive matching conditions require that the
curves intersect $V$ in the same points with matching order of contact, as was
the case in [IP1], while the enhanced ones along the singular locus require in
some sense that their slopes in the normal directions to $V$ also match. So
the enhanced matching conditions also involve the leading coefficients of the
maps in these normal directions, and so they give conditions in a certain
weighted projectivization of the normal bundle to the singular locus, a simple
form of which is described in Section 4. Luckily, this is enough to cut back
down the dimensions of the boundary to what should be expected. In retrospect,
these enhanced matching conditions already appeared in one of the key Lemmas
in our second joint paper [IP2] with Thomas H. Parker about the symplectic sum
formula, but they do not play any role in the first paper [IP1] because they
are automatically satisfied when $V$ is smooth.

The second new feature that appears when $V$ is singular is that unfortunately
one cannot avoid trivial components stuck in the neck (over the singular locus
of $V$), as we show in some simple Examples at the end of Section 4. This
makes the enhanced matching conditions much more tricky to state, essentially
because these trivial components do not have the right type of leading
coefficients. The solution to this problem is to realize that the trivial
components are there only to make the maps converge in Hausdorff distance to
their limit, and in fact they do not play any essential role in the
compactification, so one can simply collapse them in the domain, at the
expense of allowing a node of the collapsed domain to be between not
necessarily consecutive levels. The enhanced matching condition then occurs
only at nodes between two nontrivial components, but needs to take into
account this possible jump across levels. It is described more precisely in
Section 5.

This finally allows us to define in Section 6 the compactified moduli space
$\overline{{\cal M}}_{s}(X,V)$ of relatively $V$-stable maps into $X$, which
comes together with a continuous map

$\displaystyle\mbox{\rm st}\times\mbox{\rm Ev}:\overline{{\cal
M}}_{s}(X,V)\rightarrow\overline{{\cal
M}}_{\chi_{s},\ell(s)}\times\prod_{x}{\mathbb{P}}_{s(x)}(NV_{I(x)})$ (0.1)

The first factor is the usual stabilization map recording the domain of $f$,
which may be disconnected, but the new feature is the second factor Ev. It is
a refinement of the usual (naive) evaluation map ev at the points $x$ that are
mapped into the singular locus of $V$, and it also records the weighted
projectivization of the leading coefficients of $f$ at $x$ in all the normal
directions to $V$ at $f(x)$. This is precisely the map that appears in the
enhanced matching conditions.

In Section 7 we then show that for generic $V$-compatible $(J,\nu)$ the image
of $\overline{{\cal M}}_{s}(X,V)$ under the map (0.1) indeed defines a
homology class $GW_{s}(X,V)$ in dimension

$\displaystyle{\rm dim\;}\overline{{\cal M}}_{s}(X,V)=2c_{1}(TX)A_{s}+({\rm
dim\;}X-6)\frac{\chi_{s}}{2}+2\ell(s)-2A_{s}\cdot V$

called the GW invariant of $X$ relative the normal crossings divisor $V$. The
class $GW_{s}(X,V)$ is independent of the perturbation $\nu$ and is in fact an
invariant under smooth deformations of the pair $(X,V)$ and of $(\omega,J)$
though $V$-compatible structures. When $V$ is smooth these invariants agree
with the usual relative GW invariants as constructed in [IP1].

There is a string of very recent preprints that have some overlap with the
situation considered in our paper, in that they all generalize in some way the
normal crossings situation from algebraic geometry. First of all, there is
certainly an overlap between what we call a symplectic normal crossings
divisor in this paper and what fits into the exploded manifold setup
considered by Brett Parker [P]. There is also some overlap with the
logarithmic Gromov-Witten invariants [GrS] considered by Gross and Siebert in
the context of algebraic geomery (see also the Abramovich-Chen paper [AC] on a
related topic). However, the precise local structure near the divisor is very
different: log geometry vs symplectic normal crossings vs exploded structures.
Furthermore, the moduli spaces constructed in these papers and in particular
their compactifications are completely different, even when applied to the
common case when $V$ is a smooth divisor in a smooth projective variety, see
Remarks 1.15 and 1.16 for more details. This means that a priori even in this
common case each one of these other approaches many lead to different
invariants, some even different from the usual relative GW invariants.

This paper is based on notes from a talk the author gave in Sept 2006 in the
Moduli Space Program at Mittag-Leffler Institute, during a month long stay
there. The notes were expanded in the fall of 2009 during the Symplectic and
Contact Geometry and Topology program at MSRI. We thank both research
institutes for their hospitality.

##  1 Symplectic normal crossings divisors

In this section we define a notion of symplectic normal crossings divisors,
which encodes the geometrical information required for the analysis of [IP1]
and [IP2] to extend after appropriate modifications. In particular this notion
generalizes the notion of normal crossings divisor in algebraic geometry.
Clearly the local model of such divisor $V$ should be the union of $k$
coordinate planes in ${\mathbb{C}}^{n}$, where the number of planes may vary
from point to point. But we also need a local model for the symplectic form
$\omega$ and the tamed almost complex structure $J$ near such divisor. We will
therefore require that each branch of $V$ is both $\omega$ symplectic and $J$
holomorphic. This will allow us to define the order of contact of
$J$-holomorphic curves to $V$. We also need a good description of the normal
directions to the divisor, because these are going to be the directions in
which the manifold $X$ will be rescaled when components of the holomorphic
curves fall into $V$. In particular, we need to keep track of both the normal
bundle to each branch of $V$ and its inclusion into $X$ which describes the
neighborhood of that branch.

###### Definition 1.1

(local model). In ${\mathbb{C}}^{n}$ consider the union $V$ of $k\geq 0$
(distinct) coordinate hyperplanes $H_{i}=\\{x|x_{i}=0\\}$ in
${\mathbb{C}}^{n}$, together with their normal direction $N_{i}$ given by the
usual projection $\pi_{i}:{\mathbb{C}}^{n}\longrightarrow H_{i}$ onto the
coordinate plane and the usual inclusion
$\iota:N_{i}\rightarrow{\mathbb{C}}^{n}$. We say that they form a model for a
normal crossings divisor in ${\mathbb{C}}^{n}$ with respect to a pair
$(\omega,J)$ if all the divisors $H_{i}$ are both $\omega$-symplectic and
$J$-holomorphic.

###### Remark 1.2

There is a natural action of ${\mathbb{C}}^{*}$ on the model induced by
scaling by a factor of $t^{-1}$ in the normal direction to each $H_{i}$, for
$i=1,\dots,k$. This defines a rescaling map
$R_{t}:{\mathbb{C}}^{n}\rightarrow{\mathbb{C}}^{n}$ for
$t\in{\mathbb{C}}^{*}$. By construction, the $R_{t}$ leaves the divisors
$H_{i}$ invariant, but not pointwise, and may not preserve $J$. However, as
$t\rightarrow 0$, $R_{t}^{*}J$ converges uniformly on compacts to a
${\mathbb{C}}^{*}$ invariant limit $J_{0}$ which depends on the 1-jet of $J$
along the divisor.

###### Definition 1.3

Assume $(X,\omega,J)$ is a symplectic manifold with a tamed almost complex
structure. $V$ is called a normal crossings divisor in $(X,\omega,J)$ with
normal bundle $N$ if there exists a smooth manifold $\widetilde{V}$ with a
complex line bundle $\pi:N\rightarrow\widetilde{V}$ and an immersion
$\iota:U_{V}\rightarrow X$ of some disk bundle $U_{V}$ of $N_{V}$ into $X$
satisfying the following properties:

  * •

$V$ is the image of the zero section $\widetilde{V}$ of $N$

  * •

the restriction of $\iota^{*}J$ to the fiber of $N$ along the zero section
induces the complex multiplication in the bundle $N$.

  * •

at each point $p\in X$ we can find local coordinates on $X$ in which the
configuration $(X,\pi,\iota,V)$ becomes identified with one of the local
models in Definition 1.1.

Such a pair $(J,\omega)$ is called adapted to the divisor $V$.

Note that $\iota$ induces by pullback from $X$ both a symplectic structure
$\omega$ and an almost complex structure $J$ on the total space of the disk
bundle in $N$ over on $\widetilde{V}$, which will serve as a global model of
$X$ near $V$. Its zero section $\widetilde{V}$ is both symplectic and
$J$-holomorphic and serves as a smooth model of the divisor $V$ (called the
normalization of $V$). $N$ is also a complex line bundle whose complex
structure comes from the restriction of $J$ along the zero section. Thus $N$
also comes with a ${\mathbb{C}}^{*}$ action which will be used to rescale $X$
normal to $V$.

###### Remark 1.4

We are not requiring $J$ to be locally invariant under this ${\mathbb{C}}^{*}$
action. We also are not imposing the condition that the branches are
perpendicular with respect to $\omega$ or that the projections $\pi_{i}$ are
$J$-holomorphic. We also allow for self intersections of various components of
$V$. When each component of $V$ is a submanifold of $X$ the divisor is said to
have simple normal crossings singularities. Any of these assumptions would
simplify some of the arguments, but are not needed.

In this paper we will work only with $J$’s which are compatible to $V$ in the
sense of Definition 3.2 of [IP1]. This is a condition on the normal 1-jet of
$J$ along $V$:

  1. (b)

$[(\nabla_{\xi}J+J\nabla_{J\xi}J)(v)]^{N}=[\nabla_{v}J)\xi+J(\nabla_{Jv}J)\xi]^{N}$
for all $v\in TV$, $\xi\in NV$;

discussed in more detail in the Appendix. This extra condition is needed to
ensure that the stable map compactification has codimension 2 boundary strata,
so it gives an invariant, independent of parameters. A priori, even when $V$
is smooth the relatively stable map compactification may have real codimension
1 boundary without this extra assumption.

###### Example 1.5

A large class of examples of normal crossings is provided by algebraic
geometry. Assume $X$ is a smooth projective variety and $V$ a complex
codimension 1 subvariety which is locally a transverse union of smooth
divisors. In particular $V$ could be the transverse intersection of smooth
divisors, in which case $V$ is said to have simple normal crossings, but in
general the divisors may also self intersect. Then $V$ is a symplectic normal
crossings divisor for $(X,J_{0},\omega_{0})$ where $J_{0}$ is integrable
complex structure and $\omega_{0}$ the Kahler form. For example (a) $X$ could
be a Hirzebruch surface and $V$ the union of the zero section, the infinity
section and several fibers or (b) $V$ could be the union of a section and a
nodal fiber in an elliptic surface $X$.

An important example of this type is when $X$ is a toric manifold and $V$ is
its toric divisor, which is a case considered in mirror symmetry, see for
example [Au2].

###### Example 1.6

Another particular example to keep in mind is $X={\mathbb{C}}{\mathbb{P}}^{2}$
with a degree 3 normal crossings divisor $V$. For example $V$ could be a
smooth elliptic curve, or $V$ could be a nodal sphere, or finally $V$ could be
a union of 3 distinct lines. In the second case the normalization
$\widetilde{V}$ is ${\mathbb{C}}{\mathbb{P}}^{1}$ with normal bundle $O(7)$
while in the last case it is
${\mathbb{C}}{\mathbb{P}}^{1}\sqcup{\mathbb{C}}{\mathbb{P}}^{1}\sqcup{\mathbb{C}}{\mathbb{P}}^{1}$,
each component with normal bundle $O(1)$. Of course, in a complex 1-parameter
family, a smooth degree three curve can degenerate into either one of the
other two cases.

Another motivating example of this type comes from a smooth quintic 3-fold
degenerating to a union of 5 hyperplanes in ${\mathbb{C}}{\mathbb{P}}^{4}$.

###### Remark 1.7

Yet another special case is $X=\overline{{\cal M}}_{0,n}$ the Deligne-Mumford
moduli space of stable genus $0$ curves and $V$ the union of all its boundary
strata (i.e. the stratum of nodal curves). The usual description of each
boundary stratum and of its normal bundle provides the required local models
for a symplectic normal crossings divisor. This discussion can also be
extended to the orbifold setting to cover the higher genus case
$\overline{{\cal M}}_{g,n}$ and certainly covers its smooth finite cover, the
moduli space of Prym curves, constructed by Looijenga [Lo].

Of course, there are many more symplectic examples besides those coming from
algebraic geometry.

###### Example 1.8

Assume $V$ is a symplectic codimension two submanifold of $(X,\omega)$. The
symplectic neighborhood theorem then allows us to find a $J$ and a model for
the normal direction to $V$, so $V$ is normal crossings divisor in
$(X,\omega,J)$. Of course in this case $V$ is a smooth divisor, so it has
empty singular locus.

One may have hoped that the union of several transversely intersecting
codimension two symplectic submanifolds would also similarly be a normal
crossings divisor. Unfortunately, if the singular locus is not empty, that may
not be the case, as illustrated by the example below.

###### Example 1.9

Let $V_{1}$ be an exceptional divisor in a symplectic 4-manifold and $V_{2}$ a
sufficiently small generic perturbation of it, thus still a symplectic
submanifold, intersecting transversely $V_{1}$. This configuration cannot be
given the structure of a normal crossings divisor, simply because one cannot
find a $J$ which preserves both. If such a $J$ existed, then all the
intersections between $V_{1}$ and $V_{2}$ would be positive, contradicting the
fact that exceptional divisors have negative self intersection.

This example illustrates the fact that a normal crossings divisor is not a
purely symplectic notion, but rather one also needs the existence of a tamed
almost complex structure $J$ adapted to the crossings. The positivity of
intersections of all branches is a necessary condition for such a $J$ to exist
in general.

###### Remark 1.10

One could ask what are the necessary and sufficient conditions for $V$ inside
a symplectic manifold $(X,\omega)$ to be a normal crossings divisor with
respect to some $J$ on $X$. Clearly $V$ should be locally the transverse
intersection of symplectic submanifolds, and furthermore this intersection
should be positive. If we assume that the branches of $V$ are moreover
orthogonal wrt $\omega$ then the existence of a compatible $J$ is
straightforward (see the end of Appendix).

###### Example 1.11

Symplectic Lefschetz pencils or fibrations provide another source of examples
of symplectic normal crossings divisors. Assume $X$ is a symplectic manifold
which has a symplectic Lefschetz fibration with a symplectic section, for
example one coming from Donaldson Theorem [Do2] where the section comes from
blowing up the base locus. Gomph [GoS] showed that in this case there is an
adapted almost complex structure $J$ to this fibration. We could then take $V$
the union of the section with a bunch of fibers, including possibly some
singular fibers.

###### Example 1.12

(Donaldson divisors) Assume $V$ is a normal crossings divisor in
$(X,\omega,J)$ and that $[\omega]$ has rational coefficients. We can use
Donaldson theorem [Do] to obtain a smooth divisor $D$ representing the
Poincare dual of $k\omega$ for $k>>0$ sufficiently large, such that $D$ is
$\varepsilon$-$J$-holomorphic and $\eta$-transverse to $V$ (see also [Au]).
Choosing carefully the parameters $\eta$ and $\varepsilon$, one can then find
a sufficiently small deformation of $J$ such that $V\cup D$ is also a normal
crossings divisor.

###### Remark 1.13

The definition of a normal crossings divisor works well under taking products
of symplectic manifolds with divisors in them. If $V_{i}$ is a normal
crossings divisor in $X_{i}$ for $i=1,2$ then
$\pi_{1}^{-1}(V_{1})\cup\pi_{2}^{-1}(V_{2})=V_{1}\times X_{2}\cup X_{1}\times
V_{2}$ is a normal crossings divisor in $X_{1}\times X_{2}$, with normal model
$\pi_{1}^{*}N_{1}\sqcup\pi_{2}^{*}N_{2}$. Note that even if $V_{i}$ were
smooth divisors, then the divisor in the product $X_{1}\times X_{2}$ is still
singular along $V_{1}\times V_{2}$.

###### Remark 1.14

The definition of a normal crossings divisor also behaves well under
symplectic sums. Assume $U_{i}\cup V$ is a symplectic divisor in $X_{i}$ for
$i=1,2$ such that the normal bundles of $V$ in $X_{i}$ are dual. If $U_{i}$
intersect $V$ in the same divisor $W$ then Gomph’s argument [Go] shows that
the divisors $U_{i}$ glue to give a normal crossings divisor
$U_{1}\\#_{W}U_{2}$ in the symplectic sum $X_{1}\\#_{V}X_{2}$.

###### Remark 1.15

A special case of symplectic normal crossings divisor $V$ (with simple
crossings) is the union of codimension 2 symplectic submanifolds which
intersect orthogonally wrt $\omega$, and whose local model matches that of
toric divisors in a toric manifold. This is a case that fits in the exploded
manifold set-up of Brett Parker (see Example 5.3 in the recent preprint [P]),
so in principle one should be able to compare the relative invariants we
construct in this paper with the exploded ones of [P]. It is unclear to us
what is exactly the information that the exploded structure records in this
case, and what is the precise relation between the two moduli spaces. But
certainly the relatively stable map compactification we define in this paper
seems to be quite different from the exploded one, so it is unclear whether
they give the same invariants, even in the case when $V$ is smooth.

###### Remark 1.16

In a related recent preprint, Gross and Seibert define log GW invariants in
the algebraic geometry setting [GrS]. If $V$ is a normal crossings divisor in
a smooth projective variety $X$, then it induces a log structure on $X$.
However, even in the case $V$ is a smooth divisor, Gross and Siebert explain
that the stable log compactification they construct is quite different from
the relatively stable map compactification constructed earlier in that context
by J. Li [Li] (and which agrees with that of [IP1] in this context). So a
priori, even when $V$ is smooth, the usual relative GW invariants may be
different from the log GW invariants of [GrS]. The authors mention however
that in that case at least there is a map from the moduli space of stable
relative maps to that of stable log maps, which they claim could be used to
prove that the invariants are the same, though no proof of this claim is
available yet. Presumably there is also a map from the relatively stable map
compactification that we construct in this paper to the appropriate stable log
compactification in the more general case when $V$ is a normal crossings
divisor in a smooth projective variety.

In a related paper [AC] Abramovich and Chen also explain how, in the context
of algebraic geometry, the construction of a log moduli space when $V$ is a
normal crossings divisor (with simple crossings) follows from the case when
$V$ is smooth by essentially functorial reasons. Again, it is unclear to us
how exactly the two notions of log stable maps of [GrS] and [AC] are related
in this case.

###### Remark 1.17

One note about simple normal crossings vs general normal crossings: they do
complicate the topology/combinatorics of the situation, but if set up
carefully the analysis is unaffected. If the local model of $X$ is holomorphic
near $V$ (as is the case in last two examples above), even if $V$ did not have
simple crossings, one could always blow up the singular locus $W$ of $V$ to
get a total divisor $\pi^{-1}(V)=Bl(V)\cup E$ with simple normal crossings in
$Bl(X)$, where $E$ is the exceptional divisor. Blowing up in the symplectic
category is a more delicate issue, but when using the appropriate local model,
one can always express (a symplectic deformation) of the original manifold
$(X,V)$ as a symplectic sum of its blow up $(Bl(X),Bl(V))$ along the
exceptional divisor $E$ with a standard piece $({\mathbb{P}},V_{0})$ involving
the normal bundle of the blowup locus. Since we are blowing up the singular
locus of $V$, the proper transform $Bl(V)$ intersects nontrivially the
exceptional divisor $E$; the symplectic sum $Bl(X)\\#_{E}{\mathbb{P}}=X$ then
also glues $Bl(V)$ to the standard piece on the other side to produce $V$, as
in Remark 1.14. Therefore a posteriori, after proving a symplectic sum formula
for the relative GW of normal crossings divisors passing through the neck of a
symplectic sum, one could also express the relative GW invariants of the
original pair $(X,V)$ as universal expressions in the relative GW invariants
of its the blow up and those of the piece obtained from the normal bundle of
the blow-up locus.

We study some of the properties of (symplectic) normal crossings divisors in
more detail in the Appendix. This is also where we include a more detailed
description of the stratifications of the divisor $V$ which record how the
various local branches of $V$ intersect. In particular, each stratum is itself
a normal crossings divisor in the resolution of the next one, and we describe
its normalization. These provide a global way to encode the local information
about the way $J$-holomorphic curves meet various branches of $V$.

##  2 The Relative Moduli Space ${\cal M}_{s}(X,V)$

Assume now $(X,\omega,J)$ is a smooth symplectic manifold with a normal
crossings divisor $V$. We want to define the moduli space of stable
(perturbed) $J$-holomorphic maps into $X$ relative $V$, which will enter in
the definition of the relative GW invariant of the pair $(X,V)$ in the usual
way. We will follow closely the approach of [IP1] and [IP2] and explain how
many of the arguments there extend in this context.

The notion of stability always refers to finitely many automorphisms, but the
notion of automorphism is relative, that is it depends on the particular
setup. We will later on explain what exactly we mean by a stable map
$f:C\rightarrow X$ relative $V$; but any automorphism of the map will in
particular be an automorphism of its domain $C$.

###### Remark 2.1

(Stability and automorphisms) For the purpose of simplifying the discussion in
this paper, in all the local analysis arguments below we will implicitely
assume that all the domains $C$ are already stable, and have been furthermore
decorated to have non nontrivial automorphisms, as is discussed in Section 1
of [IP1]. First of all, any unstable domain components of $C$ are spheres, and
collapsing them gives an element $\mbox{\rm st}(C)$ of the Deligne-Mumford
moduli space $\overline{{\cal M}}_{g,n}$ of stable curves. If we assume to
begin with that we are in a situation where all domains are stable, then after
possibly decorating the domains by adding a finite amount of extra topological
information (like a Prym structure) we can also assume they now have no
nontrivial automorphisms; in particular, their moduli space $\overline{{\cal
M}}$ is smooth. In genus zero it is already the case that any stable domain
has no nontrivial automorphisms, and the Deligne-Mumford moduli space
$\overline{{\cal M}}_{0,n}$ is smooth; in higher genus we can replace
$\overline{{\cal M}}_{g,n}$ by its smooth finite cover, the moduli space of
Prym curves constructed by Looijenga in [Lo]. This will have the effect of
globally killing all nontrivial automorphisms of stable domains, thus making
analysis arguments like transversality much easier to setup, at the expense of
passing to a finite branched cover of the original moduli space.

Fix also a particular embedding

$\displaystyle\overline{{\cal U}}\rightarrow{\mathbb{P}}^{N}$ (2.1)

of the universal curve $\overline{{\cal U}}$ over the Deligne-Mumford moduli
space in genus zero or over the Prym moduli space in higher genus. The
embedding (2.1) gives a canonical choice of a complex structure $j$ on each
domain $C$ obtained by restricting the complex structure on ${\mathbb{P}}^{N}$
to the fiber $st(C)$ of the universal curve; the unstable domain components,
if any, already have a canonical $j_{0}$ on them. So the embedding (2.1)
provides a global slice to the action of the reparametrization group on the
moduli space of maps, as long as their domains are stable; otherwise there is
still a residual action coming from automorphisms of the unstable components.
This embedding also simultaneously gives us a simple type of global
perturbation $\nu$ of the holomorphic map equation

$\displaystyle\overline{\partial}_{jJ}f(z)=\nu(z,f(z))$ (2.2)

coming from $\overline{{\cal U}}\times X$ as it sits as a smooth submanifold
inside ${\mathbb{P}}^{N}\times X$. The perturbation $\nu$ then vanishes on all
the unstable components of the domain, so these are $J$-holomorphic. In the
case all the domains are stable to begin with, or more generally when the
restriction of $f$ to the unstable part of the domain is a simple
$J$-holomorphic map, this type of perturbation $\nu$ is enough to achieve all
the required transversality; in general there is still a problem achieving
transversality using this type of perturbation on the unstable part of the
domain that is multiply covered.

There are many ways to locally stabilize the domains to locally get oneself in
the situation described above; understanding how these local choices globally
patch together is at the heart of the construction of the virtual fundamental
cycle in GW theories, see Remark 1.9 of [IP1] and the references therein. For
the situation we discuss in this paper, we could for example add a Donaldson
divisor to $V$ as in Example 1.12; this will have the effect that all the maps
in the relatively stable map compactification will now have stable domains. It
is not a priori clear why one would get an invariant that way (independent of
the Donaldson divisor added), but that will be the topic of another upcoming
paper.

Furthermore, in this paper we will work only with $V$-compatible parameters
$(J,\nu)$ for the equation (2.2) as in [IP1]. So we will assume that $(J,\nu)$
satisfy the conditions of Definition 3.2 in [IP1] the normal direction to each
branch of $V$. These are conditions only on the 1-jet of $(J,\nu)$ along $V$,
and are used to show that the contact order to $V$ is well defined and that
generically the stable map compactification does not have real codimension one
boundary strata. See Remark A.3. in the Appendix for a discussion of the space
of $V$-compatible parameters $(J,\nu)$.

###  2.1 The relative moduli space and the contact information

The construction of the relative moduli space takes several stages; in this
section we describe the main piece

$\displaystyle{\cal M}_{s}(X,V)$

containing stable $(J,\nu)$-holomorphic maps $f:C\rightarrow X$ into $X$
without any components or nodes in $V$, such that all the points in
$f^{-1}(V)$ are marked, and come decorated by the sequence $s$ of
multiplicities recording the order of contact of $f$ to each local branch of
$V$.

There is a fair amount of discrete topological/combinatorial data that needs
to be kept track of in the definition of ${\cal M}_{s}$. We also discuss the
construction of the leading order section which keeps track of more
information than the ordinary evaluation map, and construct an enhanced
evaluation map using it. This information will then be crucial in the later
sections when we construct the relatively stable map compactification.

We start by explaining how the discussion in Section 4 of [IP1] extends to the
case when $V$ is a normal crossings divisor. Assume $f:C\rightarrow X$ is a
stable $(J,\nu)$ holomorphic map into $X$ that has no components or nodes in
$V$, such that all the points in $f^{-1}(V)$ are marked. Assume $x$ is one of
the marked points such that $f(x)$ belongs to the (open) stratum of $V$ where
$k$ local branches meet.

Choose local coordinates $z$ about the point $x$ in the domain, and locally
index the $k$ branches of $V$ meeting at $f(x)$ by some set $I$. See the
discussion in the Appendix on how to regard the indexing set $I$ in an
intrinsic manner. For each $i\in I$, choose also a local coordinate at $f(x)$
in the normal bundle to the branch labeled by $i$, see also (A.4.). Lemma 3.4
of [IP1] then implies that the normal component $f_{i}$ of the map $f$ around
$z=0$ has an expansion:

$\displaystyle f_{i}(z)=a_{i}z^{s_{i}}+O(|z|^{s_{i}})$ (2.3)

where $s_{i}$ is a positive integer and the leading coefficient $a_{i}\neq 0$.
The multiplicity $s_{i}$ is independent of the local coordinates used, and
records the order of contact of $f$ at $x$ to the $i$’s local branch of $V$.

Thus each point $x\in f^{-1}(V)$ comes with the following information:

  * (a)

a depth $k(x)\geq 1$ that records the codimension in $X$ of the open stratum
of $V$ containing $f(x)$;

  * (b)

an indexing set $I(x)$ of length $k(x)$ that keeps track of the local branches
of $V$ meeting at $f(x)$;

  * (c)

a sequence of positive multiplicities $s(x)=(s_{i}(x))$ indexed by $i\in
I(x)$, recording the order of contact of $f$ at $x$ to each of the $k(x)$
local branches of $V$.

We will think of the indexing set $I(x)$ as part of the information contained
in the sequence $s(x)$. There are several ways to encode this: one way is
regard $s(x)$ as a map $s(x):I(x)\rightarrow{\mathbb{N}}_{+}^{k(x)}$ defined
by $i\mapsto s_{i}(x)$, the multiplicity of contact at $x$ to the branch
indexed by $i\in I(x)$. Either way, the sequence $s(x)$ keeps track of how the
local intersection number of $f$ at $x$ with $V$ is partitioned into
intersection numbers to each local branch of $V$. The sequence $s$ is obtained
by putting together all the $s(x)$ for all $x\in f^{-1}(V)$ and keeps track of
all the contact information of $f$ to $V$.

###### Remark 2.2

To keep notation manageable we will also include in $s$ the information about
ordinary marked points (i.e. for which $f(x)\notin V$). For such points, we
make the convention that the depth $k=0$, $I=\emptyset$ and $s(x)=\emptyset$.
Then each marked point $x$ now has a depth $k\geq 0$: depth 0 corresponds to
an ordinary marked point, mapped into $X\setminus V$, depth 1 is mapped into
the smooth locus of $V$ and has attached just one positive multiplicity, etc.

This concludes our description of the moduli space

$\displaystyle{\cal M}_{s}(X,V)$

consisting of stable $(J,\nu)$-holomorphic maps $f:C\rightarrow X$ that have
no components or nodes in $V$ and such that all the points in the inverse
image of $V$ are marked points of $C$ and furthermore decorated according to
the sequence $s$. If $P$ denotes the collection of marked points of $C$, each
marked point $x\in P$ has an associated sequence $s(x)$ which records the
order of contact of the map $f$ at $x$ to each local branch of $V$, including
the information about the indexing set $I(x)$ of the branches. The cardinality
$k(x)=|I(x)|$ of $I(x)$ records the depth of $x$ while the degree of $s(x)$

$\displaystyle|s(x)|\;\mathop{=}\limits^{\rm def}\;\sum_{i\in I(x)}s_{i}(x)$

is the local intersection number of $f$ at $x$ with $V$. In particular, the
total degree of $s$ is purely topological:

$\displaystyle|s|=\sum_{x\in P}|s(x)|=f^{*}(V)=A\cdot V$ (2.4)

where $A\in H_{2}(X)$ is the homology class of the image of $f$. We also
denote by $\ell(s)$ the length of $s$, i.e. the total number of marked points
$x\in P$ of $C$.

###### Remark 2.3

To keep the notation simple, we will think of $s$ as recording ALL the
topological information about the stable maps $f:C\rightarrow X$; in
particular this includes the homology class  $A_{s}$ of the image of $f$ and
the topological type $\Sigma_{s}$ of the domain of $f$, and so its Euler
characteristic $\chi_{s}=\chi(\Sigma_{s})$. In the discussion above, the
domain $\Sigma_{s}$ of $f$ could be disconnected, in which case its components
are unordered. We can also include in $s$ not just the homology class of the
image of $f$, but also its appropriate relative homology class, i.e. the
information about the rim tori, see Section 5 of [IP1]. The construction there
is purely topological, so it easily extends to the case when $V$ is a normal
crossings divisor. We will not explicitly describe it in this paper.

With this notation, the arguments of Lemma 4.2 in [IP1] immediately extend in
this context to show that for generic $V$-compatible $(J,\nu)$ the moduli
space ${\cal M}_{s}(X,V)$ is a smooth orbifold of dimension

$\displaystyle{\rm dim\;}{\cal M}_{s}(X,V)=2c_{1}(TX)A_{s}+({\rm
dim}X-6)\frac{\chi(\Sigma_{s})}{2}+2\ell(s)-2A_{s}\cdot V$ (2.5)

Note that this dimension does not depend on the particular partition of the
intersection number of $V$ and $A$, only on the total number $\ell(s)$ of
marked points (which include ordinary marked points in our convention).

The ordinary evaluation map at one of the decorated marked points $x\in P$ is
defined by

$\displaystyle\mbox{\rm ev}_{x}:{\cal M}_{s}(X,V)\rightarrow V_{I(x)}$ (2.6)
$\displaystyle\mbox{\rm ev}_{x}(f)=f(x)$

where $V_{I(x)}$ is the depth $k(x)$ stratum of $V$ labeled by the indexing
set $I(x)$ of the branches of $V$. As before, this evaluation map includes the
ordinary marked points when we make the convention that $V_{\emptyset}=X$.
Putting together all the marked points $x\in P(s)$ labeled by the sequence $s$
we get the corresponding ordinary evaluation map

$\displaystyle\mbox{\rm ev}:{\cal M}_{s}(X,V)$ $\displaystyle\rightarrow$
$\displaystyle V_{s}=\mathop{\prod}\limits_{x\in P(s)}V_{I(x)}$ (2.7)

whose image is the product of all the strata corresponding to the marked
points. It is important to note the image of the evaluation map above: there
are several other choices that may seem possible (like simply $X^{\ell(s)}$),
but this is the only choice for which the evaluation map can be transverse,
without loosing important information.

###### Remark 2.4

In the above discussion for simplicity we identified the marked point $x$
which is an actual point in the domain of $f$ with its marking (index) in $P$;
usually one talks about the $p$’th marked point $x_{p}$ and about $\mbox{\rm
ev}_{p}$, the evaluation at the $p$’th marked point.

When the depth $k(x)\geq 2$, the evaluation map (2.7) still does not record
enough information to state the full matching conditions that appear in the
relatively stable map compactification. We will also need to record the
leading coefficient of the expansion (2.3). For each $f:C\rightarrow X$ in
${\cal M}_{s}(X,V)$ and each marked point $x\in f^{-1}(V)$ let $a_{i}(x)$ be
leading coefficient (2.3) of $f$ at $x$ in the normal direction $N_{i}$ to the
branch labeled by $i\in I(x)$. As explained in Section 5 of [IP2], $a_{i}(x)$
is naturally an element of $(N_{i})_{f(x)}\otimes(T_{x}^{*}C)^{s_{i}(x)}$ so
it defines a section of the bundle

$\displaystyle\mbox{\rm ev}_{x}^{*}N_{i}\otimes L_{x}^{s_{i}(x)}$ (2.8)

where $L_{x}$ is the relative cotangent bundle to the domain at the marked
point $x$. If we denote by $E_{s,x}$ the bundle over the moduli space whose
fiber at $f$ is

$\displaystyle\mathop{\bigoplus}\limits_{i\in I(x)}\mbox{\rm
ev}_{x}^{*}N_{i}\otimes L_{x}^{s_{i}(x)}$ (2.9)

then the leading order section at $x$ is defined by

$\displaystyle\sigma_{x}:{\cal M}_{s}(X,V)$ $\displaystyle\rightarrow$
$\displaystyle E_{s,x}$ (2.10) $\displaystyle\sigma_{x}(f)$ $\displaystyle=$
$\displaystyle(a_{i}(x))_{i\in I(x)}$

and records the leading term coefficient in each one of the $k(x)$ normal
directions at $x$ labeled by $I(x)$. This section will turn out to record a
lot of crucial information, and will be studied in more detailed later. It was
already used in [I] to essentially get an isomorphism between the relative
cotangent bundle of the domain and that of the target for the moduli space of
branched covers of ${\mathbb{P}}^{1}$.

###### Remark 2.5

The target of (2.10) may not be globally a direct sum of line bundles, as its
fibers may intertwine as we move along $V_{I(x)}$, the same way the $k(x)$
branches of $V$ do when the indexing set $I(x)$ has nontrivial global
monodromy, see discussion in the Appendix.

The enhanced evaluation map  is defined by

$\displaystyle\mbox{\rm Ev}_{x}:{\cal M}_{s}(X,V)$ $\displaystyle\rightarrow$
$\displaystyle{\mathbb{P}}_{s(x)}(N{V_{I(x)}})$ (2.11) $\displaystyle\mbox{\rm
Ev}_{x}(f)$ $\displaystyle=$ $\displaystyle[\sigma_{x}(f)]$

where the ${\mathbb{P}}_{s(x)}(N{V_{I(x)}})$ denotes the weighted
projectivization with weight $s(x)$ of the normal bundle $N{V_{I(x)}}$ of the
depth $k(x)$ stratum $V_{I(x)}$. More precisely,
${\mathbb{P}}_{s(x)}(N{V_{I(x)}})$ is a bundle over $V_{I(x)}$ whose fiber is
the weighted projective space obtained as the quotient by the
${\mathbb{C}}^{*}$ action with weight $s_{i}(x)$ in the normal direction
$N_{i}$ to the branch labeled by $i\in I(x)$. Globally these branches may
intertwine as discussed above in Remark 2.5.

Of course, if $\pi:{\mathbb{P}}_{s(x)}(N{V_{I(x)}})\rightarrow V_{I(x)}$ is
the projection then

$\displaystyle\pi\circ\mbox{\rm Ev}=\mbox{\rm ev}$

which explains the name; $\mbox{\rm Ev}_{x}$ is a refinement of $\mbox{\rm
ev}_{x}$ only when the depth $k(x)\geq 2$.

###### Remark 2.6

Note that by construction the leading order terms are all nonzero, so the
image of $\sigma_{x}$ is away from the zero sections of each term in (2.10).
The image of the enhanced evaluation map similarly lands in the complement of
all the coordinate hyperplanes in the target of (2.11).

So far we defined the moduli space ${\cal M}_{s}(X,V)$ of stable maps into $X$
without any components or nodes in $V$, and with fixed topological information
described by $s$. This moduli space ${\cal M}_{s}(X,V)$ may not be compact,
simply because we could have sequence of maps $f_{n}$ in ${\cal M}_{s}$ whose
limit in the stable maps compactification is a map $f$ with some components in
$V$. Then the contact information of $f$ to $V$ becomes undefined along its
components that lie in $V$, and so in particular $f$ does not belong to ${\cal
M}_{s}(X,V)$. Note that this is the only reason why ${\cal M}_{s}(X,V)$ fails
to be compact:

###### Lemma 2.7

Consider a sequence $\\{f_{n}\\}$ of maps in ${\cal M}_{s}(X,V)$ and assume
its limit $f$ in the usual stable map compactification has no components in
$V$. Then $f\in{\cal M}_{s}(X,V)$.

Proof. A priori, there are two reasons why $f$ would fail to be in ${\cal
M}_{s}$:

  1. (a)

$f$ has a node in $V$ or

  2. (b)

the contact information of $f$ is not given by $s$.

Note that case (b) includes the cases when in the limit multiplicity of
intersection jumps up or when a depth $k$ marked point falls into a higher
depth stratum of $V$.

Since ALL points in $f_{n}^{-1}(V)$ are already marked, indexed by the same
set $P$, they persist as marked points for the limit $f$, in particular they
are distinct from each other and from the nodes of $f$. On the other hand, let
$\widetilde{f}$ denote the lift of $f$ to the normalization of its domain.
Since $f$ has no components in $V$, then each point in $\widetilde{f}^{-1}(V)$
has a well defined sequence $s_{0}$ that records the local multiplicity of
intersection of $\widetilde{f}$ at that point with each local branch of $V$.
At those points of $\widetilde{f}^{-1}(V)$ which were limits of the marked
points in $f_{n}^{-1}(V)$, the multiplicity $s_{0}(x)\geq s(x)$, as the
mutiplicity could go up when the leading coefficients converge to 0. But then

$\displaystyle[f]\cdot
V=\sum_{x\in\widetilde{f}^{-1}(V)}s_{0}(x)\geq\sum_{x\in
P}s_{0}(x)\geq\sum_{x\in P}s(x)=[f_{n}]\cdot V$

Since $[f_{n}]=[f]$ then $\widetilde{f}^{-1}(V)=P$, which means that $f$ has
no nodes in $V$, rulling out (a), and that $s_{0}(x)=s(x)$ for all $x\in P$,
which rules out (b). $\Box$

##  3 Rescaling the target

Assume now $\\{f_{n}\\}$ is a sequence of maps in ${\cal M}_{s}(X,V)$ whose
limit $f$ in the stable maps compactification has some components in $V$. We
will use the methods of [IP1], to rescale the target normal to $V$ to prevent
this from happening. The analysis there is mostly done semi-locally (in a
neighborhood of $V$) and so those parts easily extend to this situation. But
as we will see below, the topology of the normal crossings divisor now enters
crucially in a couple of steps. First step is to describe the effect of
rescaling on the target $X$. It is modeled on the process of rescaling a disk
about the origin, but now performed fiberwise in the normal direction to $V$.

###  3.1 Brief review of [IP1]

We begin by briefly reviewing the situation in Section 6 of [IP1], where it
was assumed that $V$ is a smooth divisor. In local coordinates, if $x$ is a
fixed local coordinate normal to $V$, rescaling $X$ by a factor of
$\lambda\neq 0$ means we make a change of coordinates in a neighborhood of $V$
in the normal direction to $V$:

$\displaystyle x_{\lambda}=x/\lambda.$ (3.1)

Under rescaling by an appropriate amount $\lambda_{n}$, depending on the
sequence $\\{f_{n}\\}$, in the limit we will get not just a curve in $X$
(equal to the part of $f$ that did not lie in $V$), but also a curve in the
compactification of $N_{V}$, i.e. in

$\displaystyle\mathbb{F}={\mathbb{P}}(N_{V}\oplus{\mathbb{C}}).$

Here $\mathbb{F}$ is a ${\mathbb{C}}{\mathbb{P}}^{1}$ bundle over $V$, with a
zero and infinity section $V_{0}$ and $V_{\infty}$. Under the rescaling (3.1),
$x_{\lambda}$ can be thought instead as a coordinate on $\mathbb{F}$ normal to
$V_{0}$. Let $y=1/x_{\lambda}$ be the corresponding coordinate normal to
$V_{\infty}$ inside $\mathbb{F}$, so that (3.1) becomes

$\displaystyle xy=\lambda$ (3.2)

This procedure has the infinity section $V_{\infty}$ of $\mathbb{F}$ naturally
identified with $V$ in $X$ such that furthermore their normal bundles are dual
to each other, i.e.

$\displaystyle N_{V/X}\otimes N_{V_{\infty}/\mathbb{F}}\cong{\mathbb{C}}$
(3.3)

is trivial. This identification globally encodes the local equations (3.2)
because $x,y$ are local sections of $N_{V/X}$ and $N_{V_{\infty}/\mathbb{F}}$
respectively.

###### Remark 3.1

Once an identification (3.3) is fixed, then for any (small) gluing parameter
$\lambda\in{\mathbb{C}}^{*}$, equation (3.2) is exactly the local model of the
symplectic sum $X_{\lambda}$ of $X$ and $\mathbb{F}$ along $V=V_{\infty}$ with
gluing parameter $\lambda$. Of course, topologically
$X\\#_{V=V_{\infty}}\mathbb{F}=X$. This means that an equivalent point of view
to rescaling $X$ by a factor of $\lambda$ normal to $V$ is to regard $X$ as
the symplectic sum $X_{\lambda}$ of $X$ and $\mathbb{F}$ with gluing parameter
$\lambda$, with the above choice of coordinates and identifications, including
(3.3). The advantage of this perspective is that the rescaled manifolds
$X_{\lambda}$ now fit together as part of a smooth total space $Z$ as its
fibers over $\lambda$, and converge there as $\lambda\rightarrow 0$ to the
singular space

$\displaystyle X_{0}=X\mathop{\cup}\limits_{V=V_{\infty}}\mathbb{F},$ (3.4)

obtained by joining $X$ to $\mathbb{F}$ along $V=V_{\infty}$.

Denote by $U_{\lambda}$ the tubular neighborhood of $V$ in $X$ described in
coordinates by $|x|\leq|\lambda|^{1/2}$ and by $V_{\lambda}$ the complement of
the tubular neighborhood of $V_{\infty}$ in $\mathbb{F}$ described in
coordinates by $|y|\geq|\lambda|^{1/2}$. From this perspective, rescaling $X$
around $V$ by $\lambda\neq 0$ gives rise to a manifold $X_{\lambda}$ together
with a diffeomorphism

$\displaystyle R_{\lambda}:X\rightarrow X_{\lambda}$ (3.5)

which is the identity outside $U_{\lambda}$ and which identifies $U_{\lambda}$
with $V_{\lambda}$ by rescaling it by a factor of $\lambda$, or equivalently
via the equation (3.2). As $\lambda\rightarrow 0$, $U_{\lambda}$ shrinks to
$V$ inside $X$, but it expands in the rescaled version $X_{\lambda}$ to
$\mathbb{F}\setminus F_{\infty}$. So in the limit as $\lambda\rightarrow 0$,
the rescaled manifolds $X_{\lambda}$, with the induced almost complex
structures $J_{\lambda}=(R_{\lambda}^{-1})^{*}J$ converge to the singular
space $X_{0}$ defined by (3.4) with an almost complex structure $J_{0}$ which
agrees with $J$ on $X$ and is ${\mathbb{C}}^{*}$ invariant on the $\mathbb{F}$
piece.

After rescaling the sequence $\\{f_{n}\\}$ by appropriate amounts
$\lambda_{n}$, the new sequence $R_{\lambda_{n}}(f_{n})$ has a limit inside
$Z$ which is now a map into $X_{0}$ satisfying a matching condition along
$V_{\infty}=V$, described in more details later on. Of course, in general,
different components may fall in at different rates, so we need to rescale
several (but finitely many) times to catch all of them, and in the limit we
get a map into a building with several levels.

###  3.2 Rescaling the manifold $X$ normal to $V$

Assume now $V$ is a normal crossings divisor. We next describe the effect on
the manifold $X$ of rescaling around $V$ (in a tubular neighborhood of $V$).
Using our local models, we could extend the discussion above independently in
each normal direction to $V$, so normal to each open stratum of ${V^{k}}$, we
could rescale in $k$ independent directions. However, globally these
directions may intertwine, and not be independent, so one has to be careful
how to globally patch these local pictures. Here is where we use the fact that
the normal bundle $N$ was defined over the normalization $\widetilde{V}$ of
$V$. We will rescale normal to $V$ using the ${\mathbb{C}}^{*}$ action in this
normal bundle.

###### Remark 3.2

The ${\mathbb{C}}^{*}$ action in the complex line bundle $N$ induces in fact
several different actions. The one we will use in this paper is the diagonal
${\mathbb{C}}^{*}$ action in the normal bundle to each stratum $V^{k}$ of $V$.
When the normalization of $V$ has several components we have a
${\mathbb{C}}^{*}$ action for each component. In particular, when the divisor
$V$ has simple crossings, then we also have a local $({\mathbb{C}}^{*})^{k}$
action on $X$ normal to each $V^{k}$; essentially this happens only when the
crossings are simple.

When we rescale once $X$ normal to the normal crossings divisor $V$, in level
one we get several pieces, one for each piece $V^{k}$ of the stratification of
$V$ according to how many local branches meet there. The level zero unrescaled
piece is still $(X,V)$ as before. But now level one

$\displaystyle\mathbb{F}_{V}=\mathop{\sqcup}\limits_{k\geq 1}\mathbb{F}_{k}$
(3.6)

consists of several pieces $\mathbb{F}_{k}$, one for each depth $k\geq 1$. The
first piece is

$\displaystyle\mathbb{F}_{1}={\mathbb{P}}(N_{V}\oplus{\mathbb{C}}),$

a ${\mathbb{P}}^{1}$ bundle over the $\widetilde{V}$, the normalization of
$V$, obtained by compactifying the normal bundle
$N_{V}\rightarrow\widetilde{V}$ by adding an infinity section. Similarly, the
$k$’th piece

$\displaystyle\mathbb{F}_{k}\longrightarrow\widetilde{V^{k}}$

is a $({\mathbb{P}}^{1})^{k}$ bundle over the normalization
$\widetilde{V^{k}}$ of the closed stratum $V^{k}$, described in more details
in the Appendix. What is important here is that $\mathbb{F}_{k}$ is a bundle
over a smooth manifold $\widetilde{V^{k}}$ which it is obtained by separately
compactifying each of the $k$ normal directions to $V$ along $V^{k}$, see
(A.4.). This means that its fiber at a point $p\in\widetilde{V^{k}}$ is

$\displaystyle\mathbb{F}_{k}=\mathop{\times}\limits_{i\in
I}{\mathbb{P}}(N_{i}\oplus{\mathbb{C}})$

where $N_{i}$ is the normal direction to the $i$’th branch, and $I$ is an
indexing set of the $k$ local branches of $V$ meeting at $p$. Globally, these
${\mathbb{P}}^{1}$ factors intertwine as dictated by the global monodromy of
the $k$ local branches of $V$.

Each piece $\mathbb{F}_{k}$ comes with a natural normal crossings divisor

$\displaystyle W^{k}=D_{k,\infty}\cup D_{k,0}\cup F_{k}$ (3.7)

obtained by considering together its zero and infinity divisors plus the fiber
$F_{k}$ over the (inverse image of the) higher depth strata of $V^{k}$. The
construction of the divisor $W^{k}$ is described in more details the Appendix,
but here let us just mention that $D_{k,0}$ is the zero divisor in
$\mathbb{F}_{k}$ where at least one of the ${\mathbb{P}}^{1}$ coordinates is
equal to $0$, while the fiber divisor $F_{k}$ is the restriction of
$\mathbb{F}_{k}$ to the stratum of $\widetilde{V^{k}}$ coming from the higher
depth stratum $V^{k+1}$.

###### Remark 3.3

The ${\mathbb{C}}^{*}$ action in the normal bundle to $V$ induces a fiberwise,
diagonal ${\mathbb{C}}^{*}$ action on each piece $\mathbb{F}_{k}$; the action
preserves the divisor $W^{k}$, but not pointwise.

To keep notation manageable, we make the convention

$\displaystyle(\mathbb{F}_{0},W^{0})\mathop{=}\limits^{def}(X,V)$ (3.8)

This is consistent with our previous conventions that $V^{0}=X$, in which case
$\mathbb{F}_{0}={\mathbb{P}}(N_{V^{0}/X}\oplus{\mathbb{C}})=X$ and
$W_{0}=F_{0}=V$ is the fiber divisor over the lower stratum, as the zero and
infinity divisors are empty in this case. However, one difference that this
notation obscures is the fact that while $X=\mathbb{F}_{0}$ is on level 0
(unrescaled), the rest of the pieces $\mathbb{F}_{k}$ for $k\geq 1$ are all on
level 1 (all appeared as the result of rescaling once normal to $V$).

###### Definition 3.4

A level one building $X_{1}$ with zero divisor $V_{1}$ is obtained by
identifying the fiber divisor $F_{k}$ of $\mathbb{F}_{k}$ with the infinity
divisor $D_{k+1,\infty}$ of $\mathbb{F}_{k+1}$:

$\displaystyle
X_{1}=\mathop{\bigcup}\limits_{F_{k}=D_{k+1,\infty}}\mathbb{F}_{k}$

Denote by $W_{1}$ the singular locus of $X_{1}$ where all the pieces are
attached to each other, by $V_{1}$ the (singular) divisor in $X_{1}$ obtained
from the zero divisors, and let $D_{1}=W_{1}\cup V_{1}$ be the total divisor:

$\displaystyle(X_{1},D_{1})=\mathop{\bigcup}\limits_{F_{k}=D_{k+1,\infty}}(\mathbb{F}_{k},\;F_{k}\cup
D_{k,\infty}\cup D_{k,0})$ (3.9)

So the level one building comes with a resolution
$(\widetilde{X}_{1},\widetilde{D}_{1})=\mathop{\bigsqcup}\limits_{k\geq
0}(\mathbb{F}_{k},\;F_{k}\cup D_{k,\infty}\cup D_{k,0})$ and an attaching map

$\displaystyle\xi:(\widetilde{X}_{1},\widetilde{V}_{1})\rightarrow(X_{1},V_{1}).$
(3.10)

It also comes with a collapsing map to level zero $(X_{0},V_{0})=(X,V)$:

$\displaystyle p:(X_{1},V_{1})\rightarrow(X,V)$ (3.11)

which is identity on level 0, but which collapses the fiber of each level one
piece $\mathbb{F}_{k}$, $k\geq 1$.

###### Remark 3.5

The precise identifications required to construct this building are also
described in the Appendix. It is easy to see that both $F_{k}$ and
$D_{k+1,\infty}$ are normal crossings divisors, so in fact what we identify is
their normalizations, via a canonical map (A.6.). Furthermore, their normal
bundles are canonically dual to each other, see (A.7.), and the
${\mathbb{C}}^{*}$ action on $N$ induces an anti-diagonal ${\mathbb{C}}^{*}$
action in the normal bundle

$\displaystyle N_{F_{k}}\oplus N_{D_{k+1,\infty}}$

of each component $F_{k}=D_{k+1,\infty}$ of the singular divisor $W_{1}$ of
$X_{1}$, where $k\geq 0$.

###### Example 3.6

Assume $X$ has 4 real dimensions, and that the normal crossings divisor

$\displaystyle V=V_{1}\mathop{\cup}\limits_{p_{1}=p_{2}}V_{2}$

is the union of two submanifolds $V_{1}$ and $V_{2}$ intersecting only in a
point $p=p_{1}=p_{2}$. After rescaling once, we get 3 main pieces $X$,
$\mathbb{F}_{1}$ and $\mathbb{F}_{2}$ together this an attaching map, see the
left hand side of Figure 1. Here $\mathbb{F}_{1}$ is a ${\mathbb{P}}^{1}$
bundle over $\widetilde{V^{1}}=V_{1}\sqcup V_{2}$, while $\mathbb{F}_{2}$ is
just ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ (over the point
$\widetilde{V}^{2}=p$). The divisor $D_{1,\infty}\subset\mathbb{F}_{1}$ is a
copy of $\widetilde{V}=V_{1}\sqcup V_{2}$ and it is attached to
$F_{0}=V=V_{1}\mathop{\cup}\limits_{p_{1}=p_{2}}V_{2}\subset X$. Similarly,
$D_{2,\infty}\subset\mathbb{F}_{2}$ is
${\mathbb{P}}^{1}\times\infty\cup\infty\times{\mathbb{P}}^{1}$, and it is
attached to $F_{1}$, which is the disjoint union of two fibers of
$\mathbb{F}_{1}$ over the points $p_{1}$ and $p_{2}$ in $V_{1}\sqcup V_{2}$.
Note that $\mathbb{F}_{1}$ does not descend as a bundle over $V$: the two
fibers of $\mathbb{F}_{1}$ over singular locus $p_{1}$ and $p_{2}$ are not
identified with each other, but rather each gets identified with different of
fibers of $\mathbb{F}_{2}$.

Figure 1: The pieces of a level 1 building

###### Example 3.7

Assume $X$ has 4 real dimensions, but the normal crossings divisor $V$ has
only one component, self intersecting itself in just a point $p$. Locally, the
situation looks just like the one in Example 3.6, with $V$ having two local
branches meeting at $p$. The only difference is that globally now
$\widetilde{V}$ has only one connected component containing both points
$p_{1}$ and $p_{2}$, see right hand side of Figure 1.

###### Remark 3.8

In the discussion above, we had a rescaling parameter $\lambda$ normal to $V$,
which means that we considered the action of $\lambda\in{\mathbb{C}}^{*}$ on
the normal bundle $N$ over $\widetilde{V}$. If $\widetilde{V}$ has several
connected components

$\displaystyle\widetilde{V}=\mathop{\sqcup}\limits_{c\in C}\widetilde{V}_{c}$
(3.12)

then we could independently rescale normal to each one of them; this gives a
$\lambda\in({\mathbb{C}}^{*})^{C}$ action, rather then just the diagonal one
we considered before. Rescalling in all these independent directions now gives
a multi-building, where each floor has a level associated to each connected
component of $\widetilde{V}$. We could talk about a floor which is on level
one normal to some of the components, but level zero normal to other
components.

By iterating the rescaling process, we obtain level $m$ buildings where we
rescale $m$ times normal to $V$, or more generally multi-buildings with
$m_{c}$ levels in the normal direction to each connected component
$\widetilde{V}_{c}$ of $\widetilde{V}$.

###### Definition 3.9

A level $m$ building is a singular space $X_{m}$ with a singular divisor
$V_{m}$, called the zero section, that is obtained recursively from
$(X_{m-1},V_{m-1})$ by iterating the level one building procedure. In
particular, a level $m$ building comes with a resolution
$(\widetilde{X}_{m},\widetilde{D}_{m})$ and an attaching map
$\xi:(\widetilde{X}_{m},\widetilde{D}_{m})\rightarrow(X_{m},D_{m})$ that
attaches all the floors together producing the singular locus $W_{m}$ of
$X_{m}$, where $D_{m}=W_{m}\cup V_{m}$ is the total divisor. It also comes
with a collapsing map

$\displaystyle p_{m}:(X_{m},V_{m})\rightarrow(X_{m-1},V_{m-1})$

that collapses the level $m$ floor or more generally a collapsing map that
collapses any subset $J$ of the levels $\\{1,\dots,m\\}$.

For example,

$\displaystyle p:(X_{m},V_{m})\rightarrow(X_{0},V_{0})$ (3.13)

collapses all positive levels (but leaves the level zero unaffected).

Note that as we add floors, the building grows bigger in several (local)
directions. Starting with $(X_{m-1},V_{m-1})$, we then construct depth $k$
pieces

$\displaystyle\mathbb{F}_{k,m}=\mathbb{F}_{k}(V_{m-1}).$

on level $m$. Since the divisor $V_{m-1}$ has several pieces (it is itself a
rescaled version of $V_{0}$), the number of depth $k\geq 2$ pieces increases
(at least locally) as the building grows new levels.

###### Remark 3.10

Note that we used depth to measures how many local branches of $V$ meet at a
point. On the other hand, as a result of rescaling, we now also get levels,
which measure how many times the target has been rescaled.

###### Example 3.11

Consider the simplest model of a level $m$ building, the $m$ times rescaled
disk

$\displaystyle(D^{2})_{m}=D^{2}\mathop{\cup}\limits_{0=\infty}{\mathbb{P}}^{1}\mathop{\cup}\limits_{0=\infty}\dots\mathop{\cup}\limits_{0=\infty}{\mathbb{P}}^{1}$
(3.14)

on which in some sense all the other level $m$ buildings are modeled on. Its
resolution has $m+1$ components, each indexed by a level $l=0,\dots,m$ with
$D^{2}$ on level zero (unrescaled). The total divisor also has $m+1$
components, each similarly labeled by a level $l=0,\dots,m$ with the $0\in
D^{2}$ on level one. This defines a lower-semicontinuous level map $l$ on
$(D^{2})_{m}$. It is discontinuous precisely at points $y$ in the singular
divisor, where it has two limits, indexing the two lifts $y_{\pm}$ of $y$ to
consecutive levels, where $y_{+}=0$ is on the same level as $y$ while
$y_{-}=\infty$ is on the next level. Each point on the resolution comes with a
sign $\varepsilon=\pm$ or 0, keeping track of whether its coordinate is
$\infty$, 0 or neither. Then $\varepsilon=0$ corresponds to a smooth point of
$(D^{2})_{m}$, while $\varepsilon=\pm$ corresponds to a point in the
resolution of the total divisor. Intrinsically, $\varepsilon$ keeps track of
the weight of the ${\mathbb{C}}^{*}$ action on each piece.

###### Example 3.12

Suppose next we are in the situation of Example 3.6. The first level had three
components: two are ${\mathbb{P}}^{1}$ bundles over $V_{1}$ and respectively
$V_{2}$ and another is a ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ over a
point. A second level would have five components. Two of them are still
${\mathbb{P}}^{1}$ bundles over $V_{1}$ and $V_{2}$ respectively, but there
are now three copies of ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. The way they
come about is as follows. The zero divisor $V_{(1)}$ of the first floor
consists now of 4 pieces: $V_{1}$, $V_{2}$ but also two ${\mathbb{P}}^{1}$’s
intersecting in a point $p^{1}$ (coming from the zero section of depth two
piece $\mathbb{F}_{2,1}$ on the first floor). When we rescale again to get the
second floor, $\mathbb{F}_{2,2}$ still a
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ over the point $p^{1}$, but
$\mathbb{F}_{1,2}$ is now larger, it is a ${\mathbb{P}}^{1}$ bundle over
$(V_{1}\sqcup{\mathbb{P}}^{1})\sqcup(V_{2}\sqcup{\mathbb{P}}^{1})$, so it
really has four pieces: a ${\mathbb{P}}^{1}$ bundle over $V_{1}$ and
respectively $V_{2}$, as was the case in level one, but also two other
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ pieces coming from rescaling over the
two ${\mathbb{P}}^{1}$ fibers in $V_{(1)}$. All together this level 2 building
has $4$ copies of ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$; in general, such
level $m$ building will have $m^{2}$ copies of
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$, and $m$ copies of the
${\mathbb{P}}^{1}$ bundle over $V_{1}\sqcup V_{2}$.

Figure 2: A level 2 building

###### Example 3.13

Consider again the situation in Example 3.6. Because
$\widetilde{V}=V_{1}\sqcup V_{2}$ has two connected components, we can also
independently rescale normal to each of them, getting instead a multi-
building. For example, a level (2,2) multibuilding in this case looks just
like the level $2$ building described above, except that the pieces might be
on different levels when regarded in different directions. For example, the 4
pieces ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ now land one on each level
$(i,\;j)$ for $i,\;j=1$ or 2, while before one of them (the one on level
(1,1)) landed on level 1, while the other three landed on level 2. More
generally, the level $m$ building from the example above can be regarded as a
level $(m,m)$ multibuilding when we rescale independently in the two
directions, with exactly one copy of ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$
on each level $(i,\;j)$ for $i,\;j=1,\dots m$. But in this case we can also
have a multi-building with different number of levels in each direction, for
example we could have just one level normal to $V_{1}$ but three normal to
$V_{2}$.

###### Example 3.14

Consider now the situation of Example 3.7. Locally everything looks the same
as in Example 3.12 and even near $p$ we have two independent local directions
in which we could rescale as in Example 3.13. However, because $\widetilde{V}$
is now connected, globally there is only one scaling parameter
$\lambda\in{\mathbb{C}}^{*}$ normal to $V$, so the two local scaling
parameters at $p$ have to be related to each other (they are essentially
equal).

###### Remark 3.15

In general, for any level $m$ building $X_{m}$, even if globally we may not be
able to have separate directions of rescaling, at least semi-locally the
picture always looks like part of a multilevel building. More precisely, fix a
depth $k$ point $p$ in $V$, and locally index by $i\in I$ the $k$ branches of
$V$ coming together at $p$. A neighborhood $U_{p}$ of fiber $F_{p}$ over $p$
of the collapsing map $X_{m}\rightarrow X$ of (3.13) is a product of a small
neighborhood $O_{p}$ of $p$ in the depth $k$ stratum and $k$ copies of an
$m$-times rescaled disk $(D^{2})_{m}$ of (3.14), one factor for each one of
the $k$ branches of $V$ at $p$. This describes not only the tower of
$(m+1)^{k}$ pieces of the resolution $\widetilde{X}_{m}$ with its total
divisor $\widetilde{D}_{m}$, but also their attaching map, just as in Example
3.11, except that now we have $k$ directions to keep track of instead of just
one.

In particular, each point $y\in X_{m}$ (which projects to $p$ under the
collapsing map $X_{m}\rightarrow X$) comes with a multi-level map

$\displaystyle l:I\rightarrow\\{0,\dots,m\\}$ (3.15)

as if it was part of a multilevel building (with $k$ independent directions),
where we would separately keep track of the level $l_{i}$ for each one of the
directions $i\in I$. Of course, as part of $X_{m}$ this piece is on level
$l=\mathop{\max}\limits_{i}l_{i}$, but it appears as part of each one of the
local levels $l_{1},\dots,l_{k}$. Furthermore, a point $\widetilde{y}$ in the
resolution $\widetilde{X}_{m}$ comes not only with the multi-level map (3.15)
of its image in $X_{m}$, but also a multi-sign map

$\displaystyle\varepsilon:I\rightarrow\\{0,\pm 1\\}$ (3.16)

that keeps track of whether it is equal to zero, infinity or neither in the
$i$’th direction. This allows us for example to keep track of the various
branches of the total divisor, and the fact that they come in dual pairs
indexed by opposite multi-sign maps $\varepsilon$; note that in general we can
have a point which is in the zero divisor in some of the directions, on
infinity divisor in other directions, and then in some other directions on
neither, and this is precisely what $\varepsilon$ records.

Equivalently, over a depth $k$ point $p$ of $V$, we can instead index each
local piece $P$ of the resolution $\widetilde{X}_{m}$ by a

partition of $I$ into $I_{0},\dots,I_{m}$. (3.17)

The local piece $P(I_{0},\dots,I_{m})$ is on level $l$ with respect to the
branches indexed by $I_{l}$, and thus can be thought as obtained by rescaling
$l$ times $X$ near $p$ in the $I_{l}$ directions; for example, when $I_{0}=I$,
that piece is completely unrescaled and corresponds to a piece of the level
zero $X$. More generally, for any local piece indexed by such a partition, the
directions in $I_{0}$ are unrescaled, while the remaining $j$ directions are
rescaled (at least once). Then

$I_{0}$ determines a unique lift $p_{0}$ of the point $p$ to the resolution
$\widetilde{V^{j}}$ (3.18)

of the depth $j$ stratum of $V$ while $I_{1},\dots,I_{m}$ index the directions
of the

fibers
$({\mathbb{P}}^{1})^{j_{1}}\times\dots\times({\mathbb{P}}^{1})^{j_{m}}=({\mathbb{P}}^{1})^{j}$
of $\mathbb{F}_{j}$ over $\widetilde{V^{j}}$. (3.19)

This identifies the local piece $P(I_{0},\dots,I_{m})$ with a neighborhood in
$\mathbb{F}_{j}$ of the fiber over $p_{0}$. Keeping track of which coordinates
in the fiber of $\mathbb{F}_{j}$ are infinity, zero or neither (which induces
a further partition of $I$ into $I^{\pm}$ and $I^{0}$) then allows us to index
any open stratum of the total divisor $\widetilde{D}_{m}$ in
$\widetilde{X}_{m}$ over the point $p$ by a

partition of $I$ into $I_{j}^{\pm},I_{j}^{0}$ for $j=0,\dots,m$. (3.20)

Then $I^{+}$ records the branches of the zero and fiber divisor, while $I^{-}$
those of the infinity divisor.

###### Remark 3.16

The ${\mathbb{C}}^{*}$ action of $N_{V}$ induces a $({\mathbb{C}}^{*})^{m}$
action on a level $m$ building such that each factor
$\alpha_{l}\in{\mathbb{C}}^{*}$ rescales the level $l\geq 1$ piece of $X_{m}$
normal to its zero divisor, fixes the level zero pointwise, and preserves the
total divisor and its stratification, but not pointwise. As before, this is
modeled by the $({\mathbb{C}}^{*})^{m}$ action on the rescaled disk
$(D^{2})_{m}$ in which each factor $\alpha_{l}\in{\mathbb{C}}^{*}$ acts on the
${\mathbb{P}}^{1}$ component of $(D^{2})_{m}$ in level $l\geq 1$, but now
extended as the diagonal action to the product of the disks in each normal
direction to $V$. More precisely, each component of the resolution $X_{m}$
whose (local) multi-levels are $l_{1},\dots,l_{k}$, is acted by
$\alpha_{l_{i}}$ on its ${\mathbb{P}}^{1}$ factor in direction $i$ for each
$l_{i}\geq 1$.

###  3.3 Local model near the divisor

Rescaling $X$ by a factor of $\lambda\neq 0$ along a normal crossings divisor
$V$ similarly gives rise to manifold we denote $X_{\lambda}$ and an
identification

$\displaystyle R_{\lambda}:X\rightarrow X_{\lambda}$ (3.21)

exactly as in (3.5). Just as described in §3.1, the manifold $X_{\lambda}$ has
two regions, one is the complement of the $|\sqrt{\lambda}|$-tubular
neighborhood $U_{\lambda}$ of $V$ in $X$, on which $R_{\lambda}$ is the
identity, and the other one is identified with $O_{\lambda}$, the complement
of the $|\sqrt{\lambda}|$-tubular neighborhood of the singular divisor $W_{1}$
in $\mathbb{F}_{V}$. The only difference now is that we have several
overlapping local models, coming from the stratification of $V$.

Still, this perspective allows us to think of the rescaled $X$ as a sequence
of manifolds $X_{\lambda}$ with varying $J_{\lambda}=R_{\lambda}^{*}J$, which
as $\lambda\rightarrow 0$ converge to a level one building $X_{1}$. In fact,
$X_{\lambda}$ can be thought as some sort of iterated symplectic sum: fix a
level one building $X_{1}$ as in Definition 3.4, with appropriate
identifications along corresponding divisors, including fixed isomorphisms
(A.7.). For each (small) gluing parameter $\lambda\in{\mathbb{C}}^{*}$ we get
the ’symplectic sum’ $X_{\lambda}$ (diffeomorphic to $X$, but with a deformed
symplectic form) and such that $X_{\lambda}$ converges to $X_{1}$ as
$\lambda\rightarrow 0$. This approach is mentioned in Remark 7.7 of [IP1] and
expanded on in [IP2]; it turns out for this paper to be more convenient than
the approach of section 6 of [IP1].

We next describe the rescaling procedure in more detail. We will work in
regions which are obtained from neighborhoods of depth $k$ strata of $V$ after
removing neighborhoods of the higher depth stratum, where we have nice local
models. Denote by $U_{\delta}$ the $\delta$-tubular neighborhood of $V$ in
$X$, that is the image under $\iota$ of $\widetilde{U}_{\delta}$, the
$\delta$-disk bundle in the normal bundle model
$N_{V}\rightarrow\widetilde{V}$ over the resolution of $V$, and by

$\displaystyle A(r,R)=\iota(\widetilde{U}_{R}\setminus\widetilde{U}_{r})$
(3.22)

the union of annular regions about $V$ in $X$, which will give rise to the
necks. Note that the region $A$ still intersects $V$ around depth $k\geq 2$
points, so it is very different from the region $U_{R}\setminus U_{r}$, an
important distinction that will become relevant later in the paper.

The neighborhood $U_{\delta}$ has subregions $U_{\delta}^{k}$ corresponding to
a product neighborhood of the depth $k$ stratum $V^{k}$, over which $\iota$ is
$k$-to-1. Each region $U_{\delta}^{k}$ can be then regarded as a subset of
$\mathbb{F}_{k}$, where it is identified, after fiberwise rescaling by
$\lambda$ with the complement of a neighborhood of the divisor $F_{k}\cup
D_{k,\infty}$ in $\mathbb{F}_{k}$ to obtain $X_{\lambda}$.

More precisely, assume $p$ is a depth $k$ point of $V$ (but away from the
higher depth strata); locally index the $k$ branches $I$ of $V$ coming
together at $p$, and choose local coordinates $u_{1},\dots,u_{k}$ normal to
each one of these branches (so in particular the $j$’th branch is given by
$u_{j}=0$) such that the $\delta$-tubular neighborhood $U_{\delta}$ of $V$ is
given in these coordinates by $|u_{i}|\leq\delta$ for some $i\in I$. The
region $U_{\delta}^{k}$ is then described in these coordinates by

$\displaystyle|u_{i}|\leq\delta\mbox{ for all }i=1,\dots,k$ (3.23)

as long as $p\in V^{k}$ but outside the $\delta$ neighborhood of the higher
depth stratum $V^{k+1}$. This means that under the change of coordinates
$v_{i}=\lambda/u_{i}$ for each $i\in I$, the region $U_{\delta}^{k}$ is
identified with a region $O_{\delta}(\lambda)$ in $\mathbb{F}_{k}$ described
by the equations

$\displaystyle|v_{i}|\geq|\lambda|\delta^{-1}\mbox{ for all }i=1,\dots,k$
(3.24)

where $p\in V^{k}$ is outside the $\delta$ neighborhood of the higher depth
stratum. This region $O_{\delta}(\lambda)$ is nothing but complement of the
union of tubular neighborhoods of the infinity divisor $D_{k,\infty}$ and the
fiber divisor $F_{k}$ in $\mathbb{F}_{k}$, see (3.7). For
$\delta=|\sqrt{\lambda}|$ this change of coordinates therefore provides an
identification $R_{\lambda}$ of the $|\sqrt{\lambda}|$-tubular neighborhood of
$V$ in $X$ (broken into pieces as above) and the complement of the
$|\sqrt{\lambda}|$-tubular neighborhood of the singular divisor $W_{1}$ in
$\mathbb{F}_{V}$, exactly as suggested by the iterated symplectic sum
construction.

Therefore the (semi)-local model of $X_{\lambda}$ over such a point $p$ is
given by the locus of the equations

$\displaystyle u_{i}v_{i}=\lambda\quad\mbox{ where
}\quad|u_{i}|\leq\delta\quad\mbox{ for all }i\in I$ (3.25)

where $u_{i}$ is a coordinate in $N_{i}$, the normal bundle to the $i$’th
local branch of $V$ and $v_{i}$ is the dual coordinate in the dual bundle
$N_{i}^{*}$ (which is allowed to equal infinity). Note that these equations
are invariant under reordering of the branches, so they describe an intrinsic
subset of $N_{V^{k}}\times\mathbb{F}_{k}$ where our semi-local analysis will
take place.

The $\delta$-neck $N_{\lambda}(\delta)$ of $X_{\lambda}$ is the region in the
above coordinates where

$\displaystyle\mbox{ $|u_{i}|<\delta$ and $|v_{i}|<\delta$ for some }i$ (3.26)

and it globally corresponds to the annular region
$A(|\lambda|\delta^{-1},\delta)$ around $V$ in $X$. The upper hemisphere
region $H_{\lambda}(\delta)$ of $X_{\lambda}$ corresponds to the region
$A(|\lambda|,\delta)$ in $X$; in coordinates it is described by:

$\displaystyle\mbox{ $|u_{i}|<\delta$ and $|v_{i}|<1$ for some }i$ (3.27)

As $\lambda\rightarrow 0$, $X_{\lambda}$ converges to a level one building
$X_{1}$; for $\lambda$ sufficiently small, the part of $X_{\lambda}$ outside
the $\delta$-neck is canonically identified with the complement a certain
neighborhood of $W_{1}$ in $X_{1}$; as both $\lambda,\delta\rightarrow 0$,
this neighborhood expands to $X_{1}\setminus W_{1}$. The upper hemisphere
region $H_{\lambda}(\delta)$ similarly converge to the upper hemisphere of
$\mathbb{F}_{V}$. The neighborhoods $U_{\delta}^{k}\setminus U_{\delta}^{k+1}$
all fit inside $N_{V^{k}}\times\mathbb{F}_{k}$ where they are described by the
equations (3.25). In particular, over a point $p$ as above, the level one
building $X_{1}$ is described by the locus of the equations:

$\displaystyle u_{1}v_{1}=0,\quad\dots\quad,u_{k}v_{k}=0$ (3.28)

where $|u_{i}|\leq\delta$ for all $i$, regarded as an intrinsic subset of
$N_{V^{k}}\times\mathbb{F}_{k}$ (or more precisely inside the pullback of
$\mathbb{F}_{k}$ over $N_{V^{k}}$).

These describe the $2^{k}$ local pieces of $X_{1}$ coming together at $p$
along the singular locus $W_{1}$: each piece of $X_{1}$ is described by the
vanishing of exactly $k$ coordinates, but some of them may be $u_{i}\in N_{i}$
in which case the rest are the complementary indexed ones $v_{j}\in
N_{j}^{*}$. Furthermore, the divisor $W_{1}$ has several local branches, each
one described by the further vanishing of one of the remaining coordinates
$u_{i}$ or $v_{i}$, matching the description from Remark 3.15.

###### Remark 3.17

Of course, the rescaling procedure can be iterated finitely many times: start
with $X$, rescale it by $\lambda_{1}$, then rescale again the resulting
manifold by $\lambda_{2}$, etc. The limit as all $\lambda_{a}\rightarrow 0$ is
then a level $m$ building $X_{m}$, where we have similar semi-local models
described over a depth $k$ point $p$ as intrinsic subsets of
$(\mathbb{F}_{k}\times\mathbb{F}_{k})^{m}$, modeled in each direction $i$ by
the process of $m$ times rescaling a disk at the origin, see Remark 3.15.
Therefore one way to describe this iterated rescaling is to start by choosing
coordinates $u_{i,0}=u_{i}$ on $X$ normal to $V$ at $p$, together with $m$
other sets of dual normal coordinates:

$u_{i,l}\in N_{i}\cup\infty$ and $v_{i,l}\in N_{i}^{*}\cup\infty$ with
$v_{i,l}=u_{i,l}^{-1}$ (3.29)

for all $l=1,\dots,m$ and all $i\in I$. These provide semi-local coordinates
on $X_{m}$ in a neighborhood of the fiber $F_{p}$ over $p$ of the collapsing
map $X_{m}\rightarrow X$. Rescaling $X$ means we

make a change of coordinates $u_{i,l-1}v_{i,l}=\lambda_{l}$ at step $l$,
(3.30)

for each $l=0,\dots,m$ and all $i\in I$.

##  4 The refined limit of a sequence of maps

Consider now a sequence $\\{f_{n}\\}$ of maps in ${\cal M}_{s}(X,V)$ whose
limit (in the usual stable map compactification) has some components in $V$
and thus is not in ${\cal M}_{s}(X,V)$. We will describe how to rescale $X$
once around $V$ and construct a refined limit $f$ into a level one building
$X_{1}$ which has no nontrivial components in the singular locus, and fewer
nontrivial components in the zero divisor. This will be the key main step in
the inductive rescaling procedure used in the next section to construct the
relatively stable map compactification.

We also want to describe in more details the collection of such possible
limits $f$. In the case $V$ is a smooth divisor, we proved in [IP2] that the
limit satisfies a matching condition along the singular divisor. The arguments
used there are semi-local (that is they involve a further analysis of what
happens only in a tubular neighborhood of $V$), and extend to the case when
$V$ is a normal crossings divisor by similarly working in neighborhoods of
each depth $k$ stratum of $V$, where we have $k$ different normal directions
to $V$. In the case the limit $f$ has no trivial components in $W_{1}$, we
will also get a matching condition along $W_{1}$. First of all, we will see
that similarly $f^{-1}(W_{1})$ consists of nodes of the domain. The naive
condition is that at each such node, $f$ has matching intersection points with
$W_{1}$, including the multiplicities, as was the case in [IP1]. But it turns
out that at points of depth $k\geq 2$, this is not enough to define relative
GW invariants, and needs to be further refined.

First of all, according to our conventions which match those of [IP2], every
map $f:\Sigma\rightarrow X$, will be regarded as a map
$(f,\varphi):\Sigma\rightarrow X\times\overline{{\cal U}}_{g,n}$, where the
second factor is the universal curve of the domains. The energy of $f$ in a
region $N$ of the target is then defined to be

$\displaystyle E(f;N)=\frac{1}{2}\int_{f^{-1}(N)}|df|^{2}+|d\varphi|^{2}$
(4.31)

Denote by $\alpha_{V}$ the minimum quanta of energy that a stable holomorphic
map into $V$ has. Note that according to our conventions, the energy also
includes the energy of domain $\Sigma$, which is stable when
$f_{*}[\Sigma]=0$, and therefore $\alpha_{V}>0$.

The key idea of Chapter 3 of [IP2] was that when $V$ is smooth and $\delta>0$
small, the limits of maps which have energy at most $\alpha_{V}/2$ in the
$\delta$-neck cannot have any components in the singular locus $V=V_{\infty}$
and thus have well defined leading coefficients along $V=V_{\infty}$ which are
furthermore uniformly bounded away from 0 and infinity (the particular bound
depends on $\delta$ and the choice of metrics).

In the case when $V$ is a normal crossings divisor, it will no longer be true
that the limit of maps which have small energy in the neck has no components
in the singular locus. Now we could have some components in level one whose
energy is smaller than $\alpha_{V}/2$: these are stable maps into
$\mathbb{F}_{V}$, but whose projection to $V$ does not have a stable model. In
the case $V$ was smooth, these correspond precisely to what we called trivial
components in Definition 11.1 of [IP2]. When $V$ is a normal crossings
divisor, the appropriate extension of that definition is then the following:

###### Definition 4.1

A trivial component in $\mathbb{F}_{k}$ is a nonconstant holomorphic map
$f:({\mathbb{P}}^{1},0,\infty)\rightarrow(\mathbb{F}_{k},D_{0}\cup
D_{\infty})$ whose image lands in a fiber $({\mathbb{P}}^{1})^{k}$ of
$\mathbb{F}_{k}$. In coordinates

$\displaystyle f(z)=(a_{1}z^{s_{1}},\dots a_{k}z^{s_{k}})$

where some, but not all of the $a_{i}$ could be zero or infinity.

This is the only kind of stable map into $(\mathbb{F}_{k},D_{0}\cup
D_{\infty}\cup F_{k})$ which does not have a stable model when projected into
$D_{0}$: its domain is an unstable sphere with just two marked points and thus
its projection has energy zero. Note that as a result of the rescaling
process, we have uniform control only on the energy of the projection, as the
area of the fiber of $\mathbb{F}_{k}$ could be arbitrarily small. We are now
ready to state the first rescaling result:

###### Proposition 4.2

Consider $\\{f_{n}:C_{n}\rightarrow X\\}$ a sequence of maps in ${\cal
M}_{s}(X,V)$ which converge, in the usual stable map compactification, to a
limit $f_{0}:C_{0}\rightarrow X$ which has at least one component in $V$.

Then there exists a sequence $\lambda_{n}\rightarrow 0$ of rescaling
parameters such that after passing to a subsequence, $R_{\lambda_{n}}f_{n}$
have a refined limit $g:C\rightarrow X_{1}$ with the following properties:

  * (a)

$g$ is a map into a level one building $X_{1}$ refining $f_{0}$:

$\displaystyle C\;\;$ $\displaystyle\mathop{\longrightarrow}\limits^{g}$
$\displaystyle{X_{1}}$ $\displaystyle\;\;\downarrow{\mbox{\rm st}}$
$\displaystyle\downarrow{p}$ (4.32) $\displaystyle C_{0}\;\;$
$\displaystyle\mathop{\longrightarrow}\limits^{f_{0}}$ $\displaystyle X$

  * (b)

$g$ has no nontrivial components in the singular divisor $W_{1}$;

  * (c)

$g$ has at least one nontrivial component in level one which does not lie
entirely in $V_{1}$, and thus has fewer nontrivial components in the zero
divisor $V_{1}$ compared with $f_{0}$;

Any refined limit $g$ that has properties (a)-(c) is unique up to rescaling
the level one of the building by an overall factor
$\lambda\in{\mathbb{C}}^{*}$.

Proof. In the case $V$ is a smooth divisor, we proved this in Section 6 of
[IP1] (see also Section 3 of [IP2]). The arguments used there to construct the
refined limit are semi-local, in a neighborhood of $V$, and if set up right,
easily extend to the case when $V$ is a normal crossings divisor. The main
rescaling argument consists of two parts, first the construction of the
refined limit by rescaling the target near $V$, and later on a further
analysis of the properties of this refined limit. For the first part of the
argument, we work separately in neighborhoods of depth $k$ stratum $V^{k}$ but
away from the higher depth strata; for the second part of the argument, we
work in the necks, where the transition between these local models happens. As
this is one of the crucial steps in the construction of the relatively stable
map compactification, we include below the complete details of both of these
arguments.

Step 0. Preliminary considerations. Assume for simplicity that the domains
$C_{n}$ are smooth (otherwise work separately on each of their components) and
that the original limit $C_{0}$ is a stable curve (see Remark 2.1).

Around each point $x\in C_{0}$, denote by $B(x,\varepsilon)$ the ball about
$x$ of radius $\varepsilon$ in the universal curve, and by
$B_{n}(x,\varepsilon)$ its intersection with $C_{n}$. Around each node $x$ of
$C_{0}$ we can choose local coordinates $z,w$ on the universal curve such that
the domains $C_{n}$ are described by

$\displaystyle zw=\mu_{n}(x)$ (4.33)

where $\mu_{0}(x)=0$. Denote by $D$ the collection of nodes of $C_{0}$, and by
$\widetilde{C}_{0}$ the resolution of $C_{0}$. For each point
$x\in\widetilde{C}_{0}$, denote by $\gamma_{\varepsilon}(x)$ the oriented
boundary of the $\varepsilon$-disk about $x$ in $\widetilde{C}_{0}$, and by
$\gamma_{n,\varepsilon}(x)$ the corresponding boundary component of
$B_{n}(x,\varepsilon)$.

Denote by $P_{n}\subset C_{n}$ the collection of marked points of $f_{n}$,
which include all the points in $f_{n}^{-1}(V)$, with their contact
information recorded by $s$. As $n\rightarrow\infty$, they converge to the
marked points $P_{0}$ of $C_{0}$, but the original limit $f_{0}$ has only a
partially defined contact information to $V$, which we describe next. Each
component $\Sigma$ of $C_{0}$ has a depth $k(\Sigma)$, defined as the maximum
$k$ such that $f_{0}(\Sigma)\subset V^{k}$. The restriction of $f_{0}$ to
$\Sigma$ has a well defined contact multiplicity to the next stratum
$V^{k+1}$; each such contact point has an associated contact information (2.3)
along $V^{k+1}$, including its depth $k(x)>k$ (with respect to the
stratification of $V$). For each $j>k$, will denote by
$R_{k}^{j}\subset\widetilde{C}_{0}$ the collection of contact points of depth
$j$ that belong to a component of depth $k$. Denote by $R$ the union of all
these contact points, and by $R_{k}$ those that have depth $k$ while by
$R^{j}$ those that belong to a depth $j$ component. Also denote by
$f^{k}:C^{k}\rightarrow V^{k}$ the restriction of $f_{0}$ to $C^{k}$, the part
of $C_{0}$ consisting of depth $k$ components; in fact, $f^{k}\in{\cal
M}_{s^{k}}(\widetilde{V}^{k},\widetilde{V}^{k+1})$ and

$\displaystyle f_{0}^{-1}(V^{k}\setminus V^{k+1})=(C^{k}\setminus R^{k})\sqcup
R_{k}$ (4.34)

This allows us to break-up the domain of $C_{0}$ and $C_{n}$ into pieces whose
image lies in controlled regions where the rescaling happens (more precisely
in a neighborhood of $V^{k}$, but away from a neighborhood of the higher depth
stratum $V^{k+1}$). Recall that $U_{\delta}$ denotes the $\delta$-tubular
neighborhood of $V$, and $U_{\delta}^{k}$ denotes the corresponding
neighborhood of $V^{k}$, see (3.23).

The complement of the ’neck’ regions $B(x,\varepsilon)$ about all points $x\in
R$ (for $\varepsilon(x)>0$ sufficiently small) decomposes $C_{n}$ into pieces
$(C_{n}^{k})^{\prime}$, which limit as $n\rightarrow\infty$ and
$\varepsilon\rightarrow 0$ to $C^{k}\setminus R^{k}$. We will denote by
$C_{n}^{k}$ the union of the $(C_{n}^{k})^{\prime}$ piece together with all
the ’neck’ pieces around points in $R_{k}$. The boundary of $C^{k}_{n}$ then
decomposes as $\partial_{+}\sqcup-\partial_{-}$ where

$\displaystyle\partial_{+}C^{k}_{n}=\mathop{\sqcup}\limits_{x\in
R_{k}}\gamma_{n,\varepsilon}(x)\quad\mbox{ while
}\quad\partial_{-}C^{k}_{n}=\mathop{\sqcup}\limits_{x\in
R^{k}}\gamma_{n,\varepsilon}(x)$ (4.35)

The curve $C_{n}$ is obtained by joining $C_{n}^{k}$ along
$\partial_{-}C_{n}^{k}$ and $\partial_{+}C_{n}^{k}$ to lower and respectively
higher depth pieces $C_{n}^{j}$.

By construction and (4.34), $f_{0}$ maps $C^{k}_{0}$ in some neighborhood
$U^{k}_{M(\varepsilon)/2}$ of the depth $k$ stratum but away from some much
smaller neighhborhood $U^{k+1}_{2m(\varepsilon)}$ of the higher depth stratum,
such that the incoming part $f_{0}(\partial_{-}C_{0}^{k})$ of the boundary
ends up outside the neighborhood $U^{k}_{2m(\varepsilon)}$ of the depth $k$
stratum, while the outgoing part $f_{0}(\partial_{+}C_{0}^{k})$ of the
boundary ends up inside the $U^{k}_{M(\varepsilon)/2}$ neighborhood of the
depth $k+1$ stratum. The sizes of these tubular neighborhoods depend on
$f_{0}$, but can be chosen uniform in $k$ and also such that
$M(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$.

Fix $0<\delta<<1$ small enough so that, for all $k\geq 0$ the depth $k$ piece
$f^{k}$ of the original limit $f_{0}$ has very small energy in a
$\delta$-neighborhood of the higher depth stratum $V^{k+1}$:

the energy of $f^{k}$ in the region $U^{k+1}_{\delta}$ is less than
$\alpha_{V}/100$ (4.36)

Of course, we have assumed that $f_{0}$ has at least one component in $V$ thus
the energy of the restriction of $f_{0}$ to the complement of $C^{0}$ is at
least $\alpha_{V}$. Assume $\varepsilon>0$ sufficiently small so that
$2M(\varepsilon)<\delta$. Since $f_{n}\rightarrow f_{0}$ in the stable map
compactification as maps into $X$, for $n$ sufficiently large,

  1. (i)

$f_{n}(C_{n}^{k})$ lies in $U^{k}_{M}\setminus U^{k+1}_{m}$;

  2. (ii)

$f_{n}(\partial_{-}C_{n}^{k})$ lies $U^{k}_{M}\setminus U^{k}_{m}$ while
$f_{n}(\partial_{+}C_{n}^{k})$ lies in $U^{k+1}_{M}\setminus U^{k+1}_{m}$;

  3. (iii)

the restriction of $f_{n}$ to $C_{n}^{k}$ has energy at most $\alpha_{V}/50$
in the region $U_{\delta}^{k+1}$;

  4. (iv)

the restriction of $f_{n}$ to the complement $C_{n}^{0}$ has energy at least
$2\alpha_{V}/3$.

Step 1. Constructing the refined limit. Next we rescale $X$ around $V$ by a
certain amount $\lambda_{n}$ to catch a refined limit of the sequence $f_{n}$.
To find $\lambda_{n}$, we consider the energy $E(t)$ of $f_{n}$ inside the
annular region $A(t,\delta)$ of (3.22) around $V$ in $X$. Because $f_{n}$ was
assumed to be smooth, the energy $E(t)$ is a continuously decreasing function
in $t$, which is equal to 0 when $t=\delta$ and is at least $2\alpha_{V}/3$
when $t=0$ (because the restriction of $f_{n}$ to the complement of
$C_{n}^{0}$ lies in the $U_{\delta}$ by (i) and has energy at least
$2\alpha_{V}/3$ by (iv)). So for $n$ sufficiently large, we can find a
$\lambda_{n}\neq 0$ such that

the energy of $f_{n}$ in the annular region $A(|\lambda_{n}|,\delta)$ is
precisely $\alpha_{V}/2$ (4.37)

where $\delta>0$ is fixed such that (4.36). Then $\lambda_{n}\rightarrow 0$,
because if they were bounded below by $\mu>0$ then in the limit the energy in
the annular region $A(\mu,\delta)$ of $f_{0}$ and thus of $f^{0}$ would be
$\alpha_{V}/2$ which contradicts (4.36).

As the rescaling maps $R_{\lambda_{n}}:X\rightarrow X_{\lambda_{n}}$ of (3.21)
are identity outside the $|\lambda_{n}|^{1/2}$ neighborhood of $V$, the
restriction of $f_{n}$ to $C_{n}^{0}$ will be unaffected by rescaling and thus
will uniformly converge (on compacts away from the nodes) to the restriction
of $f^{0}$ to $C^{0}_{0}$. Taking also $\varepsilon\rightarrow 0$, we get
usual Gromov convergence of the restrictions of $R_{\lambda_{n}}f_{n}$ to
$C_{n}^{0}$ to a limit $g_{0}$, which is equal in this case to the original
depth zero part of the limit $f^{0}:C^{0}\rightarrow X$. In particular, the
restriction of $R_{\lambda_{n}}f_{n}$ to the outgoing part of the boundary
$\partial_{+}C_{n}^{0}$ uniformly converge to the restriction of $g^{0}$ to
$\partial_{+}C_{0}^{0}$, where we have a local model (2.3) around $V$.

Now proceed by induction on $k\geq 1$. Assume that for all
$\varepsilon_{n}\rightarrow 0$, after possibly making the $\varepsilon_{n}$
even smaller, the restrictions of $R_{\lambda_{n}}f_{n}$ to
$\mathop{\cup}\limits_{j<k}C_{n}^{j}(\varepsilon_{n})$ converge to a limit
$g:\mathop{\cup}\limits_{j<k}B_{j}\rightarrow\mathop{\cup}\limits_{j<k}\mathbb{F}_{j}$
which refines the depth less than $k$ part of the original limit$f_{0}$. This
in particular means that the boundary of $\mathop{\cup}\limits_{j<k}C_{n}^{j}$
converges to finitely many marked points of $\mathop{\cup}\limits_{j<k}B_{j}$
and that the restrictions of $R_{\lambda_{n}}f_{n}$ to this boundary converge
to the image under $g$ of these points.

We want to extend this for the restrictions to $\cup_{j\leq k}C_{n}^{j}$. Note
that by construction, $\cup_{j<k}C_{n}^{j}$ has only outgoing boundary, which
matches the incoming one of $\cup_{j\geq k}C_{n}^{j}$. Therefore it suffices
to only consider the restrictions to $C_{n}^{k}$ and show that after possibly
shrinking only the outgoing part of its boundary we get a convergent
subsequence to some $g_{k}:B_{k}\rightarrow\mathbb{F}_{k}$ which refines
$f^{k}:C^{k}\rightarrow V^{k}$. Then the restriction of sequence
$R_{\lambda_{n}}f_{n}$ to $\mathop{\cup}\limits_{j\leq k}C_{n}^{j}$ will
automatically converge to a continuous limit $g:\mathop{\cup}\limits_{j\leq
k}B_{j}\rightarrow\mathop{\cup}\limits_{j\leq k}\mathbb{F}_{j}$ with the
desired properties.

By (i), the restrictions of $f_{n}$ to $C_{n}^{k}$ lie in the
$\delta$-neighborhood $U^{k}_{\delta}$ of $V^{k}$, but outside the
$m(\varepsilon)$-tubular neighborhood of the next stratum $V^{k+1}$; after
rescaling by $\lambda_{n}$, we can identify this region with the complement of
the $|\lambda_{n}|\delta^{-1}$ neighborhood of the infinity divisor
$D_{k,\infty}$ in $\mathbb{F}_{k}$ and of the $m(\varepsilon)$ neighborhood of
the fiber divisor $F_{k}$ over the higher depth stratum, see (3.23) and
(3.23).

Next consider the restriction of $f_{n}$ to each piece
$\gamma_{n,\varepsilon}(x)$ of the boundary of $C_{n}^{k}$, which after
rescaling ends up in a corresponding annular region around $D_{k,\infty}$ and
respectively $F_{k}$. We claim that the image of $\gamma_{n,\varepsilon}(x)$
under $R_{\lambda_{n}}f_{n}$ can be capped off with a small energy disk in
$\mathbb{F}_{k}$ around this divisor. For the incoming pieces of the boundary
(i.e. for each $x\in R_{j}^{k}$ with $j<k$), this is because by induction
$R_{\lambda_{n}}(\gamma_{n,\varepsilon}(x))$ uniformly converges as
$n\rightarrow\infty$ to the restriction of $g(\gamma_{\varepsilon}(x))$ where
we already have a local model coming from $f^{j}$ for each $j<k$; in
particular, the intersection of these capping disks with the infinity section
is uniformly bounded. For the outgoing pieces of the boundary (i.e. for each
$x\in R_{k}^{j}$ with $j>k$) we have the local model of $f^{k}$ along
$V^{k+1}$, to which the unrescaled $f_{n}(\gamma_{n,\varepsilon}(x))$
converges uniformly for fixed $\varepsilon>0$. So after possibly choosing a
smaller $\varepsilon_{n}(x)>0$ (depending also on $n$ for each such outgoing
$x$), we can also cap off the image of $\gamma_{n,\varepsilon}(x)$ under
$R_{\lambda_{n}}f_{n}$ by a small energy disk about the fiber divisor over the
higher depth stratum (away from the infinity section).

The resulting homology class in $\mathbb{F}_{k}$ of the capped surface is
constant in $n$, because is determined by (i) the homology class of its
projection onto the zero section which depends only on $f_{0}$ and (ii) the
intersection with the infinity divisor, where we already have uniform control
by induction. This means a fortiori that the restriction of the maps
$R_{\lambda_{n}}f_{n}$ to $C_{n}^{k}$, which are $R_{\lambda_{n}}^{*}(J,\nu)$
holomorphic, have uniformly bounded energy in $\mathbb{F}_{k}$ (where the
symplectic area of the fibers is fixed, but small) and therefore a have Gromov
convergent subsequence to a $(J_{0},\nu_{0})$ limit
$g_{k}:B_{k}\rightarrow\mathbb{F}_{k}$, defined (after removing its
singularities) on a closed, possibly nodal curve $B_{k}$ which is obtained
from $C^{k}$ after possibly inserting some bubble trees. The convergence is
uniform on compacts away from the nodes of $B_{k}$ and in Hausdorf distance.

This constructs inductively a refined limit $g:C\rightarrow X_{1}$, defined
from $C=\cup B_{k}$ into a level one building (the depth $k$ is bounded by
${\rm dim\;}X$, so the process terminates in finitely many steps). By
construction, the refined limit $g$ has energy at most $\alpha_{V}/2$ around
the infinity section of $\mathbb{F}_{V}$, and therefore has no nontrivial
components there (as these would carry at least $\alpha_{V}$ energy), giving
(b). If $g$ had only trivial components in the first level but outside
$V_{1}$, then $g$ would have very small energy in the upper-hemisphere of
${\mathbb{P}}_{V}$, contradicting the choice (4.37). Therefore the limit
satisfies (c) as well. This concludes the proof of Lemma 4.2. $\Box$

Step 2: Further properties of the refined limit. We next want to understand in
more details the behavior of the refined limit $g:C\rightarrow X_{1}$
constructed by Lemma 4.2 around the singular divisor $W_{1}$ where the pieces
of the building are joined together; for that we will restrict ourselves to
the neck regions (both of the domain and of the target) where we will use the
local models (4.33) on the domain and (3.25) in the target.

First of all, every component $\Sigma$ of the refined limit $g:C\rightarrow
X_{1}$ that does not land in $W_{1}$ has a lift
$\widetilde{g}:\Sigma\rightarrow\widetilde{X}_{1}$ to the resolution
$\widetilde{X}_{1}$, where it has a well defined order of contact with
$\widetilde{W}_{1}$; the components that land in $V_{1}$ have only a partial
contact information to the higher depth strata of $V_{1}$; the components that
land in $W_{1}$ must project to a point under $X_{1}\rightarrow X$, and have
only a partial contact information to the higher depth strata of
$\widetilde{W}_{1}\cup V_{1}$. The contact information of $g$ refines the
partial contact information of the depth $k$ part of the original limit
$f_{0}$ to the higher depth stratum of $V$.

In this section we use the semi-local models for the target described in
Remark 3.15, and in particular describe the strata of the resolution of the
total divisor $\widetilde{D}_{1}$ in terms of strata of $V$ together with a
multilevel and a multisign map. Fix a point $x\in\Sigma$ such that $g(x)\in
W_{1}\cup V_{1}$, and denote by $p$ the image of $g(x)$ under the collapsing
map $X_{1}\rightarrow X$. The image $\widetilde{g}(x)$, which is a point in
$\widetilde{W}_{1}\cup V_{1}$, comes with both a multi-level map (3.15) and a
sign map (3.16) that associate to each local branch $i$ of $V$ at $p$ a local
level $l_{i}(x)=0$ or 1 and a sign $\varepsilon_{i}(x)=\pm 1,0$ recording
whether it is equal to infinity, zero or neither in that direction.
Furthermore, $\widetilde{g}$ has an associated contact information to
$\widetilde{W}_{1}$ and a possibly partial one to $V_{1}$ at $x$. This
information includes

a partition of $I$ into $I^{\pm}(x),I^{0}(x)$ and $I^{\infty}(x)$. (4.38)

For each $i\in I^{\pm}(x)$, $\varepsilon_{i}(x)=\pm$ (respectively) and we
also have a well defined contact multiplicity $s_{i}(x)>0$ to the total
divisor and a leading coefficient $a_{i}(x)\neq 0$ which is now regarded as an
element of

$\displaystyle a_{i}(x)\in
N_{i}^{\varepsilon_{i}(x)}\otimes(T_{x}^{*}\Sigma)^{s_{i}(x)}\quad\mbox{ for
all }i\in I^{\pm}(x)$ (4.39)

The directions $i\in I^{0}(x)$ have $\varepsilon_{i}(x)=0$ and correspond to
contact of order zero with the total divisor in those directions, so we set
$s_{i}(x)=0$ while the leading coefficient is $a_{i}(x)\in N_{i}$, the
coordinate of $\widetilde{g}(x)$ in the $i$’th direction (which is nonzero).
Finally, the directions $i\in I^{\infty}(x)$ have $\varepsilon_{i}(x)=+$ and
$l_{i}(x)=1$ and correspond to those directions in which $g(\Sigma)$ is
entirely contained in the zero divisor $V_{1}$. These count as undefined or
infinite order of contact $s_{i}(x)=\infty$ while their leading coefficient
$a_{i}(x)=0$. In local coordinates $z$ on $\Sigma$ at $x$, and normal
coordinates $u_{i,l}\in N_{i}$ to the zero divisor in each level $l$, $g(z)$
then has an expansion

$\displaystyle(u_{i,l})^{\varepsilon_{i}(x)}$ $\displaystyle=$ $\displaystyle
a_{i}(x)z^{s_{i}(x)}+O(|z|^{s_{i}(x)})\qquad\mbox{ for all $i\notin
I^{\infty}(x)$}$ (4.40)

###### Lemma 4.3

Consider a sequence $f_{n}\in{\cal M}_{s}(X,V)$ as in Lemma 4.2, and let
$g:C\rightarrow X_{1}$ denote its refined limit constructed there. Then $g$
has a lift to the resolution of the level one building, which comes with a
(possibly partial) contact information $s$ to $\widetilde{W}_{1}\cup V_{1}$ as
described above. If we denote by $C^{\prime}$ any intermediate curve
$C\rightarrow C^{\prime}\rightarrow C_{0}$, then all the contact points of $g$
descend to special points of $C^{\prime}$, and moreover:

  * •

$s(x_{-})=s(x_{+})$ and $\varepsilon(x_{-})=-\varepsilon(x_{+})$ for each node
$x_{-}=x_{+}$ of $C^{\prime}$;

  * •

$s(x)=s(x_{n})$ and $\varepsilon(x)=\varepsilon(x_{n})$ for each marked point
$x\in C^{\prime}$ which is the limit of marked points $x_{n}$ of $C_{n}$

(whenever both sides are defined). Furthermore any special fiber of
$C\rightarrow C^{\prime}$ or $C^{\prime}\rightarrow C_{0}$ is a string of
trivial components with two end points (broken cylinder).

Proof. First of all, we can decompose the domain $C$ into two pieces, $C^{0}$
and $B$ where $C^{0}$ consists of nontrivial components, and $B$ consisting of
components that get collapsed under the two maps $C\rightarrow C_{0}$ and
$X_{1}\rightarrow X$. Then each connected component $B_{i}$ of $B$ is an
unstable genus zero curve (bubble tree) with either one or two marked points,
corresponding to a special fiber of $C\rightarrow C_{0}$ over either:

  1. (a)

a non special point in the case $B_{i}$ has only one marked point

  2. (b)

a node in the case $B_{i}$ has two marked points and both belong to $C^{0}\cap
B$ or

  3. (c)

a marked point in the case $B_{i}$ has two marked points, but only one belongs
to $C^{0}\cap B$ while the other is a marked point of $C$.

We have the same description for the special fibers of $C\rightarrow
C^{\prime}$ and of $C^{\prime}\rightarrow C_{0}$ for any intermediate curve
$C\rightarrow C^{\prime}\rightarrow C_{0}$.

For any point $x\in C_{0}$, choose local coordinates at $x$ on the universal
curve of the domains (containing $C_{0}$) and normal coordinates to $V$ in the
target $X$ at $p$ as above (where $p=f_{0}(x)$ is a depth $k\geq 0$ point of
$V$). Using the notations of the proof of Lemma 4.2, for
$\varepsilon,\delta>0$ sufficiently small and $n$ sufficiently large, the
image under $f_{n}$ of $B_{n}(x,\varepsilon)$ is mapped in the $\delta$
neighborhood of $p$ in $X$ but away from the $\delta$ neighborhood of the
depth $k+1$ stratum. Furthermore, since $f_{n}^{-1}(V)=P_{n}$ then $f_{n}$
maps $B_{n}(x,\varepsilon)\setminus P_{n}$ into the annular region
$O_{\delta}$

$\displaystyle 0<|u_{i}|<\delta\quad\mbox{ for all }i\in I$ (4.41)

of $N_{V^{k}}$ around $p$ (away from the higher depth stratum). This region is
homotopic to $(S^{1})^{k}$, one factor for each branch $i$ of $V$ at $p$.

If we also fix to begin with global coordinates on all bubble components in
$B$, we have a similar story for any point $x$ in any one of the intermediate
curves $C^{\prime}$: $f_{n}$ takes a sufficiently small punctured neighborhood
$B_{n}(x,\varepsilon)\setminus P_{n}$ into the annular region (4.41), where
now $B_{n}(x,\varepsilon)$ denotes the intersection of $C_{n}$ with the ball
$B(x,\varepsilon)$ about $x$ in the local model for domains containing their
intermediate limit $C^{\prime}$. Topologically, $B_{n}(x,\varepsilon)$ is
either a disk or an annulus, depending whether $x$ is a smooth point or a node
of $C^{\prime}$. In particular, for every lift $\widetilde{x}$ of $x$ to the
resolution of $C^{\prime}$ and thus to $\widetilde{C}$, the corresponding
boundary loop $f_{n}(\gamma_{n,\varepsilon}(\widetilde{x}))$ has a well
defined winding number $s_{i,n}(\widetilde{x})$ about the branch $i$ of $V$ at
$p$.

Furthermore, for $\varepsilon>0$ sufficiently small, $B_{n}(x,\varepsilon)$
contains no points from $P_{n}$ if $x$ is not a marked point of $C^{\prime}$
and otherwise it contains precisely one point $x_{n}\in P_{n}$ which limits to
$x$ as $n\rightarrow\infty$. This implies that, for $n$ sufficiently large,
and all $i\in I$,

  1. (a)

if $x$ is not a special point of $C^{\prime}$ then the winding numbers
$s_{i,n}(\widetilde{x})=0$;

  2. (b)

if $x_{\pm}\in\widetilde{C}$ correspond to a node $x$ of $C^{\prime}$ then
$s_{i,n}(x_{+})+s_{i,n}(x_{-})=0$.

  3. (c)

if $x\in C^{\prime}$ is the limit of the marked points $x_{n}\in C_{n}$ then
$s_{i,n}(\widetilde{x})=s_{i}(x_{n})$;

But the winding numbers $s_{i,n}(\widetilde{x})$ of $f_{n}$ are related to
those of the refined limit $g$. This is simply because the winding numbers of
$f_{n}$ agree with those of the rescaled maps $R_{\lambda_{n}}f_{n}$, and
these converge uniformly on compacts away from the nodes of $C$ to the refined
limit $g:C\rightarrow X_{1}$, which has well defined winding numbers about the
zero section in all directions $i\notin I^{\infty}$. Since the loops
$\gamma_{n,\varepsilon}(\widetilde{x})$ stay away from all the nodes of $C$
(for $\varepsilon$ sufficiently small) then for $n$ sufficiently large:

$s_{i,n}(\widetilde{x})=\varepsilon_{i}(\widetilde{x})s_{i}(\widetilde{x})$
for all $i\notin I^{\infty}$ (4.42)

because of the expansion (4.40) of $g$ at $\widetilde{x}$. Note that this give
us no information about the winding numbers in the direction of $I^{\infty}$,
where the winding numbers of $g$ are undefined.

In particular, any contact point $\widetilde{x}$ of $g$ with $W_{1}\cup V_{1}$
or any of its strata has $s_{i}(\widetilde{x})>0$ in some direction $i$ which
rules out case (a): if $\widetilde{x}\in\widetilde{C}$ descends to a non
special point $x$ of $C^{\prime}$, then $s_{i,n}(\widetilde{x})=0$ by (a)
which contradicts (4.42).

In case (b), for any node $x$ of $C^{\prime}$ then
$s_{i,n}(x_{-})+s_{i,n}(x_{+})=0$ and so (4.42) implies that

$\displaystyle s_{i}(x_{-})=s_{i}(x_{+})\mbox{ and
}\varepsilon_{i}(x_{-})=-\varepsilon_{i}(x_{+})\quad\mbox{ for all }i\notin
I^{\infty}(x)\mathop{=}\limits^{def}I^{\infty}(x_{1})\cup I^{\infty}(x_{2})$

as both sides are well defined for such an $i$. If $x$ is the limit of contact
points $x_{n}$ of $f_{n}$ to $V$ then (4.42) implies that

$\displaystyle s_{i}(x)=s_{i}(x_{n})\mbox{ and
}\varepsilon_{i}(x)=\varepsilon_{i}(x_{n})\ \quad\mbox{ for all }i\notin
I^{\infty}(x)$

So for example $x$ is a contact point of $g$ with the zero divisor $V_{1}$ if
and only if $x_{n}$ was one for $f_{n}$ to $V$.

Finally, this discussion implies that there are no components of $B$ with just
one marked point (otherwise, contracting such component would give a curve
$C^{\prime}$ and a non special point $x$ on it which is impossible as case (a)
is ruled out for all intermediate curves $C^{\prime}$). Therefore all the
components of $B$ have precisely 2 special points, which must be the contact
points with the zero and the infinity divisor in their fiber, and thus are
indeed trivial components as in Definition 4.1. Furthermore, the only special
fibers of $C\rightarrow C^{\prime}$ or $C^{\prime}\rightarrow C_{0}$ are
strings of trivial components (broken cylinders). $\Box$

###### Remark 4.4

Lemma 4.3 and the discussion preceding it shows that each node $x$ of $C_{0}$
comes with a partition of the original indexing set $I$ into $I^{\infty}(x)$,
$I^{0}(x)$ and then $I(x)$. $I^{\infty}(x)$ records those directions in which
at least one of the local branches of $g$ lies in the total divisor $W_{1}$
while $I^{0}(x)$ records the directions in which the $i$’th coordinates of
both $\widetilde{g}(x_{\pm})$ are nonzero, so both branches stay away from the
total divisor in those directions. The remaining directions $i\in I(x)$ come
with a multiplicity $s_{i}(x)>0$ and opposite signs
$\varepsilon_{i}(x_{+})=-\varepsilon_{i}(x_{-})\neq 0$, recording the two
opposite sides of the level one building from which the two branches of $g$
come into the singular divisor, and also two leading coefficients
$a_{i}(x_{\pm})\neq 0$ which are naturally elements of

$\displaystyle a_{i}(x_{+})\in
N_{i}^{\varepsilon_{i}(x_{+})}\otimes(T_{x_{+}}C)^{s_{i}(x)}\quad\mbox{ and
}\quad a_{i}(x_{-})\in
N_{i}^{\varepsilon_{i}(x_{-})}\otimes(T_{x_{-}}C)^{s_{i}(x)}$ (4.43)

where $N_{i}$ is the branch of the normal bundle to $V$ indexed by $i\in
I(x)$.

###### Example 4.5

In the situation of Example 3.6 we could have two nodes of the domain mapped
to $p$, one between the components $X$ and $\mathbb{F}_{2}$ while the other
one between the two $\mathbb{F}_{1}$ components, but no node at $p$ between
say $X$ and $\mathbb{F}_{1}$ (as the branches of $g$ would not be on opposite
sides of the singular divisor in all local directions). In the first case the
node is between level 0 and level 1 (really local level (0,0) and (1,1)),
while in the second case it is between two level 1 floors, or more accurately
between a local level (0,1) and (1,0) floor.

###### Remark 4.6

As a consequence of Lemma 4.3, a refined limit $g:C\rightarrow X_{1}$ which
has no components in the total divisor $D_{1}=W_{1}\cup V_{1}$ lifts, after
labeling the nodes of $C$, to a unique map $\widetilde{g}$ into
$\widetilde{X}_{1}$, the resolution of the level one building $X_{1}$, with

$\displaystyle\widetilde{g}\in{\cal M}_{s_{\pm}\sqcup
s}(\widetilde{X}_{1},\widetilde{W}_{1}\cup\widetilde{V}_{1})$

where $s_{\pm}(x)$ records the extra contact information $s(x_{\pm})$ at the
pair of points $x_{\pm}$ corresponding to a node $x\in C$. The domain
$\widetilde{C}$ of $\widetilde{g}$ is a resolution of the domain $C$ of $f$.
The combined attaching map identifies pairs of marked points of
$\widetilde{C}$ to produce the nodes of $C$, and simultaneously attaches the
targets together to produce the singular locus $W_{1}$. This describes the
limit $g$ in terms of its normalization
$\widetilde{g}:\widetilde{C}\rightarrow\widetilde{X}_{1}$.

The attaching map $\xi$ that attaches the pieces of $\widetilde{X}_{1}$, when
restricted to a level $k$ stratum is a degree $2^{k}$ cover of $W_{1}$. At
each node $x_{1}=x_{2}$ mapped into this stratum we also have a partition of
the $2k$ normal directions $N_{V^{k}}\oplus N^{*}_{V^{k}}$ into two length $k$
dual indexing sets $I_{W}(x_{1})$ and $I_{W}(x_{2})$ that record the two
opposite local pieces of $\widetilde{X}_{1}$ containing the two local branches
of $g$ at that node. According to our setup, this information is already
encoded by $s_{\pm}$, as is the topological information about the domains and
the homology of the images, see Remark 2.3. Note that $C$ may have some usual
nodes (i.e. not mapped into $W_{1}$), but then according to our conventions
these give rise to marked points with depth $k=0$, also recorded in $s_{\pm}$,
see Remark 2.2.

Furthermore, the image of the normalization
$\widetilde{g}:\widetilde{C}\rightarrow\widetilde{X}_{1}$ under the evaluation
map (2.7) at the pairs of marked points giving the nodes:

$\displaystyle\mbox{\rm ev}_{s_{+}}\times\mbox{\rm ev}_{s_{-}}:{\cal
M}_{s\sqcup
s_{\pm}}(\widetilde{X}_{1},\widetilde{W}_{1}\cup\widetilde{V}_{1})\longrightarrow(W_{1})_{s_{+}}\times(W_{1})_{s_{-}}$
(4.44)

lands in the diagonal $\Delta$. We will call these the naive matching
conditions because when $s_{\pm}$ contains depth $\geq 2$ points, the
dimension of this stratum is in general bigger than the dimension of ${\cal
M}_{s}(X,V)$!

The refined limit $g$ is only well defined up to an overall rescaling
parameter $\lambda\in{\mathbb{C}}^{*}$ that acts by rescaling on the level one
of the building and by construction the limit $g$ has at least one component
which is not fixed by this action. This ${\mathbb{C}}^{*}$ action induces an
action on the moduli space of maps into $\widetilde{X}_{1}$, and thus the
normalization of such a limit is really in the inverse image of the diagonal
$\Delta$ under the map

$\displaystyle\mbox{\rm ev}_{s_{+}}\times\mbox{\rm ev}_{s_{-}}:{\cal
M}_{s\sqcup
s_{\pm}}(\widetilde{X}_{1},\widetilde{W}_{1}\cup\widetilde{V}_{1})/{\mathbb{C}}^{*}\longrightarrow(W_{1})_{s_{+}}\times(W_{1})_{s_{-}}$
(4.45)

We will include this information later on in this section. For now, let us
notice that:

###### Lemma 4.7

The difference between the expected dimensions of ${\cal M}_{s}(X,V)$ and that
of the stratum $\mbox{\rm ev}_{s_{\pm}}^{-1}(\Delta)$ of (4.44) is equal to

$\displaystyle{\rm dim\;}{\cal M}_{s}(X,V)-{\rm dim\;}\mbox{\rm
ev}_{s_{\pm}}^{-1}(\Delta)=2\sum_{x\in P(s_{+})}(1-k(x))$

where $P(s_{+})$ denotes the collection of marked points associated with the
sequence $s_{+}$.

Proof. This is a simple adaptation of the calculations of [IP2] to this
context. The expected dimension of $\mbox{\rm ev}_{s_{\pm}}^{-1}(\Delta)$ is

$\displaystyle{\rm dim\;}\mbox{\rm ev}_{s_{\pm}}^{-1}(\Delta)$
$\displaystyle=$ $\displaystyle{\rm dim\;}{\cal
M}_{\widetilde{s}}(\widetilde{X}_{1},\widetilde{W}_{1}\cup\widetilde{V}_{1})-{\rm
dim\;}(W_{1})_{s_{+}}$

where $\widetilde{s}=s\sqcup s_{\pm}$. So the difference is

$\displaystyle{\rm dim\;}{\cal M}_{s}(X,V)-{\rm dim\;}\mbox{\rm
ev}_{s_{\pm}}^{-1}(\Delta)={\rm dim\;}{\cal M}_{s}(X,V)-{\rm dim\;}{\cal
M}_{\widetilde{s}}(\widetilde{X}_{1},\widetilde{W}_{1}\cup\widetilde{V}_{1})+{\rm
dim\;}(W_{1})_{s_{+}}$

where

$\displaystyle{\rm dim\;}{\cal M}_{s}(X,V)=$ $\displaystyle=$ $\displaystyle
2c_{1}(TX)A_{s}+({\rm dim\;}X-6)\frac{\chi}{2}+2\ell(s)-2A_{s}V$
$\displaystyle{\rm dim\;}{\cal
M}_{\widetilde{s}}(\widetilde{X}_{1},\widetilde{W}_{1}\cup\widetilde{V}_{1})$
$\displaystyle=$ $\displaystyle
2c_{1}(T\widetilde{X}_{1})A_{\widetilde{s}}+({\rm
dim\;}X-6)\frac{\widetilde{\chi}}{2}+2\ell(\widetilde{s})-2A_{\widetilde{s}}\widetilde{W}_{1}-2A_{\widetilde{s}}V_{1}$

But $\chi=\widetilde{\chi}-2\ell(s_{+})$,
$\ell(\widetilde{s})=\ell(s)+2\ell(s_{+})$ and
$A_{s}V=|s|=A_{\widetilde{s}}V_{1}$, while Lemma 2.4 of [IP2] adapted to this
context gives

$\displaystyle
c_{1}(TX)A_{s}=c_{1}(T\widetilde{X}_{1})A_{\widetilde{s}}-2A_{\widetilde{s}}\widetilde{W}_{1}$

Therefore

$\displaystyle{\rm dim\;}{\cal M}_{s}(X,V)-{\rm dim\;}{\cal
M}_{\widetilde{s}}(\widetilde{X}_{1},\widetilde{W}_{1}\cup\widetilde{V}_{1})=({\rm
dim\;}X-2)\ell(s_{+})$

On the other hand, since the image under the evaluation map of each depth $k$
point lands in a codimension $2k$ stratum of $X_{1}$ then

$\displaystyle{\rm dim\;}(W_{1})_{s_{+}}=\sum_{x\in P(s_{+})}({\rm
dim\;}X-2k(x))$

and thus the difference in dimensions is exactly as stated. $\Box$

Even after dividing by the ${\mathbb{C}}^{*}$ action on the moduli space,
Lemma 4.7 still implies that if we want to construct a relatively stable map
compactification in the case when the normal divisor is singular (has depth
$k\geq 2$) pieces, then we would need some refined matching condition,
otherwise the boundary stratum is larger dimensonal than the interior.
Luckily, the existence of such refined compactification follows after a
careful examination of the arguments in [IP2].

It turns out that when $k\geq 2$, not all the maps $g:C\rightarrow X_{1}$
(without components in $W_{1}\cup V_{1}$) whose resolution $\widetilde{g}$
satisfies the naive matching condition can occur as limits after rescaling of
maps $f_{n}:C_{n}\rightarrow X$ in ${\cal M}_{s}(X,V)$. To describe those that
occur as limits, we use the results of Section 5 of [IP2]. For each node $x$
of $C$, we will work in the local models (4.33) on the domain and (3.25) in
the target, using the local coordinates as described. The results of Lemmas
4.2 and 4.3 can then be strengthened as follows:

###### Lemma 4.8

Consider $f_{n}:C_{n}\rightarrow X$ a sequence of maps in ${\cal M}_{s}(X,V)$
as in Lemma 4.2, and further assume that its refined limit $f:C\rightarrow
X_{1}$ constructed there has no components in the total divisor $D_{1}$. Then
for each node $x_{-}=x_{+}$ of $C$, we have the following relation:

$\displaystyle\lim_{n\rightarrow\infty}\frac{\lambda_{n}}{\mu_{n}(x)^{s_{i}(x)}}=a_{i}(x_{-})a_{i}(x_{+})\quad\mbox{
for each }i\in I(x)$ (4.46)

where $\mu_{n}(x)$ are the gluing parameters (4.33) describing $C_{n}$ at $x$
in terms of $C$, $\lambda_{n}$ is the sequence of rescaling parameters in the
target, while $s_{i}(x)>0$ and $a_{i}(x_{\pm})\neq 0$ are the contact
multiplicities and respectively the leading coefficients of the expansion
(4.40) of the refined limit $f$ at $x_{\pm}$ .

Proof. Note that according to our conventions, the condition (4.46) is vacuous
($I(x)=\emptyset$) unless the image of the node $x$ is mapped to the singular
locus. For each node $x$ of $C$ that is mapped to a depth $k(x)\geq 1$ stratum
of $W_{1}$ with matching multiplicities $s(x)$, we will work in the local
models described above, where we can separately project our sequence into each
direction $i\in I(x)$, where the local model is that of rescaling a disk
around the origin, as explained in more details in the next section. The
projections are now maps into a rescaled family of disks, precisely the
situation to which Lemma 5.3 of [IP2] applies to give (4.46) in each
direction, after using the expansions (4.40). $\Box$

###### Remark 4.9

The local model of ${\cal M}_{s}(X,V)$ near a limit point $f:C\rightarrow
X_{1}$ which has no components in $W_{1}$ is described by tuples
$(\widetilde{f},\mu,\lambda)$ satisfying an enhanced matching condition
(4.47). Here $\widetilde{f}:\widetilde{C}\rightarrow\widetilde{X}_{1}$ is an
element of $\mbox{\rm ev}_{s_{\pm}}^{-1}(\Delta)\subset{\cal M}_{s\sqcup
s_{\pm}}(\widetilde{X},\widetilde{W}_{1})$ satisfying the naive matching
condition along $W_{1}$ described in Remark 4.6, $\lambda\in{\mathbb{C}}$ is
the gluing (rescaling) parameter of the target, and
$\mu\in\mathop{\bigoplus}\limits_{x\in D}T^{*}_{x_{-}}C\otimes T^{*}_{x_{+}}C$
is a gluing parameter of the domain such that they also satisfy the condition

$\displaystyle a_{i}(x_{-})a_{i}(x_{+})\mu(x)^{s_{i}(x)}=\lambda\qquad\mbox{
for all }i\in I(x)$ (4.47)

at each node $x\in D$ of the domain, where $a_{i}(x_{\pm})$ are the two
leading coefficients (4.43) of $\widetilde{f}$ in the $i$’th normal direction
$i\in I(x)$ at the node $x\in D$.

Intrinsically, the gluing parameters $\mu$ in the domain are sections of the
bundle

$\displaystyle\mathop{\bigoplus}\limits_{x\in D}L_{x_{-}}\otimes L_{x_{+}}$

while the gluing parameter $\lambda$ in the target is naturally a section of
the bundle $N\otimes N^{*}\cong{\mathbb{C}}$. The condition (4.47) can be
expressed as $k(x)$ conditions on the leading coefficients:

$\displaystyle a_{i}(x_{-})a_{i}(x_{+})=\lambda\cdot\mu(x)^{-s_{i}(x)}$

at each node. If we fix a small $\lambda\neq 0$, the existence of a
$\mu(x)\neq 0$ satisfying these relations imposes a $2k(x)-2$ dimensional
condition on the leading coefficients, which is exactly what was missing in
Lemma 4.7.

###### Remark 4.10

Notice that the enhanced matching conditions become linear if we take their
log:

$\displaystyle\log a_{i}^{+}(x)+\log a_{i}^{-}(x)=\log\lambda-
s_{i}(x)\log\mu(x)$ (4.48)

which makes the transversality of this condition easier to prove, and also
hints to the connection with log geometry. Here $\log$ is the appropriate
extension of the map $\log:{\mathbb{C}}^{*}\rightarrow{\mathbb{R}}\times
S^{1}$ defined by $\log z=\log|z|+i\arg z$ to the intrinsic bundles in
question.

###### Remark 4.11

There is another other way to read (4.46). It implies that if indeed the limit
$f:C\rightarrow X_{1}$ does not have any components in $W_{1}\cup V_{1}$ then
of course all its leading coefficients $a_{i}(x_{\pm})\neq 0,\infty$.
Therefore for each fixed node $x$ of $C$ its contact multiplicities in the
normal directions must be equal to each other:

$\displaystyle s_{i}(x)=s_{j}(x)\mbox{ for all }i,j\in I(x).$

Furthermore, the normalization
$\widetilde{f}:\widetilde{C}\rightarrow\widetilde{X}_{1}$ of the limit $f$
must satisfy the enhanced matching condition at each node $x\in D$ i.e. the
image of $\widetilde{f}$ under the enhanced evaluation map

$\displaystyle\mbox{\rm Ev}_{x_{\pm}}:{\cal M}_{s\cup
s_{\pm}}(\widetilde{X}_{1},\widetilde{W}_{1})\rightarrow{\mathbb{P}}_{s(x)}(NW_{I(x_{+})})\times{\mathbb{P}}_{s(x)}(NW_{I(x_{-})})$
(4.49)

lands in the antidiagonal

$\displaystyle\Delta_{\pm}=\\{\;([a_{i}],[a_{i}^{-1}])\;|\;a_{i}\neq
0,\infty\mbox{ for }i\in I(x)\;\\}$

Recall that for each node, the two normal bundles in the target are
canonically dual to each other, or more precisely that the normal directions
to the singular locus $W_{1}$ come in dual pairs, and that $k$ of these
directions are indexed by $I(x_{-})$ and the other $k$ dual ones are indexed
by $I(x_{+})$.

Note that the limit $f$ is well defined only up to an overall rescaling
parameter $\lambda\in{\mathbb{C}}^{*}$ that acts on the level 1 of the
building, and in fact the enhanced evaluation map descends to a map on the
quotient

$\displaystyle\mbox{\rm Ev}_{\pm}:{\cal M}_{s\cup
s_{\pm}}(\widetilde{X}_{1},\widetilde{W}_{1})/{\mathbb{C}}^{*}\rightarrow\mathop{\prod}\limits_{x\in
D}{\mathbb{P}}_{s(x)}(NW_{I(x_{+})})\times{\mathbb{P}}_{s(x)}(NW_{I(x_{-})})$
(4.50)

that combines together the enhanced evaluation maps at all the the nodes $D$
of $C$.

###### Lemma 4.12

For generic $V$-compatible perturbations $(J,\nu)$, the dimension of the
inverse image of the antidiagonal $\Delta_{\pm}$ under the enhanced evaluation
map (4.50) is

$\displaystyle{\rm dim\;}\mbox{\rm Ev}_{x_{\pm}}^{-1}(\Delta_{\pm})={\rm
dim\;}{\cal M}_{s}(X,V)-2.$

Proof. By construction $f$ has at least one nontrivial component in level 1,
which means that the ${\mathbb{C}}^{*}$ action on the level one is without
fixed points. It is then straightforward to check that for generic
$V$-compatible perturbations $(J,\nu)$ the enhanced evaluation map (4.50) is
transverse to the antidiagonal (at least when assuming the domain $C$ is
stable, see Remark 2.1). The calculations in Lemma 4.7 together with the fact
that the enhanced matching conditions impose an extra $2k(x)-2$ dimensional
condition for each node imply immediately the result. $\Box$

Unfortunately, in the presence of a depth $k\geq 2$ point, we cannot rescale
the target such that the limit $f$ has no components in the singular locus
$W_{1}$. The most we can do is to make sure it has no nontrivial components
there, but at the price of getting several trivial components stuck in the
singular divisor. Below are a couple of simple examples that illustrate this
behavior.

###### Example 4.13

Consider the situation of Example 3.7. Assume $f$ is a fixed stable map into
$X$ which has two contact points $x_{1}$ and $x_{2}$ with $V$, both mapped
into the singular locus $p$ of $X$, but such that $x_{1}$ has multiplicity
(1,1) while $x_{2}$ has multiplicity (1, 2) to the two local branches of $V$
at $p$. This means that in local coordinates $z_{i}$ around $x_{i}$ on the
domain and $u_{1},u_{2}$ on the target around $p$, the map $f$ has the
expansions

$\displaystyle f(z_{1})=(a_{11}z_{1},\;a_{21}z_{1})\qquad
f(z_{2})=(a_{12}z_{2},\;a_{22}z_{2}^{2})$

with finite, nonzero leading coefficients $a_{ij}$. Now add another marked
point $x_{0}$ to the domain. As either $x_{0}\rightarrow x_{1}$ or
$x_{0}\rightarrow x_{2}$ a constant component of $f$ is falling into $p$.

Let’s look at the case $x_{0}\rightarrow x_{1}$. Assume $x_{0}$ has coordinate
$z_{1}=\varepsilon$ so $f(x_{0})=(a_{11}\varepsilon,\;a_{21}\varepsilon)$ and
$\varepsilon\rightarrow 0$. Following the prescription of [IP2], we need to
rescale the target by $\lambda=\varepsilon$ to catch the constant component
falling in. So in coordinates $u_{11}=u_{1}/\lambda$ and
$u_{21}=u_{2}/\lambda$ we get

$\displaystyle
f_{1\lambda}(z_{1})=(a_{11}z_{1}/\lambda,a_{21}z_{1}/\lambda)\qquad
f_{2\lambda}(z_{2})=(a_{12}z_{2}/\lambda,a_{22}z_{2}^{2}/\lambda)\ $

In the domain, when we let $w_{1}=z_{1}/\varepsilon=z_{1}/\lambda$ then
$f_{1\lambda}$ converges to a level one nontrivial component which in the
coordinates $u_{11}$ and $u_{21}$ has the expansion

$\displaystyle f_{1}(w_{1})=(a_{11}w_{1},\;a_{21}w_{1})$

This component lands in $\mathbb{F}_{2}$ and contains the marked points
$x_{0}$ and $x_{1}$ (with coordinates $w_{1}=1$ and respectively $w_{1}=0$),