# Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program

Lawrence J. Barrott  Charles F. Doran111The second author was supported by the
Natural Sciences and Engineering Research Council of Canada (NSERC)

###### Abstract

The Doran-Harder-Thompson “gluing/splitting” conjecture unifies mirror
symmetry conjectures for Calabi-Yau and Fano varieties, relating fibration
structures on Calabi-Yau varieties to the existence of certain types of
degenerations on their mirrors. This was studied for the case of Calabi-Yau
complete intersections in toric varieties in [12] for the Hori-Vafa mirror
construction. In this paper we prove one direction of the conjecture using a
modified version of the Gross-Siebert program. This involves a careful study
of the implications within tropical geometry and applying modern deformation
theory for singular Calabi-Yau varieties.

Mirror symmetry started with the study of Calabi-Yau manifolds, predicting a
duality associating to any Calabi-Yau $X$ a mirror partner $\check{X}$. As it
grew it came to incorporate other statements, most notably it predicts a
similar duality associating to any Fano variety and anti-canonical divisor
$(Z,D)$ a Landau-Ginzburg model, a variety $\check{W}$ and a super-potential
$w:\check{W}\to\mathbb{C}$. The theory of Landau-Ginzburg models was
formalised in [1] where these were required to be a fibration of $\check{X}$
by lower dimensional Calabi-Yau varieties.

The two worlds, Calabi-Yau and Fano, meet in the example of Tyurin
degenerations. A Tyurin degeneration is a semi-stable degeneration
$\mathfrak{X}_{t}\to\Delta$ such that the central fibre is the union of two
smooth components $\mathfrak{X}_{0}=\mathfrak{Z}_{1}\cup\mathfrak{Z}_{2}$
meeting along a smooth divisor $\mathfrak{D}$. From the work of Kawamata-
Nakayama $\mathfrak{D}$ is an effective anti-canonical divisor on the
components $\mathfrak{Z}_{i}$, and the normal bundles are dual,
$\mathcal{N}_{\mathfrak{D}/\mathfrak{Z}_{0}}\otimes\mathcal{N}_{\mathfrak{D}/\mathfrak{Z}_{0}}\cong\mathcal{O}_{\mathfrak{D}}$,
and this is a necessary and sufficient condition for such smoothings to exist.
This was expanded on by Kato in [13] where he proved that this condition is
equivalent to the existence of a log structure of semi-stable type (in modern
language locally free) on the central fibre, log smooth over the standard log
point. This generalises the Kawamata-Nakayama condition since there are
surgeries permitted by log geometry which produce central fibres necessarily
smoothable but not of the type they consider, where the central fibre is a
chain of components. The most basic example would be to blow up $D$ inside the
total space, to obtain a smoothable central fibre with three smooth
components, $\mathcal{Z}_{0}\cup\mathcal{Z}_{1}\cup\mathcal{Z}_{2}$ meeting
along two copies of $\mathcal{D}$, say $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$. Note that it is not true that we have a balancing condition
$\mathcal{N}_{\mathcal{D}_{1}/\mathcal{Z}_{0}}\otimes\mathcal{N}_{\mathcal{D}_{1}/\mathcal{Z}_{1}}\cong\mathcal{O}_{\mathcal{D}_{1}}$,
instead we only have
$\mathcal{N}_{\mathcal{D}_{1}/\mathcal{Z}_{0}}\otimes\mathcal{N}_{\mathcal{D}_{1}/\mathcal{Z}_{1}}\otimes\mathcal{N}_{\mathcal{D}_{2}/\mathcal{Z}_{1}}\otimes\mathcal{N}_{\mathcal{D}_{2}/\mathcal{Z}_{2}}\cong\mathcal{O}_{\mathcal{D}}$.

We will study higher order degenerations extending the definition of Tyurin
degenerations. This produces a notion of higher rank Landau-Ginzburg model,
appearing in situations where we have not a smooth divisor pair, but an snc
divisor pair.

######  Definition 0.1.

Let $\mathfrak{X}\to\mathbb{A}^{1}$ be a family such that $\mathfrak{X}_{gen}$
and $\mathfrak{X}$ are both smooth and the central fibre is an snc divisor
whose smallest strata is codimension $k$. We call such families $k$-Tyurin
degenerations, in analogy to type $k$ degenerations of Calabi-Yau varieties.

Let $\mathcal{S}$ be a smooth variety of dimension $k$ and
$w:\mathfrak{W}\to\mathcal{S}$ a proper family with $\mathfrak{W}$ smooth and
generic fibre a smooth Calabi-Yau. We call such a structure a rank $k$ Landau-
Ginzburg model.

There is relatively little literature on higher rank LG models. A sensible
restriction is to look at those families such that $\mathcal{S}$ admits an snc
compactification and such that $\mathfrak{W}$ admits a relatively log smooth
compactification whose fibres over an $n$ dimensional strata of $\mathcal{S}$
has a strata of dimension $n$. This recreates the expectation that the
compactification of an LG model has a MUM point at infinity. We will see how
this condition naturally arises. For the remains of the introduction we
exposit the Tyurin case where $k=1$.

Under these conditions we can form the mirror to $\mathfrak{X}_{gen}$,
$\widecheck{\mathfrak{X}_{gen}}$, and the mirror to the pairs
$(\mathfrak{Z}_{0},\mathfrak{D})$ and $(\mathfrak{Z}_{1},\mathfrak{D})$,
$W_{0}:\widecheck{\mathfrak{Z}}_{0}\to\mathbb{C}$ and
$W_{1}:\widecheck{\mathfrak{Z}}_{1}\to\mathbb{C}$. One direction of the Doran-
Harder-Thompson conjecture is the statement that
$\widecheck{\mathfrak{X}_{gen}}$ is fibred over $\mathbb{P}^{1}$, which we
prove given a compatible toric degeneration:

######  Conjecture 0.2 (“Degenerations to Fibrations”).

[Corollary 2.14] If $\mathfrak{X}$ admits a Tyurin degeneration to
$\mathfrak{Z}_{0}\cup_{\mathfrak{D}}\mathfrak{Z}_{1}$ then the mirror
$\widecheck{\mathfrak{X}_{gen}}$ admits a map to $\mathbb{P}^{1}$ with fibres
smooth Calabi-Yau varieties of one dimension lower and further $\mathfrak{D}$
is mirror to a generic fibre of this fibration.

Our proof naturally extends to $k$-Tyurin degenerations, and demonstrates that
they are fibred over a $k$-dimensional base.

A reconstruction theorem of the following flavour was found to hold in [11],
[21]. These cases have the property that the fibres have at most one
dimensional complex moduli, and this seems to be a necessary condition to have
a holomorphic gluing, forming a second part of the conjecture:

######  Conjecture 0.3 (“Reconstruction via gluing”).

[Theorem 3.8] Suppose that the fibres of
$\widecheck{\mathfrak{Z}}_{0}\to\mathbb{A}^{1}$ have one dimensional complex
moduli. Then the Calabi-Yau $\widecheck{\mathfrak{X}_{gen}}$ is obtained by
gluing the LG models $\widecheck{\mathfrak{Z}}_{0}$ to
$\widecheck{\mathfrak{Z}}_{1}$ along the fibres over
$\mathbb{C}^{*}\subset\mathbb{C}$ under a deformation of the map $t\mapsto
t^{-1}$.

This conjecture has geometric interpretations on the level of periods for
Calabi-Yau complete intersections in toric varieties using techniques from
[33]. As stated the conjecture is agnostic of how we construct the mirror, one
typically uses the Batyrev-Borisov construction of [34] or the Hori-Vafa
construction of [19]. We will use an older version of the Gross-Siebert
program, which assumes the existence of a toric degeneration:

######  Theorem 0.4.

Suppose that $\mathfrak{X}\to\mathbb{A}^{1}$ is a simple toric degeneration.
Then the mirror to $\mathfrak{X}_{gen}$ is given by an explicit log smoothing
of a combinatorially determined central fibre. It is a smoothing of a singular
space over a monoid ring of effective curve classes in $\mathfrak{X}$.

We define a compatibility of toric degenerations and Tyurin degenerations, and
in general between toric and type $k$ degenerations with smooth total space.
From this we find a fibration of a component of the mirror before we smooth.
It is not clear at the moment how to simultaneously smooth both
$\widecheck{\mathfrak{X}_{0}}$ and a Calabi-Yau subvariety inside it purely
using the techniques of the program. The algebra map on the central fibre is
clear, but there are quantum corrections that occur during the smoothing
procedure which we cannot control.

Thankfully there has been a large amount of progress in the past year on how
to smooth log Calabi-Yau varieties using scattering diagrams. For us the most
notable are recent advances first by Chan-Leung-Ma in [5] and then by Felten-
Filip-Ruddat in [31]. These perturb the Maurer-Cartan equation in an iterative
way to construct a formal, and then analytic, smoothing. The problem of
smoothing complexes was studied in [7] where the authors give a smoothing
criterion for locally free complexes, which is not satisfied by our examples.
Following classical deformation theory we prove the smoothability of a Calabi-
Yau variety and a Calabi-Yau subvariety with trivial normal bundle by directly
relating the BV-algebras used in the correction process. This also shows that
if the normal bundle is not trivial then the subspace is at least stable. This
result is perhaps of independent interest for constructing subvarieties
tropically:

######  Theorem 0.5 (Theorem 2.12).

Let $\mathfrak{X}$ be a log toroidal family over
$\operatorname{Spec}k^{\dagger}$, and $\mathfrak{D}\subset\mathfrak{X}$ a
strictly embedded complete intersection Calabi-Yau subvariety transverse to
the singularities of the log structure on $\mathfrak{X}$. Then there are
smoothings of the pair $\mathfrak{X},\mathfrak{D}$ over
$\operatorname{Spec}k[\\![t]\\!]$

This statement could be proved by other means, showing that the normal bundle
in this case describes locally isotrivial smoothings. Our eventual goal would
be to construct a scattering process that extends this result to the intrinsic
construction of [25]. This requires that we study an appropriate differential
graded Lie algebra (dgLa) via this more complicated machinery.

Finally we apply relative deformation theory to the interior of the LG models
and the fibration. This process only applies on the level of formal geometry,
as relative deformation theory over an affine base is not very well behaved in
the algebraic or analytic categories. We prove that
$\widecheck{\mathfrak{X}_{0}}$ restricted to the interior
$\mathbb{G}_{m}\subset\mathbb{A}^{1}$ is a versal deformation space for
families of $\widecheck{\mathfrak{D}_{0}}$-bundles under a surjectivity
assumption that can be checked on the mirror. If
$\widecheck{\mathfrak{D}_{0}}$ has one dimensional deformation space this
allows us to produce an open formal embedding of $\mathfrak{W}_{0}$ into
$\mathfrak{X}$.

In the case of $k$-Tyurin degenerations the analogous statement requires a
surjectivity $H^{1}(\Theta_{\widecheck{\mathfrak{X}}})\to
H^{1}(\Theta_{\widecheck{\mathfrak{D}}})$. For the simple toric degenerations
we consider this can be studied on the mirror.

#### Comments on restrictions

There are four comments we must make about this work. Firstly we do not know
how to construct a quasi-Fano variety directly from a Landau-Ginzburg model.
There is the work of Prince in [28] which has partial solutions but this does
not apply in the generality we are looking for. This makes the converse of the
theorem impossible to resolve using these techniques in complete generality.

On the other hand the statement that the mirror to a fibred Calabi-Yau admits
a Tyurin degeneration is tractable. In this case there is a canonical affine
direction on the base of the mirror, corresponding to pairing with a fibre
class. Once one smooths perpendicularly to this one should obtain a log scheme
which is log smooth over the standard log point, which by [13] is enough to
produce a Tyurin degeneration. This requires a much stronger analysis of the
local structure of partial smoothings of the log structure than appears in say
[17].

The third comment is that we are restricted here to working in the case where
$\mathfrak{X}$ is not dimension two. This is to avoid a polarisation issue
that is intrinsic to the geometry of $K3$ surfaces. So long as the
tropicalisation $\Sigma(\mathfrak{X})$ has vanishing $H^{2}$ every choice of
open gluing data for the fan picture gives rise to a choice of gluing data for
the cone picture and our construction is written directly in terms of the cone
picture. A more detailed study of the $K3$ surface case and its subtleties
will be performed in [15].

Since there are already subtleties of mirror symmetry for $K3$ surfaces that
have not been addressed in the wider literature we pass on addressing this.

The final comment is that it is possible to prove that the mirror is fibred
directly from the recent paper [25]. Unfortunately it is not immediately clear
what it is fibred by. Essentially the toric degeneration assumption here
forces enough local rigidity to allow us to study the scattering diagram
through other deformation theoretic means. Historically this has precedent,
the work of [25] came only after the work of [17]. In future work we intend to
use the gluing calculations of [9] and [35] and the period calculations of
[29] to prove the period gluing formulae of [12] from the perspective of
tropical curves, even though we do not know a precise method to prove that
fibres are mirror to $\mathfrak{D}$.

#### Acknowledgement

We thank Mark Gross, Jordan Kostiuk, Helge Ruddat, Bernd Siebert, Alan
Thompson, Yixian Wu and Fenglong You for helpful discussions during the
development of this project. L. J. B. was supported by Boston College. C. F.
D. was supported by the University of Alberta, the Center for Mathematical
Sciences and Applications at Harvard University and NSERC.

##  1 From the general fibre inwards

We begin with an adapted version of toric Calabi-Yau degenerations to provide
compatibility between toric and Tyurin degenerations.

######  Definition 1.1.

Let $\mathcal{S}=\operatorname{Spec}k[\\![t_{0},t_{1}]\\!]$ be the formal
spectrum of a two dimensional local ring with the log structure induced by the
divisors $t_{0}$ and $t_{1}$. A toric connected $k$-Tyurin degeneration over
$\mathcal{S}$ is a proper normal log algebraic space over flat over
$\mathcal{S}$ with the following properties:

  1. (i)

The generic fibre $\mathfrak{X}_{\eta}$ is an irreducible normal variety.

  2. (ii)

The fibre over $t_{1}=0$ is reducible and the components are in bijection with
the components over a closed point $t_{0}=a\neq 0$.

  3. (iii)

The restriction of $\mathfrak{X}$ to the open subscheme
$\operatorname{Spec}k((t_{0}))[\\![t_{1}]\\!]$ is a $k$-Tyurin degeneration.
In particular the closed fibre of this family
$\operatorname{Spec}k((t_{0}))[\\![t_{1}]\\!]/\langle t_{1}\rangle$ contains a
minimal smooth strata $\mathfrak{D}_{gen}$ of codimension $k$, the
intersection of components $\mathfrak{Z}^{0}_{gen}$ up to
$\mathfrak{Z}^{k}_{gen}$. These have well defined boundary divisors, which we
write $\partial\mathfrak{Z}^{i}_{gen}$

  4. (iv)

The central fibre $\mathfrak{X}_{0}$ satisfies the assumptions (2) and (3) of
[24] Definition 4.1.

  5. (v)

The map $\mathfrak{X}\to\mathcal{S}$ is a saturated and log smooth morphism
away from a set $\mathfrak{S}\subset\mathfrak{X}$ which is relative
codimension $\leq 2$ and does not contain any toric strata of the central
fibre $\mathfrak{X}_{0}$.

The fact that the general fibre is a $k$-Tyurin degeneration forces a very
particular structure on the log structure, at least near points of
$\mathfrak{D}$.

######  Lemma 1.2.

There are charts for the log structure on $\mathfrak{X}$ in étale
neighbourhoods of the points of the closure of $\mathfrak{D}_{gen}$ of the
form:

$t_{1}\mapsto p\quad t_{2}\mapsto y_{0}\ldots y_{k}$

for $p,y_{0},\ldots,y_{k}$ generating a monoid subring and such that
$y_{0},\ldots,y_{k}$ are irreducible elements cutting out
$\mathfrak{Z}_{gen}^{0}$ up to $\mathfrak{Z}_{gen}^{k}$ generating an
$\mathbb{N}^{k}$ submonoid and not dividing $p$.

###### Proof.

Since $\mathfrak{X}\to\mathcal{S}$ is generically log smooth we can find
charts:

${\mathfrak{X}}$${\operatorname{Spec}k[P]\times_{\operatorname{Spec}k[\mathbb{N}^{2}]}\mathcal{S}}$${\operatorname{Spec}k[P]}$${\mathcal{S}}$${\operatorname{Spec}k[\mathbb{N}^{2}]}$$\scriptstyle{\phi}$

where the map $\mathbb{N}^{2}\to P$ is an inclusion, $\phi$ is smooth and
$t_{1}$ and $t_{2}$ correspond to the monomials $(1,0)$ and $(0,1)$
respectively and $\mathcal{S}\to\operatorname{Spec}k[\mathbb{N}^{2}]$ is
smooth. Suppose that we can write $f^{\\#}(t_{1})$ as a sum of irreducible
elements $\sum a_{i}p_{i}$. These irreducible elements give components of the
fibre over $t_{2}\neq 0$, with $a_{i}$ being the generic length of the
subscheme $t_{i}=0$ over each component. But we assumed that on this locus the
degeneration had smooth total space, which enforces the above description of
the monoid map, where $y_{i}$ is the monomial $z^{p_{i}}$.

Now if $p$ were divisible by any $p_{i}$ then there would be an entire divisor
contained in the fibre over $t_{1}=t_{2}=0$. But by assumption $f$ is flat.
The ring elements $y_{i}$ are the pullbacks of $p_{i}$.

∎

######  Definition 1.3.

We set $\mathfrak{D}$ to be the subscheme of $\mathfrak{X}$ obtained locally
by the equations $y_{i}=0$ for all $i$. We write $\mathfrak{Z}^{i}$ for the
subscheme obtained locally by the equation $y_{i}=0$.

We need to further restrict, so that the central fibre has degenerate
properties similar to a Tyurin degeneration. For us this means that
$\mathfrak{D}$ is a normal integral scheme, and this is satisfied so long as
$\mathfrak{X}$ has property $S_{k+1}$ near $\mathfrak{D}$, in particular if
the total space of the degeneration is Cohen-Macaulay since we know
$\mathfrak{D}$ is $R_{0}$ and $R_{0}$ plus $S_{1}$ implies reduced. This is
also satisfied for our examples arising as complete intersections in toric
varieties since toric varieties are always Cohen-Macaulay.

######  Definition 1.4.

We say that $\mathfrak{X}$ is a toric $k$-Tyurin degeneration if the induced
toric degeneration of $\mathfrak{D}$ has normal integral total space and the
Tyurin singular locus $\mathfrak{D}$ is transverse to the singular locus of
the total space. We say that $\mathfrak{Z}^{i}$ is a _toric degenerating
component_ if the induced degeneration of $\mathfrak{Z}^{i}$ is toric.

Many of the schemes appearing here are Cohen-Macaulay where being normal is
equivalent to being regular in codimension one, which is substantially easier
to check, we will use this observation without mention.

Taking the fibre of $\mathfrak{X}$ over the closed subset $t_{0}=a\>t_{1}$ for
generic choice of $a$ (which we may take to be 1) we obtain a toric
degeneration in the sense of [24] Definition 4.1 over a DVR. We write this
degeneration $\mathfrak{X}_{\Delta}$. We can also extract from this the data
of a toric degeneration of the components of the Tyurin degeneration.

######  Lemma 1.5.

The scheme $\mathfrak{D}$ is a toric degeneration under the induced maps to
$V(t_{0}-a\>t_{1})$.

###### Proof.

Condition (1) of [24] Definition 4.1 is trivial, the generic fibre
$\mathfrak{D}_{gen}$ is smooth.

To prove condition (2) we have to recall the construction and definition of
the normalisation from [32]. We proceed in two steps. The first restricts to
the vanishing of one of the $y_{i}$, which interacts with the normalisation in
an essentially trivial way. The second step is then a stability under passing
along a regular sequence.

We fix $y_{1}$ say, and consider the associated degeneration of
$\mathfrak{Z}_{1}$. The central fibre $\mathfrak{X}_{0}$ has normalisation the
disjoint union of toric varieties $X_{\Xi_{i}}$ for $i\in I$ and the central
fibre of $\mathfrak{Z}_{i}$ selects a subset $J\subset I$ of the components.
The connected and irreducible components $X_{\Xi_{i}}$ are in bijection with
the irreducible components of $\mathfrak{X}_{0}$ via a birational map with
function field $K_{i}$. $X_{\Xi_{i}}$ is locally the integral closure of the
corresponding component of $\mathfrak{X}_{0}$ inside the function field
$K_{i}$.

By the universal property of[32, Lemma 035Q] the normalisation of
$\mathfrak{Z}$ is the disjoint union $\coprod_{j\in J}X_{\Xi_{j}}$. The
conductor locus is precisely the union of the toric Weil divisors not lying on
an intersection between the components of $\mathfrak{Z}_{1}$. In particular it
is reduced, the map $C_{\mathfrak{Z}_{1}}\to\nu(C_{\mathfrak{Z}_{1}})$
unramified and generically two-to-one. That the square:

${C_{\mathfrak{Z}_{1}}}$${\prod_{j\in
J}X_{\Xi_{j}}}$${\nu(C_{\mathfrak{Z}_{1}})}$${\mathfrak{Z}_{1}}$

is Cartesian follows by the description of the conductor ideals appearing
in[32, Lemma 035Q] and co-Cartesian since all the objects are reduced, and
embed into the corresponding diagram for $\mathfrak{X}$. Hence it is enough to
see that they have the same closed points as the desired pushout, which they
do by construction.

Condition (3) follows from firstly the fact that $\mathfrak{D}_{0}$ is a
complete intersection inside a Gorenstein scheme, hence is Gorenstein, using
theorem 1 of [22]. The second statement follows from adjunction applied to the
sequence of intersections $V(y_{1}),V(y_{1},y_{2}),\ldots,\mathfrak{D}_{0}$.

Condition (4) is precisely encoded by our assumptions on the structure of the
log morphism $\mathfrak{X}\to\mathcal{S}$.

∎

The final restriction that we require is that the family be simple. In the
context of the Gross-Siebert program this is a monodromy requirement on the
affine manifold. It constrains the monodromy polytopes associated to each
strata to be elementary simplices, so the only integral points are the
vertices.

######  Definition 1.6.

A toric $k$-Tyurin degeneration is called _simple_ if the induced degeneration
of $\mathfrak{X}$ is simple.

We could not answer satisfactorily if the induced degeneration of
$\mathfrak{D}$ is necessarily also simple. If it is then we know that our
construction here agrees with that appearing in [24]. If it is not then we
will see that it still carries a simple and generically log smooth log
structure by restriction, and so we did not pursue this. All the examples we
give of toric complete intersections indeed have this property.

Of course one is interested in providing examples which satisfy these
conditions. The easiest examples of Tyurin degenerations of Calabi-Yau
varieties are obtained as a $(k+1,2)$ hypersurface in
$\mathbb{P}^{k}\times\mathbb{P}^{1}$ and applying the subdivision techniques
of [20]. This corresponds to the polytope $(k+1)\Delta^{k}\times 2\Delta^{1}$,
the product of dilations of standard simplices.

######  Example 1.7.

We partially follow [14] and look for a product of MPCP desingularizations of
the total family. Fix an MPCP subdivision of
$(k+1)\Delta^{k}\subset\mathbb{R}^{k}$ with associated strictly convex
function $h_{\mathbb{P}^{k}}$ and we implicitly pull this back to
$\mathbb{R}^{k}\oplus\mathbb{R}$. On the other hand we have the piecewise
linear function on $\mathbb{R}$ given by $h_{\mathbb{P}^{1}}=\max(0,x)$ where
$x$ is a coordinate on the final factor, giving an MPCP desingularization of
the polytope for $\mathbb{P}^{1}$. Again we use the same notation for the
pullback. The sum
$h=h_{\mathbb{P}^{k}}+h_{\mathbb{P}^{1}}+\phi_{\mathbb{P}^{k}\times\mathbb{P}^{1}}$
gives a polytopal decomposition of $(k+1)\Delta^{k}\times 2\Delta^{1}$
together with a strictly convex piecewise linear function on the refinement,
such that $h-\phi_{\mathbb{P}^{k}\times\mathbb{P}^{1}}$ is nef. Rather than
taking a one parameter graph to obtain a log smooth map to $\mathbb{A}^{1}$ we
take the two dimensional graph over $(h_{\mathbb{P}^{k}},h_{\mathbb{P}^{1}})$,
following section 4 of [20] and for the same reasons outlined in example 4.2
of [24] or theorem 3.10 of [14] to obtain a generically log smooth morphism to
$\mathbb{A}^{2}$ throughout. The restriction of the two parameter degeneration
to the diagonal is by construction the single parameter degeneration
associated to $h$. On the dual side we choose the data of any MPCP
desingularization none of whose faces have interior intersecting the
hyperplane $H$ given by the vanishing of the final coordinate. This is not
quite the data of an MPCP desingularization on $(k+1)\Delta^{k}\times
2\Delta^{1}$, there are some “square” polytopes appearing.

Nonetheless we follow the proof of simplicity given in Theorem 3.17 of [14],
translated back to be working on $\Delta$ rather than $\nabla$. First we note
that the local monodromy description the simplifies the barycentric
subdivision description of the singular locus, it is supported on the set

$\mathcal{S}ing((k+1)\Delta^{k+1})\times[0,1]\cup(k+1)\Delta^{k+1}\times\\{-1/2,1/2\\}$

In this decomposition there are four types of $\tau\in\Delta^{\prime}$. The
types are:

  1. (i)

Faces intersecting the interior of the the top and bottom copies of
$(k+1)\Delta^{k}$.

  2. (ii)

Faces contained inside the boundary of the top and bottom copies of
$(k+1)\Delta^{k}$.

  3. (iii)

Faces of the side which are horizontal contained inside the vanishing of the
final coordinate.

  4. (iv)

Faces of the side which are vertical projecting to the boundary of
$(k+1)\Delta^{k}$.

The first type are locally given by an MPCP desingularization and in
particular are guaranteed to be simple. The second and third types of faces
contain no vertical edges, and any horizontal codimension one component can be
ignored when constructing the choices of $\Omega_{i}$ since there is not any
monodromy around them. In the third case this reduces the description of the
monodromy to that induced by the MPCP desingularization given above,
restricted to the boundary, which remains MPCP.

The second type has two types of divisors, the horizontal divisors of
$(k+1)\Delta^{k}$ and the vertical fibres over codimension two strata in the
decomposition of $(k+1)\Delta^{k}$. A local computation shows that the
monodromy is only non-trivial for the vertical divisors contained inside the
boundary, all the other loops occur entirely inside the interior of each face
where there is trivial monodromy. Then we are back in the above situation.

The final type of faces are those which are vertical. These are not contained
inside any horizontal divisor, and hence we may ignore horizontal edges in the
calculation of the monodromy since they will be monodromy invariant. The new
singularities that it meets have monodromy:

$n\mapsto n+\langle m,n\rangle\>u$

where $u$ is a unit vector in the vertical direction and $m$ is the normal
vector to the vanishing of $x$ in the $u$-direction. It is clear from this
description that the associated monodromy polytopes are elementary simplices.

In the case of a degeneration of a $K3$ surface embedded as an anti-canonical
hypersurface inside $\mathbb{P}^{2}\times\mathbb{P}^{1}$ we can draw
explicitly the picture, seen in the figure below with the degenerating
hyperplane dotted, and the 16 visible singularities of the 24 focus-focus
singularities are marked in red. It is clear that the induced subdivision is
simple, although we avoid specifically this surface case later:

This is the limit of the cases where explicit examples are tractable, and in
dimension higher than two computer algebra packages would be needed to attempt
any calculations. For a general refinement of a nef partition there are
choices of lifts involved, we handle the symmetric blow-up case from [12]
below, along with a non-symmetric example for the quintic threefold. For this
we need to use an actual MPCP desingularization, but one well adapted to the
choice of resolution.

######  Theorem 1.8.

Let $P\subset M$ be a reflexive polytope with a nef partition
$P=\sum_{0}^{n}\Delta_{i}$ and let $\Delta_{0}=\Delta_{a}+\Delta_{b}$ be a
refinement of this nef partition. Let $\tilde{P}$ be the polytope associated
to blowing up $\Delta_{a}$ and $\Delta_{b}$, and $\tilde{\Delta_{i}}$ the
associated partitions of this polytope.

Then there is a simple toric-Tyurin degeneration associated to this data, with
generic fibre a complete intersection in $P$ given by the nef partition
$\tilde{\Delta}_{a},\tilde{\Delta}_{b},\tilde{\Delta}_{1},\ldots,\tilde{\Delta}_{n}$.

###### Proof.

The degeneration may be embedded into the total space associated to
$P\times[-1,1]$ embedded inside $M\oplus\mathbb{R}$ and we write $H$ for the
hyperplane given by the vanishing of the final coordinate. The associated
refined polytopes $\Delta_{i}$ are the trivial product
$\Delta_{i}\times\\{0\\}$ except for $\tilde{\Delta}_{a},\tilde{\Delta}_{b}$
and $\tilde{\Delta}_{0}$. The polytope $\tilde{\Delta}_{a}$ is the product
$\Delta_{a}\times[0,1]$ with $\tilde{P}$ while $\tilde{\Delta_{b}}$ is the
product $\Delta_{b}\times[-1,0]$. The intersection with $H$ gives a reflexive
polytope of dimension one lower, in which the degeneration for $\mathfrak{D}$
is embedded.

Choose an MPCP desingularization $(\Sigma^{\prime},h)$ of which none of the
interiors of the maximal polytopes intersect $H$. This can be obtained by
taking any MPCP desingularization and taking the Minkowski sum of the dual
polytope to $h$ and a large multiple of the degenerate polytope
$\\{(0,1),(0,-1)\\}$. This produces a new strictly convex function on a new
refinement of $\Sigma$, $\tilde{\Sigma}$. There are vertical edges under the
projection to the final factor and a small perturbation horizontally of these
ensures that $\tilde{\Sigma}$ is an elementary triangulation of the
intersection with $H$ as proved in [33]. A perturbation of the points not
lying as end points of the vertical edges then ensures that the rest of the
triangulation is elementary. We replace $h$ and $\Sigma^{\prime}$ by these
choices.

Then we form the two parameter family given by taking the graph over
$h^{\prime}$ constructed in [14] and $h_{\mathbb{P}^{1}}$. This gives a log
smooth degeneration of the total space. Restricting this to the diagonal we
obtain the MPCP desingularization given by $h^{\prime}+h_{\mathbb{P}^{1}}$.
The construction of the complete intersection is modified to produce a semi-
stable degeneration given by $h^{\prime}$ as done in section 4 of [20]. The
induced two parameter family then has the above toric degeneration properties
by the same arguments that the one parameter family does. The induced
degeneration of $\mathfrak{D}$ can already be seen to be simple, induced by
the toric degeneration obtained by intersecting the above data with the
hyperplane $H$. The MPCP desingularization restricts on each side of $H$ to
give compatible MPCP desingularizations.

The piecewise linear functions $\phi_{*}$ are defined to be the maximum over
the points in the corresponding dual polytope. By construction these are
linear across $H$ and so there is no monodromy across these faces. Thus the
Tyurin singular locus $\mathfrak{D}$ is transverse to the singular locus of
the degenerating family.

∎

######  Example 1.9.

Let $Q_{t}\subset\mathbb{P}^{4}$ be a family of generic quintic threefolds
degenerating to the union of a degree $i$ and a degree $5-i$ hypersurface. We
construct the family produced by blowing up the degree $i$ hyperplane.

Recall from [20] the construction of degenerations of toric varieties given by
subdividing polytopes. The polytope $\Delta$ for $\mathbb{P}^{4}$ is of course
given by the convex hull of the columns of the following matrix contained in
the vector space spanned by $u_{1},\ldots,u_{4}$:

$\begin{bmatrix}-1&&4&&-1&&-1&&-1\\\ -1&&-1&&4&&-1&&-1\\\ -1&&-1&&-1&&4&&-1\\\
-1&&-1&&-1&&-1&&4\\\ \end{bmatrix}$

We subdivide this polytope along the hyperplane $H$ given by $u_{1}=i-1$ and
take the piecewise linear function $\min(0,i-1-u_{i})$. The associated
degeneration of the quintic is into a degree $5-i$ hypersurface and the blow
up of a degree $i$ hypersurface, recalling the description of the fan for the
total space of a projective bundle from Example 7.3.5 of [8] and the
classification of section 4.2 of [20]. It carries an embedding into a non-
compact toric variety with polytope $\tilde{\Delta}$. The polytope
$\tilde{\Delta}$ further carries a piecewise linear function $\phi$ pulled
back from $\Delta$ which generically fibrewise gives a quintic hypersurface
inside $\mathbb{P}^{4}$. One can use an auxiliary choice of toric
compactification and the proof of [33] to construct an MPCP desingularization
of $\Delta$ whose non-linear locus contains $H\cap\partial\Delta$. This
condition ensures that it pulls back to give an MPCP desingularization of the
bottom face of $\tilde{\Delta}$, though possibly with singularities along the
side. As in the previous two examples gives a two parameter family log smooth
over $\mathbb{A}^{2}$, where one factor is given by the new coordinate
introduced by Hu’s construction (section 4.1.2 of [20]) and the other factor
is the toric degeneration parameter (the section after Observation 3.9 of
[14]). Since $\phi$ is linear across $H$ the Tyurin singular locus is
transverse to the singularities of the total space. It is a toric-Tyurin
degeneration for the same reasons that the one parameter family is a toric
degeneration. The corresponding degeneration of $\mathfrak{D}$ is given by
restricting the polytope to the preimage of $H$, in particular it is a toric
degeneration.

Note that this is a slightly different construction from the one appearing in
[12] which makes explicit the target in which the degeneration is embedded.
The above example also produces torically degenerating components of the two
components of the Tyurin degeneration, fitting into the examples considered in
our third section.

Examples of toric $k$-Tyurin degenerations can be constructed as a
$(k+1,2,\ldots 2)$ hypersurface in
$\mathbb{P}^{k}\times\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}$, or as
more complicated subdivisions arising in the above example. It seems to be an
essentially hard problem to tell if a given Tyurin degeneration, connected in
moduli to a large complex structure limit point, admits such a toric model.
This is similar to the case for toric degenerations arising in the Gross-
Siebert program more generally.

We want to deduce statements via tropical geometry, and so we need to recall
the definition of the dual intersection complex and tropicalisation.

######  Definition 1.10.

Let $\mathfrak{X}$ be a log scheme. Let $\mathcal{C}$ be the category whose
objects are in bijection with the log strata of $\mathfrak{X}$ and whose
morphisms are given by inclusions of strata. There is a functor sending each
object to the dual cone of the stalk of
$\overline{\mathcal{M}}_{\mathfrak{X}}$ over the corresponding strata and each
morphism to the corresponding inclusion of faces. The cone complex
$\Sigma_{\mathcal{C}}(\mathfrak{X})$ is defined to be the complex obtained by
this gluing data.

Given a map $\mathfrak{X}\to\operatorname{Spec}k^{\dagger}$ there is an
induced map $\Sigma_{\mathcal{C}}(\mathfrak{X})\to\mathbb{R}_{\geq 0}$ and the
_tropicalisation_ or _dual intersection complex_ , $\Sigma(\mathfrak{X})$ is
defined to be the fibre over $1$. At the moment it only carries an affine
structure on the interior of each cell. To provide an affine structure away
from a set of codimension two we equip this with a fan structure around each
zero-dimensional strata using the corresponding toric variety as done in the
proof of Proposition 4.10 of [24].

From this data and a choice of closed gluing data in the cone picture (see
[24] Definition 2.3) there is a projective scheme obtained by gluing spectra
of the affine tangent spaces, as outlined in [17]. We will write this
$\operatorname{Proj}k[\Sigma(\mathfrak{X})]$.

We can identify an embedding of dual intersection complexes of
$\Sigma(\mathfrak{D})$ into the dual intersection complex
$\Sigma(\mathfrak{X}_{\Delta})$ and by doing so build an explicit surjection
on the level of rings.

######  Lemma 1.11.

There is a closed embedding of $\Sigma(\mathfrak{D})$ into
$\Sigma(\mathfrak{X})$ inducing a surjection on integral tangent spaces and
compatible with the fan structures at zero-dimensional points of
$\Sigma(\mathfrak{D})$. Further for any toric degenerating component
$\mathfrak{Z}$ of the Tyurin degeneration there is an affine linear open
embedding of an open subset of
$\Sigma(\mathfrak{Z},\partial\mathfrak{Z})^{\circ}$ into
$\Sigma(\mathfrak{X}_{\Delta})^{\circ}$ compatible with the fan structure
around each point of $\Sigma(\mathfrak{Z},\partial\mathfrak{Z})$.

###### Proof.

We calculate at a point $x\in\mathfrak{D}$, comparing the local structure to
the local structure induced by treating these as points of $\mathfrak{X}$. The
let $P$ be the cone of $\mathfrak{X}$ at $x$, it admits a collection of maps
$\mathbb{N}^{k}\oplus\mathbb{N}\to P$. Taking the dual these give a collection
of maps $\rho_{\Sigma}:P^{\vee}_{\mathbb{R}}\to\mathbb{R}_{\geq
0}^{k}\oplus\mathbb{R}_{\geq 0}$. Let $+:\mathbb{R}_{\geq
0}^{k}\oplus\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}^{2}$ be the map sending
$(p_{1},\ldots p_{k},q)$ to $(\sum p_{i},q)$. The composite
$+\cdot\rho_{\Sigma}$ is the structure map to $\mathbb{R}_{\geq 0}^{2}$. By
construction the set $\Sigma_{\mathfrak{D}}$ is locally precisely the fibre of
$\rho_{\Sigma}$ over $(1,1,\ldots,1)$, whilst $\Sigma_{\mathfrak{X}}$ is
locally the fibre of $+\cdot\rho_{\Sigma}$ over $(1,1)$. This map is integral
affine linear, and the ordering of $p_{1},\ldots p_{k}$ is fixed since the
monodromy action on the divisors $\mathfrak{Z}^{i}$ is trivial.

This gives a description of $\Sigma_{\mathfrak{X}}$ as a fibration over a
$k+1$ dilation of a $k$-simplex, $\Delta_{x}$, with $\Sigma_{\mathfrak{D}}$
the fibre over the central point. As a warning we point out that this does not
need to be a trivial fibration, for instance it might rescale the integral
affine structure.

The statement for $\mathfrak{Z}$ is the following. At a point $x$ of
$\mathfrak{X}$ lying on $\mathfrak{Z}$ there is a canonical inclusion
$P_{\mathfrak{Z}}\to P_{\mathfrak{X}_{\Delta}}$ such that the image of the
elements $2p$ and $\sum y_{i}+p$ are equal (here it is important that we are
using $P_{\mathfrak{X}_{\Delta}}$ and not $P_{\mathfrak{X}}$). This
corresponds to a diagram on the duals:

${P_{\mathfrak{Z}}^{\vee}}$${P_{\mathfrak{X}_{\Delta}}^{\vee}}$${\mathbb{R}_{\geq
0}}$

The local structure of $\Sigma(\mathfrak{Z},\partial\mathfrak{Z})$ and
$\Sigma(\mathfrak{X}_{\Delta})$ is given as the fibre over $1$ of each of
these spaces. Since they are the same dimension and the map is a continuous
embedding it is locally an isomorphism. Now in a neighbourhood of each point
lying on $\mathfrak{Z}$ this map is in fact an isomorphism by a comparison of
the local fan structures. It extends then in a neighbourhood of
$\Sigma(\partial\mathfrak{Z})$, which by compactness we may take to include a
positive integral neighbourhood of each point. This gives the desired
statement.

∎

The reason that it is not a global isomorphism is because of the existence of
phenomena of the type pictured below:

$\Sigma(\mathfrak{X}_{\Delta})$$\Sigma(\mathfrak{Z},\partial\mathfrak{Z})$

The picture shows the conical complexes, together with the level set
corresponding to the tropicalisation. The tropicalisation of
$\Sigma(\mathfrak{X}_{\Delta})$ is only locally a closed subset, and it may
have some other components missing that do not appear in
$\Sigma(\mathfrak{Z},\partial\mathfrak{Z})$, but on the left hand side it is a
local isomorphism.

Associated to this is a choice to $0^{th}$ order of an embedding of
$\widecheck{\mathfrak{D}}$ into $\widecheck{\mathfrak{X}_{\Delta}}$.

######  Lemma 1.12.

There is an open subset of
$\operatorname{Proj}k[\Sigma(\mathfrak{X}_{\Delta})]$ fibred over
$\mathbb{G}_{m}^{k}$ with fibres inducing potentially different log structures
on $\operatorname{Proj}k[\Sigma(\mathfrak{D})]$. Over an open subset the
fibres have simple log structure in the sense of [16].

###### Proof.

Open gluing data for $\mathfrak{X}$ gives rise to open gluing data for
$\mathfrak{D}$ by restriction. In turn this gives cone gluing data compatibly
under restriction. Defining equations for
$\operatorname{Proj}k[\Sigma(\mathfrak{D}_{\Delta})]$ can locally be
constructed by pulling back the integral tangent vectors from $\Delta_{k}$
using the fact that $\Sigma(\mathfrak{D})$ is transverse to the walls of
$\Sigma(\mathfrak{X}_{\Delta})$. Locally this gives a choice of monomial along
each face, and any two choices differ by a function invertible on the polytope
cell connecting them. Given a choice of gluing data in the cone picture (as
defined in [24] Definition 2.3) on
$\operatorname{Proj}k[\Sigma(\mathfrak{X}_{\Delta})]$ we get induced gluing
data on $\operatorname{Proj}k[\Sigma(\mathfrak{D})]$ compatible with these
choices. Note that a monoid relation $m=0$ becomes after passing to the
associated monoid ring $z^{m}=1$, not $z^{m}=0$. Fix one chart corresponding
to a $k$-dimensional face of $\Sigma(\mathfrak{X}_{\Delta})$ corresponding to
a zero-dimensional strata of $\Sigma(\mathfrak{D})$ and a relation $z^{m}=1$,
twisting by the gluing data gives a collection of relations
$z^{m_{\sigma}}=a_{\sigma}$ in each chart compatible with one another under
restriction. Since the monodromy action on $\Sigma(\mathfrak{X})$ restricts to
the monodromy action on $\Sigma(\mathfrak{D})$ the associated sections of the
normal bundle are non-vanishing on $\Sigma_{\mathfrak{D}}$ and linearly
independent.

On an initial chart rather than taking $z^{m}=1$ as the defining relation one
can choose $z^{m}=a$ to obtain a different embedding. Doing this separately
for each of the $k+1$ generators gives a $\mathbb{G}_{m}^{k+1}$ worth of
choices, but a common rescaling identifies the associated ideals and so we get
the desired claim. A generic choice is transverse to the singular locus and
any other log strata and so the strict induced log structure is indeed simple
and generically log smooth.

∎

A similar statement holds for the LG model:

######  Lemma 1.13.

Let $\mathfrak{Z}_{i}$ be a toric degenerating component. There is an open
embedding of
$\operatorname{Proj}k[\Sigma(\mathfrak{Z}_{i},\partial\mathfrak{Z}_{i})]$ into
$\operatorname{Proj}k[\Sigma(\mathfrak{X})]$.

###### Proof.

This is a trivial consequence of the above open embedding on tropical spaces.

∎

##  2 Logarithmic deformation of pairs

We now bootstrap the smoothing results of [5] and [31] from smoothing Calabi-
Yau varieties to smoothing pairs of such varieties, and higher codimension.
This uses an explicit description of the canonical class to calculate how one
corrects the Maurer-Cartan equations and show compatibility. We should explain
that this is not a trivial application of deformation theory since none of the
deformations are locally trivial, and it applies in the case where
$\mathfrak{D}$ is an elliptic curve. The classical statement is the following:

######  Theorem 2.1.

Suppose that $(\widecheck{\mathfrak{X}},\widecheck{\mathfrak{D}})$ are a pair
with $\widecheck{\mathfrak{X}}$ a Calabi-Yau variety of dimension at least 3,
and $\widecheck{\mathfrak{D}}$ a smooth Cartier divisor on
$\widecheck{\mathfrak{X}}$ which is itself Calabi-Yau. Equivalently the normal
bundle of $\widecheck{\mathfrak{D}}$,
$\mathcal{N}_{\widecheck{\mathfrak{D}}/\widecheck{\mathfrak{X}}}$, is
$\mathcal{O}_{\widecheck{\mathfrak{D}}}$. Then $\widecheck{\mathfrak{X}}$ is
fibred over $\mathbb{P}^{1}$ with fibre class $\widecheck{\mathfrak{D}}$ and
any deformation of $\widecheck{\mathfrak{X}}$ induces an unobstructed
deformation of $(\widecheck{\mathfrak{X}},\widecheck{\mathfrak{D}})$. Hence
any smooth Calabi-Yau obtained from deforming $\widecheck{\mathfrak{X}}$ is
also fibred by Calabi-Yau divisors.

A similar statement holds for $\widecheck{\mathfrak{D}}$ any smooth Calabi-Yau
subvariety of $\widecheck{\mathfrak{X}}$ with trivial normal bundle. Under
these conditions $\widecheck{\mathfrak{X}}$ is generically fibred by smooth
Calabi-Yau varieties and the same remains true for deformations of
$\widecheck{\mathfrak{X}}$.

In [5] the authors prove formal smoothability for any log scheme for which two
Hodge-theoretic conditions hold. These are known to hold for maximal
degenerations of Calabi-Yau manifolds, and [31] extended this to much larger
classes of singular spaces, those which are toroidal families. In future work
they intend to extend this to families not just over the standard log point,
but more general bases but still with underlying scheme a point. Our technique
here would extend in parallel to their work.

To begin with we have
$\widecheck{\mathfrak{X}}\to\operatorname{Spec}k^{\dagger}$ a log toroidal
family and a strictly embedded codim $k$ log subscheme
$\widecheck{\mathfrak{D}}\subset\widecheck{\mathfrak{X}}$ such that the
induced map $\widecheck{\mathfrak{D}}\to\operatorname{Spec}k^{\dagger}$ is log
toroidal and the normal bundle of $\widecheck{\mathfrak{D}}$ inside
$\widecheck{\mathfrak{X}}$ is trivial, isomorphic to
$\mathcal{O}_{\widecheck{\mathfrak{D}}}^{k}$. In particular local to
$\widecheck{\mathfrak{D}}$ the log scheme $\widecheck{\mathfrak{X}}$ is
isomorphic to $\widecheck{\mathfrak{D}}\times\mathbb{A}^{k}$. This will give
local models for $\widecheck{\mathfrak{X}}$ near $\widecheck{\mathfrak{D}}$
compatible with the local models of $\widecheck{\mathfrak{D}}$. We consider
divisorial deformations of $\mathfrak{X}$ and $\mathfrak{D}$ with a compatible
closed embedding.

######  Lemma 2.2.

Let $V\to\operatorname{Spec}k^{\dagger}$ be an affine log toroidal family,
$W\subset V$ a strictly embedded codimension $k$ closed subscheme transverse
to all the log strata and such that the induced map to
$\operatorname{Spec}k^{\dagger}$ is also a log toroidal family. Then any two
first order deformations of this data are isomorphic.

###### Proof.

One étale locally factors the problem with $V=\mathbb{A}^{k}\times Y$ and
$W=\\{0\\}\times Y$ where $Y\to\operatorname{Spec}k^{\dagger}$ is a log
toroidal family. Then this works precisely as in [16].

∎

###  2.1 Compatible BV structures

The key idea of the proof of our main theorem is that there are compatible BV
structures between the deformation theories of
$\operatorname{Proj}k[\Sigma(\widecheck{\mathfrak{X}}_{0})]$,
$\operatorname{Proj}k[\Sigma(\widecheck{\mathfrak{D}})]$, and
$(\operatorname{Proj}k[\Sigma(\widecheck{\mathfrak{X}}_{0})]$,
$\operatorname{Proj}k[\Sigma(\widecheck{\mathfrak{D}})])$. We follow [30] in
defining the log vector fields on $\widecheck{\mathfrak{X}}$ tangent to
$\widecheck{\mathfrak{D}}$. In the case $k=1$ we could describe this using an
additional log structure on $\widecheck{\mathfrak{X}}$, but since we want to
handle all the cases at once we will stick to a classical description. For
this subsection we will assume that $\widecheck{\mathfrak{X}}$ and
$\widecheck{\mathfrak{D}}$ are globally log smooth, and will lift this
restriction later.

######  Definition 2.3.

Let $\widecheck{\mathfrak{D}}$ locally be defined by $V(t)$ inside
$\widecheck{\mathfrak{X}}$. Strict étale locally we have that
$\widecheck{\mathfrak{X}}$ is isomorphic to
$\widecheck{\mathfrak{D}}\times\mathbb{A}^{1}$ with $t$ the final coordinate.
We take a local basis
$\frac{d}{dt},\frac{d}{dx_{1}}...\frac{d}{dx_{k-1}}\ldots\frac{d}{d\;log\>x_{k}}\ldots\frac{d}{d\;log\>x_{n-1}}$
of $\Theta^{1}_{\widecheck{\mathfrak{X}}}$. The algebra of polyvector fields
on $\widecheck{\mathfrak{X}}$ tangent to $\widecheck{\mathfrak{D}}$ is the
subalgebra generated locally to $\widecheck{\mathfrak{D}}$ by
$t\frac{d}{dt},\frac{d}{dx_{1}}...\frac{d}{dx_{k-1}}\ldots\frac{d}{d\;log\>x_{k}}\ldots\frac{d}{d\;log\>x_{n-1}}$
and the full algebra over other open sets. We write this
$\Theta^{i}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$
for the degree $i$ term, and
$\bigwedge\Theta_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$
for the whole polyvector field algebra.

In general if $\widecheck{\mathfrak{D}}$ is an equisingular complete
intersection inside $\widecheck{\mathfrak{X}}$ then we analogously take the
algebra of polyvector fields on $\widecheck{\mathfrak{X}}$ tangent to
$\widecheck{\mathfrak{D}}$ to be the subalgebra generated locally to
$\widecheck{\mathfrak{D}}$ by
$t_{1}\frac{d}{dt_{1}},\ldots,t_{1}\frac{d}{dt_{r}},\ldots
t_{r}\frac{d}{dt_{r}},\frac{d}{d\;log\>x_{r+1}}...\frac{d}{d\;log\>x_{k}}\ldots\frac{d}{dx_{k}}\ldots\frac{d}{dx_{n}}$
and the full algebra over other open sets. We write this
$\Theta^{i}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$
for the degree $i$ term, and
$\bigwedge\Theta_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$
for the whole polyvector field algebra.

This structure controls the deformations of the local models as described in
[31], and for the same reason as their Theorem 6.13, combined with our local
rigidity.

From now on let us assume that the surjectivity and Hodge theoretic
assumptions necessary to apply techniques of [5] hold for
$\widecheck{\mathfrak{X}}$ and $\widecheck{\mathfrak{D}}$. In the case we are
interested in both are maximal and so by [18] Theorem 4.1 we will see that
these assumptions hold.

Note that if $\widecheck{\mathfrak{D}}$ is codimension greater than one this
is not a collection of bundles, rather of sheaves. One worries at this point
that much of the argument involved in proving smoothness works by studying
cotangent rather than tangent vectors. Amazingly this is not a problem as we
shall see. An easy local calculation shows that
$\bigwedge\Theta_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$
is a subalgebra of $\bigwedge\Theta_{\widecheck{\mathfrak{X}}}$ and the
inclusion is compatible with the BV operator:

######  Lemma 2.4.

The complex
$\bigwedge\Theta_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$
with the Schouten-Nijenhuis bracket is a sub-Gerstenhaber algebra of
$\bigwedge\Theta_{\widecheck{\mathfrak{X}}}$. The BV operator on
$\bigwedge\Theta_{\widecheck{\mathfrak{X}}}$ restricts to a BV operator on
$\bigwedge\Theta_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$.

###### Proof.

The statements are all local in nature, so we may reduce to the case where we
have local equations for $\widecheck{\mathfrak{D}}$. To show that the
Schouten-Nijenhuis bracket restricts to a bracket on this sub-space we need
only check it for degree one elements, so for vector fields. Let $t_{1}\ldots
t_{r}$ be local equations for $\widecheck{\mathfrak{D}}$, then we must check
that the brackets $[t_{i}\frac{d}{dt_{j}},t_{l}\frac{d}{dt_{m}}]$ lie inside
the span of these:

$\left[t_{i}\frac{d}{dt_{j}},t_{l}\frac{d}{dt_{m}}\right]=t_{i}\delta_{j,l}\frac{d}{dt_{m}}-t_{l}\delta_{m,i}\frac{d}{dt_{j}}$

And so this forms a sub-Gerstenhaber algebra. Now let $\omega$ be a global top
form $\bigwedge dt_{i}\wedge\bigwedge dx_{i}\wedge\bigwedge d\>log\;x_{i}$
inducing the BV operator. Then the composite $\llcorner\>\omega\cdot
d\cdot\llcorner\>\omega$ maps:

$\displaystyle\prod_{j\in J}t_{j}$ $\displaystyle
f(t_{1},\ldots,x_{n-1})\bigwedge_{j\in J}\frac{d}{dt_{j}}\bigwedge_{i\in
I}\frac{d}{dx_{i}}\bigwedge_{k\in K}\frac{d}{d\;log\>x_{k}}\mapsto\prod_{j\in
J}t_{j}f\bigwedge_{j\in\hat{J}}dt_{j}\bigwedge_{i\in\hat{I}}dx_{i}\bigwedge_{k\in\hat{K}}d\>log\;x_{k}$
$\displaystyle\mapsto\sum_{l\in J}\left(f\>dt_{l}\prod_{j\in J,j\neq
l}t_{j}\right)\bigwedge_{j\in\hat{J}}dt_{j}\bigwedge_{i\in\hat{I}}dx_{i}+\prod_{j\in
J}t_{j}\sum_{l\in
J}\left(\frac{df}{dt_{l}}dt_{l}\right)\bigwedge_{j\in\hat{J}}dt_{j}\bigwedge_{i\in\hat{I}}dx_{i}\bigwedge_{k\in\hat{K}}d\>log\;x_{k}$
$\displaystyle\quad+\prod_{j\in
J}t_{j}\bigwedge_{j\in\hat{J}}dt_{j}\left(\sum_{i\in
I}\frac{df}{dx_{i}}dx_{i}+\sum_{k\in
K}\frac{df}{d\>log\;x_{i}}d\>log\;x_{i}\right)\bigwedge_{i\in\hat{I}}dx_{i}\bigwedge_{k\in\hat{K}}d\>log\;x_{k}$
$\displaystyle\mapsto\sum_{l\in J}\left(\left(f\prod_{j\in J,j\neq
l}t_{j}+\frac{df}{dt_{l}}\prod_{j\in J}t_{j}\right)\bigwedge_{j\in J,j\neq
l}\frac{d}{dt_{j}}\right)\bigwedge_{i\in I}\frac{d}{dx_{i}}\bigwedge_{k\in
K}\frac{d}{d\;log\>x_{k}}$ $\displaystyle\quad+\left(\sum_{l\in
I}\frac{df}{dx_{l}}\bigwedge_{i\in I,i\neq l}\frac{d}{dx_{i}}\bigwedge_{k\in
K}\frac{d}{d\;log\>x_{k}}+\sum_{l\in K}\frac{df}{d\>log\;x_{l}}\bigwedge_{i\in
I}\frac{d}{dx_{i}}\bigwedge_{k\in K,k\neq
l}\frac{d}{d\;log\>x_{k}}\right)\prod_{j\in J}t_{j}\bigwedge_{j\in
J}\frac{d}{dt_{j}}$

where $\hat{A}$ is the complement of the indexing set $A$. So this BV operator
preserves the subalgebra
$\bigwedge\Theta_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$.
∎

The definition of
$\Theta^{i}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$
is precisely set up so that the restriction
$\Theta^{i}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}\mid_{\widecheck{\mathfrak{D}}}$
admits a map to $\Theta_{\widecheck{\mathfrak{D}}}$, which sends any section
deforming away from $\widecheck{\mathfrak{D}}$ to zero. We claim that there is
a commutative diagram of BV algebras:

######  Lemma 2.5.

The following diagram commutes as a diagram of BV algebras and the outer
square is Cartesian as a diagram of complexes of sheaves on
$\widecheck{\mathfrak{X}}$:

${\Theta_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}}$${\Theta_{\widecheck{\mathfrak{X}}}}$${\Theta_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}\mid_{\widecheck{\mathfrak{D}}}}$${\Theta_{\widecheck{\mathfrak{X}}}\mid_{\widecheck{\mathfrak{D}}}}$${\Theta_{\widecheck{\mathfrak{D}}}}$${\Theta_{\widecheck{\mathfrak{X}}}\mid_{\widecheck{\mathfrak{D}}}}$

###### Proof.

This is a local calculation and follows directly from the above coordinate
description.

∎

We now make the assumption that the normal bundle of
$\widecheck{\mathfrak{D}}$ inside $\widecheck{\mathfrak{X}}$ is globally
trivial. This is a strong assumption satisfied by fibrations and without it we
could only show that $\widecheck{\mathfrak{D}}$ is stable inside
$\widecheck{\mathfrak{X}}$. Since we assume that $\widecheck{\mathfrak{D}}$ is
Calabi-Yau and dimension at least two this assumption is also stable under
infinitesimal deformations of $\widecheck{\mathfrak{D}}$ (since
$\operatorname{Ext}^{1}(\mathcal{O}_{\widecheck{\mathfrak{D}}}^{r},\mathcal{O}_{\widecheck{\mathfrak{D}}}^{r})=0$).
Note that by strictness of the embedding of $\widecheck{\mathfrak{D}}$ into
$\widecheck{\mathfrak{X}}$ the log normal bundle coincides with the normal
bundle.

######  Definition 2.6.

Let $\widecheck{\mathfrak{D}}\subset\widecheck{\mathfrak{X}}$ be a regularly
embedded subscheme. Let $f_{1}\ldots f_{r}$ be local equations cutting out
$\widecheck{\mathfrak{D}}$. We define the sheaf of graded algebras
$\bigwedge\mathcal{N}_{\widecheck{\mathfrak{D}}/\widecheck{\mathfrak{X}}}$ to
be the exterior algebra generated in degree one by
$\frac{d}{df_{1}}\ldots\frac{d}{df_{r}}$ supported on
$\widecheck{\mathfrak{D}}$.

######  Lemma 2.7.

There is a split exact sequence on the level of sheaves:

$0\to\Theta^{1}_{\widecheck{\mathfrak{D}}}\to\Theta^{1}_{\widecheck{\mathfrak{X}}}\mid_{\widecheck{\mathfrak{D}}}\to\mathcal{N}_{\widecheck{\mathfrak{D}}/\widecheck{\mathfrak{X}}}\to
0$ (1)

This induces a split exact sequence of sheaves of graded algebras:

$0\to\bigwedge\Theta_{\widecheck{\mathfrak{D}}}\to\bigwedge\Theta_{\widecheck{\mathfrak{X}}}\mid_{\widecheck{\mathfrak{D}}}\to\bigwedge\mathcal{N}_{\widecheck{\mathfrak{D}}/\widecheck{\mathfrak{X}}}\otimes\bigwedge\Theta_{\widecheck{\mathfrak{D}}}\to
0$ (2)

The left hand map of this second sequence is a map of Gerstenhaber algebras.
The induced map
$\bigwedge\Theta_{\widecheck{\mathfrak{X}}}\to\bigwedge\Theta_{\widecheck{\mathfrak{D}}}$
is a surjective map of BV algebras.

###### Proof.

Again we can pass to a local chart where we have coordinates $t_{i}$ defining
$\widecheck{\mathfrak{D}}$ inside $\widecheck{\mathfrak{X}}$ and
$t_{1}\frac{d}{dt_{1}},\ldots,t_{1}\frac{d}{dt_{r}},\ldots
t_{r}\frac{d}{dt_{r}},\frac{d}{d\;log\>x_{r+1}}...\frac{d}{d\;log\>x_{k}}\ldots\frac{d}{dx_{k}}\ldots\frac{d}{dx_{n}}$
local generators on $\widecheck{\mathfrak{X}}$. There is a clear injection of
$\Theta_{\widecheck{\mathfrak{D}}}$ into
$\Theta_{\widecheck{\mathfrak{X}}}\mid_{\widecheck{\mathfrak{D}}}$ with
cokernel spanned by the $\frac{d}{dt_{i}}$. This produces the exact sequence.
It is split via the obvious map.

∎

It is not immediate that the “Hodge numbers”
$H^{i}(\Theta^{j}_{\widecheck{\mathfrak{X}}})$ are invariant under pullback
along the Frobenius map
$Fr:\widecheck{\mathfrak{X}}^{\prime}\to\widecheck{\mathfrak{X}}$. To see why
note that although the Frobenius map on $S$ is an isomorphism of schemes for a
perfect field, it can never be an isomorphism of fine saturated log schemes
for divisibility reasons. Fortunately the maps $\widecheck{\mathfrak{X}}\to S$
and $\widecheck{\mathfrak{D}}\to S$ are saturated and the Frobenius map
integral, we will see in the next section that this is enough to ensure
independence.

###  2.2 Away from the smooth locus

We have described these sheaves in the case that $\widecheck{\mathfrak{X}}$
and $\widecheck{\mathfrak{D}}$ are log smooth over $S$. We are about to apply
this to the mirrors where there is no guarantee that the log structure is
smooth or even globally defined. Instead we know that
$\widecheck{\mathfrak{X}}$ and $\widecheck{\mathfrak{D}}$ are central fibres
of log toroidal families. Let us introduce some notation:

Let $\widecheck{\mathfrak{X}}\to S$ be a log toroidal morphism with
generically smooth locus
$i_{\widecheck{\mathfrak{X}}}:U_{\widecheck{\mathfrak{X}}}\subset\widecheck{\mathfrak{X}}$
and complement $Z_{\widecheck{\mathfrak{X}}}$, $\widecheck{\mathfrak{D}}$ a
Calabi-Yau cycle on $\widecheck{\mathfrak{X}}$ which is itself a log toroidal
family over $S$ with respect to the induced log structure and
$i_{\widecheck{\mathfrak{D}}}:U_{\widecheck{\mathfrak{D}}}\subset\widecheck{\mathfrak{D}}$
the generically smooth locus. Write $Fr_{S}:S\to S$ for the Frobenius map,
$\widecheck{\mathfrak{X}}^{\prime}$ for the fibre product
$S\times_{S}\widecheck{\mathfrak{X}}$, $\widecheck{\mathfrak{D}}^{\prime}$ for
the product $S\times_{S}\widecheck{\mathfrak{D}}$. These are themselves log
toroidal over $S$, with open smooth locus
$U_{\widecheck{\mathfrak{X}}^{\prime}}$ and
$U_{\widecheck{\mathfrak{D}}^{\prime}}$ respectively, which we may assume are
the pullbacks of $U_{\widecheck{\mathfrak{X}}}$ and
$U_{\widecheck{\mathfrak{D}}}$ respectively. They admit induced pullback maps
$Fr_{\widecheck{\mathfrak{X}}^{\prime}}:\widecheck{\mathfrak{X}}^{\prime}\to\widecheck{\mathfrak{X}}$
and
$Fr_{\widecheck{\mathfrak{D}}^{\prime}}:\widecheck{\mathfrak{D}}^{\prime}\to\widecheck{\mathfrak{D}}$.
Since the maps are saturated the underlying scheme maps of
$Fr_{\widecheck{\mathfrak{X}}^{\prime}}$ and
$Fr_{\widecheck{\mathfrak{D}}^{\prime}}$ are isomorphisms of schemes (though
not over $S$).

As in [31] we can take the pushforward of these sheaves from the log smooth
locus to the whole space and show that the desired properties hold for them,
even once pushed forward. In particular we know already that
$\bigwedge\Theta_{\widecheck{\mathfrak{X}}/S}$ and
$\bigwedge\Theta_{\widecheck{\mathfrak{D}}/S}$ are
$Z_{\widecheck{\mathfrak{X}}}$-closed. This follows from the fact that the
open inclusion of the smooth locus is affine and so in both cases pushforward
and pullback are exact. In particular this gives for free that
$\Theta^{\bullet}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle/S}$
is $Z_{\widecheck{\mathfrak{X}}}$-closed since it is the kernel of a
surjective map between two $Z_{\widecheck{\mathfrak{X}}}$-closed sheaves. All
the constructions above for split exact sequences continue to hold by
exactness. We finally prove that Frobenius independence holds.

######  Lemma 2.8.

There are isomorphisms
$H^{i}(\widecheck{\mathfrak{X}},\Theta_{\widecheck{\mathfrak{X}}/S})\cong
H^{i}(\widecheck{\mathfrak{X}}^{\prime},\Theta_{\widecheck{\mathfrak{X}}^{\prime}/S})$,
$H^{i}(\widecheck{\mathfrak{D}},\Theta_{\widecheck{\mathfrak{D}}/S})\cong
H^{i}(\widecheck{\mathfrak{D}}^{\prime},\Theta_{\widecheck{\mathfrak{D}}^{\prime}/S})$
and
$H^{i}(\widecheck{\mathfrak{X}},\Theta_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle/S})\cong
H^{i}(\widecheck{\mathfrak{X}}^{\prime},\Theta_{\widecheck{\mathfrak{X}}^{\prime}\langle\widecheck{\mathfrak{D}}^{\prime}\rangle/S})$
for all $i$.

###### Proof.

By [27] there are canonical isomorphisms
$i_{U_{\widecheck{\mathfrak{X}}^{\prime}}}^{*}\Omega^{i}_{U_{\widecheck{\mathfrak{X}}}/S}\cong\Omega^{i}_{U_{\widecheck{\mathfrak{X}}^{\prime}}/S}$
and
$i_{U_{\widecheck{\mathfrak{D}}^{\prime}}}^{*}\Omega^{i}_{U_{\widecheck{\mathfrak{D}}}/S}\cong\Omega^{i}_{U_{\widecheck{\mathfrak{D}}^{\prime}}/S}$.
By[31] these remain isomorphisms under pushforward to
$\widecheck{\mathfrak{X}}^{\prime}$ and $\widecheck{\mathfrak{D}}^{\prime}$
respectively. Since $Fr_{\widecheck{\mathfrak{X}}^{\prime}}$ and
$Fr_{\widecheck{\mathfrak{D}}^{\prime}}$ are isomorphisms on schemes the first
two isomorphisms now hold. The final isomorphism holds then by applying the
five lemma to the exact sequence appearing in 2.

∎

To apply this we construct the pre-sheaves
${}^{k}PV^{n}_{\widecheck{\mathfrak{X}}}$,
${}^{k}PV^{n}_{\widecheck{\mathfrak{D}}}$ and
${}^{k}PV^{n}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$
as outlined in [31] section 13.1. There are canonical maps
${}^{k}PV^{n}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}\to^{k}PV^{n}_{\widecheck{\mathfrak{X}}}$
and
${}^{k}PV^{n}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}\to^{k}PV^{n}_{\widecheck{\mathfrak{D}}}$
which are morphisms of BV algebras. Given a ${}^{k}\phi$ solving the Maurer-
Cartan equation on
${}^{k}PV^{n}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle}$
the image under each of these maps solves the corresponding Maurer-Cartan
equations and we have compatible morphisms of Gerstenhaber algebras between
the induced cohomology pre-sheaves.

######  Corollary 2.9.

The induced morphisms
$H^{\bullet}(^{k}PV_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle})\to
H^{\bullet}(^{k}PV_{\widecheck{\mathfrak{D}}})$ and
$H^{\bullet}(^{k}PV_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle})\to
H^{\bullet}(^{k}PV_{\widecheck{\mathfrak{X}}})$ are morphisms of presheaves of
Gerstenhaber algebras.

So the final thing we need to check are the assumptions of [5] Theorem 5.5, or
equivalently [31] Theorem 13.4 and show that one can use the BV operator to
construct solutions to the Maurer-Cartan equation. The components are
degeneration of the Hodge-to-de-Rham spectral sequence at $E_{1}$ and a
surjection $H^{i}(^{k}PV^{\bullet})\to H^{i}(^{k-1}PV^{\bullet})$.

######  Theorem 2.10.

The ranks of
$H^{i}(\Theta^{j}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle})$
are invariant in one dimensional log toroidal families.

###### Proof.

We will make use of the technology developed in [31] together with some
homological algebra. First we spread out to finite characteristic which we can
do compatibly for $\widecheck{\mathfrak{X}}$ and $\widecheck{\mathfrak{D}}$
preserving the inclusion. We can therefore form the Frobenius twists
$F_{\widecheck{\mathfrak{X}}}:\widecheck{\mathfrak{X}}\to\widecheck{\mathfrak{X}}^{\prime}$
and
$F_{\widecheck{\mathfrak{D}}}:\widecheck{\mathfrak{D}}\to\widecheck{\mathfrak{D}}^{\prime}$
with
$\widecheck{\mathfrak{D}}^{\prime}\subset\widecheck{\mathfrak{X}}^{\prime}$ a
closed subscheme. Note that
$\widecheck{\mathfrak{X}},\widecheck{\mathfrak{X}}^{\prime},\widecheck{\mathfrak{D}}$
and $\widecheck{\mathfrak{D}}^{\prime}$ all have trivial dualizing sheaf and
$F$ is a finite map. Then taking derived sheaf $\mathcal{H}om$ and using the
fact that
$F_{\widecheck{\mathfrak{X}}}^{!}\mathcal{O}_{\widecheck{\mathfrak{X}}^{\prime}}=\omega^{\circ}_{\widecheck{\mathfrak{X}}}\otimes(F_{\widecheck{\mathfrak{X}}}^{*}\omega^{\circ}_{\widecheck{\mathfrak{X}}^{\prime}})^{\vee}\otimes
F_{\widecheck{\mathfrak{X}}}^{*}\mathcal{O}_{\widecheck{\mathfrak{X}}^{\prime}}[\dim\widecheck{\mathfrak{X}}-\dim\widecheck{\mathfrak{X}}^{\prime}]=\mathcal{O}_{\widecheck{\mathfrak{X}}}$
we have:

$\displaystyle
F_{\widecheck{\mathfrak{X}}\>*}\Theta^{\bullet}_{\widecheck{\mathfrak{X}}/S}$
$\displaystyle=\mathcal{H}^{0}RF_{\widecheck{\mathfrak{X}}\>*}\Theta^{\bullet}_{\widecheck{\mathfrak{X}}/S}$
$\displaystyle=\mathcal{H}^{0}RF_{\widecheck{\mathfrak{X}}\>*}R\mathcal{H}om_{\widecheck{\mathfrak{X}}}(W^{\bullet}_{\widecheck{\mathfrak{X}}/S},\mathcal{O}_{\widecheck{\mathfrak{X}}})$
$\displaystyle=\mathcal{H}^{0}RF_{\widecheck{\mathfrak{X}}\>*}R\mathcal{H}om_{\widecheck{\mathfrak{X}}}(W^{\bullet}_{\widecheck{\mathfrak{X}}/S},F_{\widecheck{\mathfrak{X}}}^{!}\mathcal{O}_{\widecheck{\mathfrak{X}}^{\prime}})$
$\displaystyle=\mathcal{H}^{0}R\mathcal{H}om_{\widecheck{\mathfrak{X}}^{\prime}}(RF_{\widecheck{\mathfrak{X}}\>*}W^{\bullet}_{\widecheck{\mathfrak{X}}/S},\mathcal{O}_{\widecheck{\mathfrak{X}}^{\prime}})$
$\displaystyle=\mathcal{H}om_{\widecheck{\mathfrak{X}}^{\prime}}(F_{\widecheck{\mathfrak{X}}\>*}W^{\bullet}_{\widecheck{\mathfrak{X}}/S},\mathcal{O}_{\widecheck{\mathfrak{X}}^{\prime}})$

and similarly for $\widecheck{\mathfrak{D}}$. Thus we have that pushforward
and dualizing commute for $\Theta_{\widecheck{\mathfrak{X}}/S}$ and
$\Theta_{\widecheck{\mathfrak{D}}/S}$ on $\widecheck{\mathfrak{X}}$ and
$\widecheck{\mathfrak{D}}$ respectively. Unfortunately the dual of
$\Theta^{\bullet}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle/S}$
would have to be the derived dual, and the above argument would not apply. The
Cartier isomorphisms for $\widecheck{\mathfrak{X}}$ and
$\widecheck{\mathfrak{D}}$ induce dual decompositions in the derived category
of $\widecheck{\mathfrak{X}}$:

$F_{\widecheck{\mathfrak{X}}\>*}(\Theta_{\widecheck{\mathfrak{X}}/S}^{\bullet})\cong\bigoplus\Theta^{i}_{\widecheck{\mathfrak{X}}^{\prime}/S}[-i]\text{\quad
and
\quad}F_{\widecheck{\mathfrak{D}}\>*}(\Theta_{\widecheck{\mathfrak{D}}/S}^{\bullet})\cong\bigoplus\Theta^{i}_{\widecheck{\mathfrak{D}}^{\prime}/S}[-i]$

Then there is a partial morphism of distinguished triangles which we may
complete to a morphism of distinguished triangles:

${F_{\widecheck{\mathfrak{X}}\>*}\Theta^{\bullet}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle/S}}$${F_{\widecheck{\mathfrak{X}}\>*}\Theta^{\bullet}_{\widecheck{\mathfrak{X}}/S}}$${F_{\widecheck{\mathfrak{D}}\>*}\Theta^{\bullet}_{\widecheck{\mathfrak{D}}/S}}$
${\bigoplus\Theta^{i}_{\widecheck{\mathfrak{X}}^{\prime}\langle\widecheck{\mathfrak{D}^{\prime}}\rangle/S}[-i]}$${\bigoplus\Theta^{i}_{\widecheck{\mathfrak{X}}^{\prime}/S}[-i]}$${\bigoplus\Theta^{i}_{\widecheck{\mathfrak{D}}^{\prime}/S}[-i]\otimes\bigwedge\mathcal{N}_{\widecheck{\mathfrak{D}}/\widecheck{\mathfrak{X}}}}$
$\scriptstyle{\iota}$

The right hand square commutes by the explicit description of the Cartier
isomorphism. The left hand map is a quasi-isomorphism since the other two
vertical maps are. The map $\iota$ is the natural inclusion, hence inducing
the trivial differential on the complex
$\bigoplus\Theta^{i}_{\widecheck{\mathfrak{X}}^{\prime}\langle\widecheck{\mathfrak{D}}^{\prime}\rangle/S}[-i]$.
Thus we have a decomposition of
$F_{\widecheck{\mathfrak{X}}\>*}\Theta^{\bullet}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle/S}$
as a complex with trivial differential without having to find the derived
dual. Now we apply the same argument as found in section 4 of [10] to this
complex using the two spectral sequences, the “Hodge spectral sequence” and
the “conjugate spectral sequence”:

$E_{1}^{i,j}:\quad
R^{i}f_{*}\Theta^{j}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle/S}\Rightarrow
R^{i+j}f_{*}\Theta^{\bullet}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle/S}$

and

${}_{c}E_{2}^{i,j}:\quad
R^{i}f^{\prime}_{*}\mathcal{H}^{j}F_{\widecheck{\mathfrak{X}}\>*}\Theta^{\bullet}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle/S}\Rightarrow
R^{i+j}f_{*}\Theta^{\bullet}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle/S}$

obtained as the Cartan-Eilenberg spectral sequence and the Grothendieck
spectral sequence respectively. Note that this argument crucially relies on
three facts, that
$\Theta^{i}_{\widecheck{\mathfrak{X}}^{\prime}\langle\widecheck{\mathfrak{D}}\rangle/S}$
are flat over $S$, formation of cohomology is compatible with base change for
these sheaves and that the “Hodge numbers”
$R^{i}f_{*}\Theta^{j}_{\widecheck{\mathfrak{X}}\langle\widecheck{\mathfrak{D}}\rangle/S}$
are invariant under Frobenius. The first is true when $S$ is a one-dimensional
thickening of a point since there are no sections which are torsion over $S$.
The second follows from the same results in large enough characteristic for
$\Theta^{i}_{\widecheck{\mathfrak{X}}/S}$ and
$\Theta^{i}_{\widecheck{\mathfrak{D}}/S}$. The last one we have proved
earlier.

This gives us a proof whenever there is a Frobenius lifting, and the standard
spreading out argument then proves the claim.

∎

For the sake of completeness we translate the proof of section 4 of [10] into
English. We found it both surprising that this result has not been exposited
elsewhere and was relatively unknown in the community, for instance it acts to
replace the local Poincaré lemma assumption in [5] and reduces the work
required in section 11 of [31]. Although we have rephrased the argument
without mentioning the de Rham complex we have kept the thematic namings of
“Hodge”, “conjugate” and “Cartier” as they explain where the terms came from.

######  Theorem 2.11.

Let $f:X\to S$ be a flat finite type morphism of schemes of characteristic
$p$, $S$ local Artinian with closed point $s$ and residue field $k$ and
$\mathcal{F}^{\bullet}_{X_{T}/T}$ a function associating to every object $T$
of $Sch/S$ a bounded complex on $X_{T}=X\times_{S}T$ of $S$-flat coherent
sheaves and to every morphism $g:(f^{\prime}:T^{\prime}\to S)\to(f:T\to S)$ an
isomorphism

$g^{*}\mathcal{F}^{\bullet}_{X_{T}/T}\to\mathcal{F}^{\bullet}_{X_{T^{\prime}}/T^{\prime}}$

Suppose further that $F_{*}(\mathcal{F}^{\bullet}_{X/S})$ is quasi-isomorphic
to a complex with trivial differential, where $F$ is the relative Frobenius
map $X\to X^{\prime}$ and that there is a Cartier isomorphism
$C^{-1}:\mathcal{F}^{i}_{X^{\prime}/S}\to
H^{i}F_{*}(\mathcal{F}^{\bullet}_{X/S})$ for every $i$.

Then under these hypotheses the “Hodge spectral sequence” formed as the
Grothendieck spectral sequence of the complex verifies
$E_{1}^{ij}=E_{\infty}^{ij}$ for $i+j=n$ and the sheaves
$E^{ij}_{1}=R^{j}f_{*}\mathcal{F}_{X/S}^{i}$ are locally free of formation
compatible with all base change for $i+j=n$.

###### Proof.

First note that the statement that $F_{*}(\mathcal{F}^{\bullet}_{X/S})$ is
quasi-isomorphic to a complex with trivial differential implies that the
“conjugate spectral sequence” obtained as the Grothendieck spectral sequence
for the composite of proper pushforward along the composable maps $X\to
X^{\prime}\to S$ degenerates at page $E_{2}$.

Write $S=\operatorname{Spec}A$, since the sheaves $\mathcal{F}^{i}_{X/S}$ are
$S$-flat and $f$ is finite type, the complexes $Rf_{*}\mathcal{F}^{i}_{X/S}$
are isomorphic to a bounded complex $K_{i}^{\bullet}$ of free $A$-modules. We
write $f_{T}$ for the base change of $f$ to $T$. For every subscheme $T\subset
S$ the complex $Rf_{T\>*}\mathcal{F}_{X_{T}/T}^{i}$ is the base change of
$Rf_{*}\mathcal{F}_{X/S}^{i}$, and in particular restricting to $s\in S$ we
have that $H^{j}(X_{s},\mathcal{F}_{X_{s}/s}^{i})$ is equal to
$H^{j}(K_{i}\otimes_{A}k)$. It follows by devissage that

$\text{length}Rf_{*}\mathcal{F}_{X/S}^{i}\leq h^{ij}\>\text{length}\>A$ (3)

where $h^{ij}=\dim_{k}H^{j}(X_{s},\mathcal{F}_{X_{s}/s}^{i})$ and we take
length as an $A$-module, which are all finite as they are the proper
pushforward of finite complexes of coherent sheaves. We may assume that
$K_{i}^{\bullet}$ is minimal, so that the differentials on
$K_{i}^{\bullet}\otimes_{A}k$ are all zero. Once we know that the inequality
in 3 is an equality then all the differentials in $K_{i}^{\bullet}$ are
trivial and we have the desired local freeness and compatibility with base
change.

The convergence $E_{1}^{ij}=E_{\infty}^{ij}$ is equivalent to the equality:

$\text{length}R^{n}f_{*}\mathcal{F}^{\bullet}_{X/S}=\sum_{i+j=n}\text{length}R^{j}f_{*}\mathcal{F}^{i}_{X/S}$
(4)

On the other hand $X_{s}^{\prime}$, constructed from $X_{s}$ by an extension
of the base field, has the same “Hodge” numbers $h^{ij}$ as $X_{s}$. Hence one
obtains an equality

$\text{length}R^{n}f_{*}\mathcal{F}^{\bullet}_{X/S}=\sum_{i+j=n}\text{length}R^{j}f^{\prime}_{*}\mathcal{F}^{i}_{X^{\prime}/S}$
(5)

Already if $S$ is the spectrum of a field this equality implies the desired
equality in 3. Now we prove the general case by induction on the nilpotency of
the maximal ideal $\mathfrak{m}$ of $A$. Suppose that $N$ is positive integer
and $\mathfrak{m}^{N}=0$, and by the above we are free to assume that $N$ is
at least two. Hence there exists $N^{\prime}$ with $1\leq N^{\prime}<N$ and
$pN^{\prime}\geq N$. For such an $N^{\prime}$ the Frobenius endomorphism of
$S$, $F$, factorises over $T=\operatorname{Spec}(A_{1})$,
$A_{1}=A/\mathfrak{m}^{N^{\prime}}$:

${X}$${X^{\prime}}$${X_{T}}$${X}$${S}$${T}$${S}$$\scriptstyle{F}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f_{T}}$$\scriptstyle{f}$

Applying the induction hypothesis to $X_{T}/T$ we see that
$R^{j}f_{*}^{\prime}\mathcal{F}^{i}_{X^{\prime}/S}$ are locally free of rank
$h^{ij}$. The degeneration of the “conjugate spectral sequence” ensures in
turn that $R^{n}f_{*}\mathcal{F}^{\bullet}_{X/S}$ is locally free of rank
$\sum_{i+j=n}h^{ij}$. Hence we must have equality in 3 and the desired result.

∎

The surjectivity assumption follows as argued in Theorem 13.1 of [31] once one
knows the compatibility of these sheaves with base change shown above.

######  Corollary 2.12.

For $\widecheck{\mathfrak{X}}$ and $\widecheck{\mathfrak{D}}$ described above
there is an analytic log toroidal family smoothing this pair.

###### Proof.

By the above compatibility results, together with the description of how to
build the deformed Maurer-Cartan equation this follows from [31] section 13.

∎

This proof actually works without assuming that $\widecheck{\mathfrak{D}}$ has
the cohomology type of a Calabi-Yau so if
$H^{i}(\mathcal{O}_{\widecheck{\mathfrak{D}}})\neq 0$ for $i\neq 0,n-1$, it
only requires the existence of a global top form. In our case since
$\widecheck{\mathfrak{D}}$ arises as the central fibre of a toric degeneration
of Calabi-Yau varieties, and so does have the cohomology type of a Calabi-Yau
by [31]. The same proof applies however to subschemes which are étale covers
of products of Abelian varieties and Calabi-Yau varieties. Since we have an
analytic smoothing we can then apply the classical deformation argument to a
smooth fibre to obtain:

######  Corollary 2.13.

A smooth generic fibre of the smoothing of $\widecheck{\mathfrak{X}}$ is
fibred by Calabi-Yau varieties in codimension $d$.

In particular this applies to the above examples of toric $k$-Tyurin
degenerations from which we deduce the following corollary.

######  Corollary 2.14.

Let $\mathfrak{X}\to\mathcal{S}$ be a simple toric $k$-Tyurin degeneration,
then the mirror $\widecheck{\mathfrak{X}_{\Delta}}$ is fibred by Calabi-Yau
subvarieties in codimension $k$. Further if the degeneration of $\mathfrak{D}$
is simple then the fibres are mirror to $\mathfrak{D}$.

The dgLa described above suggests that there is a version of the scattering
diagrams described in [26] where the coefficients lie in this subalgebra.
There is a notion of consistency based on these structures and the above
section implies that there is an appropriate iterated method to ensure
consistency. It would be a fundamental development to define a canonical
scattering diagram for such a subalgebra.

This result also implies a (non-effective) variant of a quantum-Lefschetz
theorem. Take the asymptotic limit of the Maurer-Cartan equation as outlined
in [6] and relating the terms to counts of log invariants on $\mathfrak{X}$
and $\mathfrak{D}$ as outlined in [25] we see that this expresses the log
Gromov-Witten invariants for $\mathfrak{D}$ in terms of $\mathfrak{X}$. This
is a broader context than that considered in [3], there are no ampleness
assumptions around. We would rather develop fully the theory of these
scattering diagrams than make this explicit.

##  3 Gluing Landau-Ginzburg models

We now consider what happens for $k=1$ if we formed the Landau-Ginzburg mirror
to $\mathfrak{Z}^{0}$ and $\mathfrak{Z}^{1}$ and then attempt to glue them,
considering each as a divisor pair relative to $\mathfrak{D}$. We solve the
problem satisfactorily and prove that this recreates the mirror to
$\mathfrak{X}_{\Delta}$ under a necessary compatibility condition.

###  3.1 Formation of Landau-Ginzburg models

The deformation theoretic Gross-Siebert program described above is sadly not
very well developed for the case of LG models. We believe that such a theory
will appear in tandem with a comparison to the modern approach. To avoid
reliance on forthcoming work we will give a partial description here of how to
construct Landau-Ginzburg models using the old theory, which is convoluted to
avoid issues of boundedness of the cells. As an upside we see a natural
occurrence of a phenomenon mentioned in [23].

We have the tropicalisation of $(\mathfrak{Z}^{0},\mathfrak{D})$, an unbounded
polytope with a well defined affine direction $u$ and a proper piecewise
linear map to $[0,\infty)$. Assuming that $\mathfrak{D}$ is nef on
$\mathfrak{Z}^{0}$ all the initial slab functions on this point non-negatively
in the $u$ direction. We can form an affine manifold with boundary by
restricting to the closed subset over $[0,1]$. This admits an affine linear
embedding into $\Sigma(\mathfrak{X})$ by 1.11.

We choose the slab structure given by restriction, so there are no slabs
contained inside the fibre over $1$. But by the above observations the
scattering diagram is consistent to order zero. Therefore applying the
construction of [17] to this now bounded affine manifold we obtain a pair
$(\overline{\mathfrak{W}_{0}},\partial\mathfrak{W}_{0})$ and the open
subscheme $\overline{\mathfrak{W}_{0}}\setminus\partial\mathfrak{W}_{0}$
admits a map to $\mathbb{A}^{1}$ given by $u$. This fits into the philosophy
of [23] where one expects to see a compactification at infinity with a
canonical choice of differential form on the interior.

Such a compactification exists for higher rank Tyurin degenerations at least
so long as each of the components themselves have nef anti-canonical bundle.
We believe that the restriction to this case is a necessary first step to
understanding the “Mirror $P=W$ conjecture” of [2] through the Gross-Siebert
construction. The compactifications have the property that there are no broken
lines passing in from infinity. In particular one obtains a compactification
of the base $\mathcal{B}$, $\overline{\mathcal{B}}$ by an $snc$ divisor and a
relative compactification of $\mathfrak{W}$ which is relatively log divisorial
over $\overline{\mathcal{B}}$ and possesses a strata of dimension $d$ over a
$d$-dimensional strata of $\overline{\mathcal{B}}$.

###  3.2 Relative deformation theory

To prove the existence of the desired embedding we construct a relative
deformation theory for the interior of the Landau-Ginzburg models. This is
simple on the central fibre where we know precisely where the singular fibres
are, the same techniques apply to relatively smooth families of Calabi-Yau
varieties with minor modifications.

To begin with we have
$\widecheck{\mathfrak{X}}\to\mathbb{P}^{1}\times\operatorname{Spec}\widehat{k[P]}$
a formally versal smoothing of a toric Calabi-Yau space fibred over
$\mathbb{P}^{1}$ and a fixed $\mathbb{G}_{m}\subset\mathbb{P}^{1}$ a dense
affine open with fibres generically divisorial smoothings of
$\mathfrak{D}_{0}$. We have an LG model
$w_{0}:\mathfrak{W}_{0}\to\mathbb{A}^{1}\times\operatorname{Spec}\widehat{k[P]}$
and again a dense open $\mathbb{G}_{m}\subset\mathbb{A}^{1}$ whose fibres are
generically divisorial smoothings of $\mathfrak{D}_{0}$. Over
$0\in\operatorname{Spec}\widehat{k[P]}$ the superpotential
$\mathfrak{W}_{0}\to\mathbb{A}^{1}$ admits compatible open embeddings into
$\widecheck{\mathfrak{X}}_{0}\to\mathbb{P}^{1}$. We write $.^{\circ}$ for the
restriction of an object to the corresponding open $\mathbb{G}_{m}$.

Our goal is to produce an open embedding of $\mathfrak{W}_{0}$ into
$\widecheck{\mathfrak{X}}$. We begin by introducing relative toric CY spaces
and our choice of formal deformation functor.

######  Definition 3.1.

Let $\mathcal{C}_{\mathfrak{B}}$ be the category of log schemes strict over
$\operatorname{Spec}k[\mathbb{N}]$ whose underlying scheme is an infinitesimal
extension of $\mathfrak{B}$ and $\mathfrak{B}$ a fixed affine scheme. This
category inherits a notion of small extension from the notion for the
underlying schemes, noting that all morphisms are strict.

Let
$\rho:\mathfrak{Y}\to\mathfrak{B}^{\dagger}:=\mathfrak{B}\times\operatorname{Spec}k^{\dagger}$
be a morphism of log schemes such that:

  1. (i)

The log scheme $\mathfrak{Y}$ is a toric Calabi-Yau space.

  2. (ii)

$\rho$ is flat as a morphism of schemes and $\rho$ is log smooth on the log
smooth locus of $\mathfrak{Y}$

  3. (iii)

There are diagrams as appearing in [24] (2.1) but with $X^{\dagger}$ replaced
by $X^{\dagger}\times\mathfrak{B}$.

A relative divisorial deformation of $\rho$ over
$\mathcal{A}\in\mathcal{C}_{\mathfrak{B}}$ is a map
$\rho_{\mathcal{A}}:\mathfrak{Y}_{\mathcal{A}}\to\mathcal{A}$ and an
isomorphism of the central fibre
$\mathfrak{Y}_{\mathcal{A}}\times_{\mathcal{A}}\operatorname{Spec}k^{\dagger}\cong\mathfrak{Y}$
such that:

  1. (i)

$\rho_{\mathcal{A}}$ is flat as a morphism of schemes and log smooth on the
restriction to the log smooth locus of $\mathfrak{Y}$.

  2. (ii)

There are diagrams as appearing in [24] (2.2) but with
$X^{\dagger}_{A}:=Y^{\dagger}\times_{\operatorname{Spec}k[\mathbb{N}]}\operatorname{Spec}A^{\dagger}$
replaced by
$X^{\dagger}_{\mathcal{A}}:=Y^{\dagger}\times_{\operatorname{Spec}k[\mathbb{N}]}\mathcal{A}$.

The deformation functor $F$ we consider will be have
$\mathfrak{B}\subset\mathbb{G}_{m}$ the common dense open sending an element
$\mathcal{A}$ to the set of relative divisorial deformations over it, where
two relative divisorial deformations are isomorphic if there is an isomorphism
fixing the central fibre. The action on morphisms is given by pullback.

We first of all claim that this deformation problem has a deformation and
obstruction sheaf on $\mathfrak{B}$. To be precise we mean that the analogue
of Theorem 2.11 of [24] holds, but with $\Theta_{\mathfrak{Y}/k^{\dagger}}$
replaced everywhere with $\Theta_{\mathfrak{Y}/\mathfrak{B}^{\dagger}}$. The
proof is simply compatibility of these sheaves with base change and the
previous result.

######  Theorem 3.2.

Let $\mathcal{A}^{\prime}$ and $\mathcal{A}$ be elements of
$\mathcal{C}_{\mathfrak{B}}$, with $\mathcal{A}^{\prime}$ a small extension of
$\mathcal{A}$ by an ideal $I$, finite over $\mathfrak{B}$. Let $a\in
F(\mathcal{A})$ be a divisorial deformation of
$\mathfrak{Y}\to\mathfrak{B}^{\dagger}$. Then:

  1. (i)

Let $\mathfrak{Y}_{\mathcal{A}^{\prime}}$ be a lift of
$\mathfrak{Y}_{\mathcal{A}}$ to $\mathcal{A}^{\prime}$. Then the set of log
automorphisms of $\mathfrak{Y}_{\mathcal{A}^{\prime}}$ fixing
$\mathfrak{Y}_{\mathcal{A}}$ is:

$R^{0}\rho_{*}(\mathfrak{Y},\Theta_{\mathfrak{Y}/\mathfrak{B}}\otimes_{\mathfrak{B}}I)$

  2. (ii)

The set of equivalence classes of lifts
$\mathfrak{Y}_{\mathcal{A}^{\prime}}\to\mathcal{A}^{\prime}$ if non-empty is a
torsor over:

$R^{1}\rho_{*}(\mathfrak{Y},\Theta_{\mathfrak{Y}/\mathfrak{B}}\otimes_{\mathfrak{B}}I)$

  3. (iii)

An obstruction sheaf for the existence of liftings is:

$R^{2}\rho_{*}(\mathfrak{Y},\Theta_{\mathfrak{Y}/\mathfrak{B}}\otimes_{\mathfrak{B}}I)$

Let us check that the relative log versions of the Schlessinger axioms hold.

######  Theorem 3.3.

The following three axioms hold:

  1. (i)

Let $\mathcal{A}^{\prime}\to\mathcal{A}$ be a small extension and
$\mathcal{B}\to\mathcal{A}$ be a surjective morphism, then the canonical map:

$F(\mathcal{A}^{\prime}\times_{\mathcal{A}}\mathcal{B})\to
F(\mathcal{A}^{\prime})\times_{F(\mathcal{A})}F(\mathcal{B})$

is surjective.

  2. (ii)

If $\mathcal{A}=\mathfrak{B}$ and
$\mathcal{A}^{\prime}=\mathfrak{B}+\mathcal{M}$ for a finite module over
$\mathfrak{B}$. Then the above map is bijective.

  3. (iii)

For a finite $\mathfrak{B}$ module $M$ the set $F(\mathfrak{B}\oplus M)$ is a
finite $\mathfrak{B}$-module.

###### Proof.

The third item follows from the above theorem together with relative
properness of $\rho$. The second follows from the first once one notices that
isomorphisms are preserved under deformation, see Lemma 9.1 of [13]. Therefore
it remains to prove the first. Suppose we are given the data of
$\mathcal{A}^{\prime}\to\mathcal{A}$ a small extension and
$\mathcal{B}\to\mathcal{A}$ a surjective morphism. For objects $X^{\prime}\in
F(\mathcal{A}^{\prime})$ and $X^{\prime\prime}\in F(\mathcal{B})$, both
restricting to $X\in F(\mathcal{A})$. Then we can fix closed immersions $X\to
X^{\prime}$ and $X\to X^{\prime\prime}$. We then define $\underline{X^{*}}$ to
be the scheme whose structure sheaf is
$\mathcal{O}_{\underline{X^{\prime}}}\times_{\mathcal{O}_{\underline{X}}}\mathcal{O}_{\underline{X^{\prime\prime}}}$.
We give this the log structure given by
$\mathcal{M}_{X^{\prime}}\times_{\mathcal{M}_{X}}\mathcal{M}_{M^{\prime\prime}}$,
by the universal property of products it admits a map to
$\mathcal{O}_{\underline{X^{\prime}}}\times_{\mathcal{O}_{\underline{X}}}\mathcal{O}_{\underline{X^{\prime\prime}}}$.
That this is a log scheme with the desired restriction properties is proved in
section 9 of [13]. Condition i) of the definition of relative divisorial
deformations is trivial, flat deformations of log smooth morphisms remain log
smooth. We briefly prove condition ii) of the definition, $X^{*}$ admits étale
locally maps to
$X^{\dagger}_{\mathcal{A}^{\prime}\times_{\mathcal{A}}\mathcal{B}}$ whose
central fibres are étale. But flat deformations of an étale map remain étale
and so we have the result.

∎

We cannot apply the machinery built by Artin directly, even if it were
translated into logarithmic geometry for the simple reason that we do not know
openness of versality for log deformations. Raffaele Caputo has some results
in this direction using analytic techniques in the absolute case, see [4].
Instead we argue directly, that we can construct a versal family over an open
affine subset of $\mathbb{G}_{m}$ by inducing one from the deformations of
$\widecheck{\mathfrak{X}}$.

######  Lemma 3.4.

Suppose that
$H^{1}(\Theta_{\widecheck{\mathfrak{X}}_{0}/\operatorname{Spec}k})\to
H^{1}(\Theta_{\widecheck{\mathfrak{D}}_{0}/\operatorname{Spec}k})$ is
surjective under a splitting of the morphism
$\Theta_{\widecheck{\mathfrak{X}}_{0}/\operatorname{Spec}k}\to\Theta_{\widecheck{\mathfrak{X}}_{0}/\operatorname{Spec}k}\mid_{\mathfrak{D}_{0}}\to\Theta_{\mathfrak{D}_{0}/\operatorname{Spec}k}$
for some initial choice of fibre. Then the space
$\widecheck{\mathfrak{X}}\to\operatorname{Spec}k[\\![P]\\!]$ induces a versal
family over some choice of $\mathfrak{B}\subset\mathbb{G}_{m}$.

We prove this in several steps, first we relate the absolute and relative
deformation theories (depending of course on a choice of splitting).

######  Lemma 3.5.

There is a non-canonical morphism
$H^{1}(\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k})\otimes_{k}k[x^{\pm}]\to
R^{1}\pi_{*}(\Theta_{\mathfrak{W}_{i}/\mathcal{T}_{i}})^{\circ}$.

###### Proof.

First take the relative tangent triangle for the triple of morphisms
$\widecheck{\mathfrak{X}}\to\mathbb{P}^{1}\to\operatorname{Spec}k^{\dagger}$,
and push it forward under $\rho$. This produces an exact sequence on
$\mathbb{P}^{1}$:

$R\rho_{*}\Theta_{\widecheck{\mathfrak{X}}/\mathbb{P}^{1}}\to
R\rho_{*}\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k^{\dagger}}\to
R\rho_{*}\rho^{*}\Theta_{\mathbb{P}^{1}/\operatorname{Spec}k^{\dagger}}$

applying the projection formula to the last term we see that the cokernel of
$R^{1}\rho_{*}\Theta_{\widecheck{\mathfrak{X}}/\mathbb{P}^{1}}\to
R^{1}\rho_{*}\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k^{\dagger}}$
is torsion supported on the singular fibres. In particular over
$\mathbb{G}_{m}$ it is split. Pulling back to $\mathbb{G}_{m}$ and applying
cohomology and base change we obtain a map:

$(R^{1}\rho_{*}\Theta_{\widecheck{\mathfrak{X}}/\mathbb{P}^{1}})^{\circ}\cong
R^{1}\rho_{*}^{\circ}\Theta_{\widecheck{\mathfrak{X}}^{\circ}/\mathbb{G}_{m}}$

The Grothendieck spectral sequence gives a map
$H^{1}(\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k^{\dagger}})\to\Gamma(R^{1}\rho_{*}(\Theta_{\widecheck{\mathfrak{X}}/\mathbb{P}^{1}}))$,
and since $\mathbb{G}_{m}$ is affine composing with a choice of splitting we
get the desired map.

∎

By our assumptions on the surjectivity of the map
$H^{1}(\Theta_{\widecheck{\mathfrak{X}}_{0}/\operatorname{Spec}k})\to
H^{1}(\Theta_{\mathfrak{D}_{0}/\operatorname{Spec}k})$ there is an open dense
$\mathfrak{B}\subset\mathbb{G}_{m}$ where the above map is surjective on
fibres. This is our choice of $\mathfrak{B}$.

We lift this formal statement about differentials to show that we can glue the
formal smoothings:

######  Lemma 3.6.

Let $\mathfrak{F}\to\mathfrak{B}\times\operatorname{Spec}k[\\![t]\\!]$ be a
divisorial family deforming a trivial $\mathfrak{D}_{0}$ bundle. Then there is
a non-canonical map
$\mathfrak{B}\times\operatorname{Spec}k[\\![t]\\!]\to\mathfrak{B}\times\mathbb{A}(H^{1}(\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k^{\dagger}}))$.

###### Proof.

For $t=0$ there is a canonical such map. We now induct on the order, suppose
that we have a choice of map
$\mathfrak{B}\times\operatorname{Spec}k[\\![t]\\!]/\langle
t^{k}\rangle\to\mathfrak{B}\times\mathbb{A}(H^{1}(\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k}))$
for some $k$. Then from the above analysis the lift of $\mathfrak{F}$ to
$\mathfrak{B}\times\operatorname{Spec}k[\\![t]\\!]/\langle t^{k}\rangle$
defines an element of
$R^{1}\rho_{*}(\mathfrak{Y},\Theta_{\mathfrak{Y}/\mathfrak{B}}\otimes_{\mathfrak{B}}\mathfrak{B})$,
up to an element of
$R^{0}\rho_{*}(\mathfrak{Y},\Theta_{\mathfrak{Y}/\mathfrak{B}}\otimes_{\mathfrak{B}}\mathfrak{B})$.
But the map
$H^{1}(\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k})\otimes_{k}k[x^{\pm}]\to
R^{1}\rho_{*}(\mathfrak{Y},\Theta_{\mathfrak{Y}/\mathfrak{B}}\otimes_{\mathfrak{B}}\mathfrak{B})$
is surjective and so we may lift it to an element of
$H^{1}(\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k})\otimes_{k}k[x^{\pm}]$.

But this gives a lift of $\mathfrak{F}$ inside $\widecheck{\mathfrak{X}}$,
hence an embedding $\mathfrak{B}\times\operatorname{Spec}k[\\![t]\\!]/\langle
t^{k+1}\rangle\to\mathfrak{B}\times\mathbb{A}(H^{1}(\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k}))$.

∎

We need only one more tool to prove the claimed theorem.

######  Lemma 3.7.

Suppose that
$h^{1}(\Theta_{\widecheck{\mathfrak{D}}/\operatorname{Spec}k})=1$. Then there
is a formal automorphism of
$\mathfrak{B}\times\operatorname{Spec}k[\\![t]\\!]$ such that the induced map
$\mathfrak{B}\times\operatorname{Spec}k[\\![t]\\!]\to\mathfrak{B}\times\mathbb{A}(H^{1}(\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k}))$
sends elements of
$H^{1}(\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k})$ to pure powers
of $t$.

###### Proof.

This follows from the formal version of the inverse function theorem applied
to generators of
$H^{1}(\Theta_{\widecheck{\mathfrak{X}}/\operatorname{Spec}k^{\dagger}})$.

∎

We can now prove the existence of the embedding:

######  Theorem 3.8.

There is a formal open embedding of $\mathfrak{W}^{0}$ into
$\widecheck{\mathfrak{X}}$ compatible with the fibration structure.

###### Proof.

Suppose we have a divisorial smoothing of $\mathfrak{W}^{0}$ over
$\operatorname{Spec}k[\\![t]\\!]$. The same trick as 3.6 produces a relative
deformation of the fibres. Then we can find an embedding of the restriction of
this smoothing to $\mathfrak{B}$ horizontally into the smoothings of
$\widecheck{\mathfrak{X}}$ restricted to $\mathfrak{B}$. We take the
corresponding degeneration of $\widecheck{\mathfrak{X}}$ over
$\operatorname{Spec}k[\\![t]\\!]$. Then there is a common dense open
$\mathfrak{U}$ of $\mathfrak{W}^{i}$ and $\widecheck{\mathfrak{X}}$ which we
can identify.

Remove from $\mathbb{P}^{1}$ the set
$\mathbb{A}^{1}\setminus\mathfrak{B}\cup\\{0\\}$ to obtain an open subset of
$\widecheck{\mathfrak{X}}$ and glue along the open dense set found above. This
produces a family which is locally everywhere a divisorial deformation of
$\widecheck{\mathfrak{X}}_{0}$ over $\operatorname{Spec}k[\\![t]\\!]$. But the
family $\widecheck{\mathfrak{X}}$ is a versal deformation space for divisorial
deformations of $\widecheck{\mathfrak{X}}_{0}$, and hence this lifts to an
open embedding of $\mathfrak{W}^{i}$ into a family of deformations of
$\widecheck{\mathfrak{X}}$, which is the desired gluing statement.

∎

######  Example 3.9.

The mirror to a quintic threefold, a sextic fourfold, etc, can be constructed
as a specialisation of a family of Calabi-Yau’s obtained by gluing the two
Landau-Ginzburg models obtained as mirrors to the components of the Tyurin
degeneration constructed by blowing up both sides evenly.

The same technique works for higher rank LG models under some comparable
restrictions so long as one works with formal complex analytic spaces. The
restriction is introduced part way through the construction, after we have
introduced some terminology.

######  Construction 3.10.

Let $\mathfrak{X}\to\mathcal{S}$ be a type $k+1$ toric-Tyurin degeneration.
There are two associated tropical structures. The first is the tropicalisation
of the type $k+1$ locus which defines an affine manifold with singularities
$\Xi_{gen}=\Sigma(\mathfrak{X}_{gen})$ but without a fan structure around the
vertices. The second is the tropicalisation of the central fibre, defining for
us an affine manifold with singularities $\Xi_{0}=\Sigma(\mathfrak{X}_{0})$
with a fan structure around the vertices. There is a canonical map
$\rho_{\Xi}:\Xi_{gen}\to\Xi_{0}$ given by specialisation. We take the vanilla
gluing data on $\Xi_{0}$ and $\Xi_{gen}$

To this data we have a reduced reducible scheme given by taking the cone
picture on $\Xi_{gen}$. We write
$\mathcal{B}_{gen}=\operatorname{Proj}k[\Xi_{gen}]$ for this scheme, noting
that without any more integral affine structure we cannot deform this even to
first order. The map $\rho_{\Xi}$ lifts to a map
$\rho_{\mathfrak{X}}:\operatorname{Proj}k[\Xi_{0}]\to\operatorname{Proj}k[\Xi_{gen}]$.

We are now ready for our assumption. Choose a zero dimensional strata
$b\in\Xi_{gen}$ corresponding to a component of the general fibre over the
Tyurin degenerate locus $\mathfrak{Z}^{0}$. Suppose that the induced
degeneration of $\mathfrak{Z}^{0}$ is toric, simple and the boundary divisors
of $\mathfrak{Z}^{0}$ are nef. This furnishes us with a collection of top
dimensional strata of $\Xi_{gen}$, $\sigma_{1},\ldots,\sigma_{m}$
corresponding to the zero-dimensional strata of $\mathfrak{Z}^{0}$ and
intersecting at $b$. We write $\mathfrak{D}_{i}$ for the Calabi-Yau variety
corresponding to $\sigma_{i}$, and we assume that all the induced
degenerations are simple. Locally there is a morphism of affine manifolds with
singularities
$\rho_{\Sigma}:\Xi_{gen}\to\Sigma(\mathfrak{Z}^{0},\partial\mathfrak{Z}^{0})$.

We can construct a higher rank $LG$ model as the mirror to
$(\mathfrak{Z}^{0},\partial\mathfrak{Z}^{0})$, by taking the induced tropical
map to be evaluation against the irreducible strata of
$\partial\mathfrak{Z}^{0}$ and we write this $\mathfrak{W}\to\mathfrak{B}$.
The base $\mathfrak{B}$ is obtained by gluing a collection of toric varieties
along toric strata, and each $\sigma_{i}$ corresponds to a dense torus in a
toric component. By assumption there is a unique zero dimensional strata which
we are free to conflate with $b$. Over the interior of each $\sigma_{i}$ the
map $\rho_{\mathfrak{X}}$ is a locally trivial $\mathfrak{D}_{i}$ bundle. In
particular it is a relatively divisorial family. We assume that the map
$H^{1}(\Theta_{\widecheck{\mathfrak{X}}_{0}/\operatorname{Spec}k})\to\oplus
H^{1}(\Theta_{\mathfrak{D}_{i}/\operatorname{Spec}k})$ is surjective and
$\dim\operatorname{Spec}k[P]\geq\sum
h^{1}(\Theta_{\mathfrak{D}_{i}/\operatorname{Spec}k})$.

Restrict the higher rank LG model mirror to $\mathfrak{W}_{0}\to\mathfrak{B}$
to an analytic open set $U_{P}$ containing $b$. Take an open set of
$\mathcal{B}$ overlapping $U_{P}$ only inside the $\sigma_{i}$, and so that
the complement $\delta_{P}$ is codimension two. The fibre of
$\rho_{\mathfrak{X}}$ over this set is then codimension also at least two
inside the total space $\widecheck{\mathfrak{X}}_{0}$. The same argument above
would show that we can glue these two complex analytic varieties except over a
closed set of codimension two. This uniquely defines a global deformation of
$\mathfrak{X}$ compatible with the above gluing. Hence we can extend it to a
globally defined gluing since differentials extend uniquely across codimension
two sets.

Our restriction on
$H^{1}(\Theta_{\widecheck{\mathfrak{X}}_{0}/\operatorname{Spec}k^{\dagger}})\to
H^{1}(\Theta_{\mathfrak{D}_{0}/\operatorname{Spec}k^{\dagger}})$ has an
interpretation on the mirror side which is more easily checkable. Assume that
$\mathfrak{X}$ is an ample complete intersection inside a Fano toric variety
$T$ of dimension at least 3. Up to torsion we have an isomorphism
$H^{2}(\mathfrak{X})\cong H^{1}(\Omega_{\mathfrak{X}})$. Then by the Lefschetz
hyperplane theorem there is an isomorphism $H^{2}_{\mathbb{Q}}(T)\cong
H^{2}_{\mathbb{Q}}(\mathfrak{X})$. Via composition with restriction there is
therefore a canonical morphism $H^{2}_{\mathbb{Q}}(\mathfrak{X})\to\oplus
H^{2}_{\mathbb{Q}}(\mathfrak{D}_{i})$.

######  Lemma 3.11.

In this situation the above condition holds so long as each degeneration of
$\mathfrak{D}_{i}$ is simple and $H^{2}_{\mathbb{Q}}(\mathfrak{X})\to\oplus
H^{2}_{\mathbb{Q}}(\mathfrak{D}_{i})$ is surjective.

###### Proof.

We apply Theorem 3.22 of [16].

∎

Constructing examples of such degenerations seems to be a hard problem, and
the initial assumptions on surjectivity are certainly not sharp, it rarely
includes the case of even toric hypersurfaces! One family of examples can be
constructed via subdividing the hypercube $[-1,1]^{k}$ along the coordinate
hyperplanes and studying an anti-canonical section. This produces a $k$-Tyurin
degeneration whose minimal strata are therefore points. There is a further
toric degeneration given by scaling the toric parameters.

A better solution than classifying examples would be to extend the work of [5]
and [31] to the case of log toroidal families over bases where the log
structure on the base is allowed to change. How much of the theory can be
extended from the fibrewise theory is at present unclear. Nonetheless there
are some forthcoming results of Doran, Kostiuk and You in which they prove
gluing formulae for periods under such higher rank degenerations. Since their
results do not require such strong assumptions one would hope that better
technology would allow us to lift the restrictions here.

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Lawrence J. Barrott
Boston College, Boston, USA
<EMAIL_ADDRESS>

Charles F. Doran
University of Alberta, Edmonton, Canada
Center for Mathematical Sciences and Applications, Cambridge, USA
<EMAIL_ADDRESS>