Intersection theory of toroidal compactifications of A_4

Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces

It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily-Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line.

Stable cohomology of the perfect cone toroidal compactification of ${\mathcal A}_g$

We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification of the moduli space of complex principally polarized abelian varieties stabilizes, in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification stabilizes (in low degree). For degree up to …

Stable Betti numbers of (partial) toroidal compactifications of the moduli space of abelian varieties

We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of ${\mathcal A}_g$, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers i…

Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties

The main purpose of this paper is to present a conceptual approach to understanding the extension of the Prym map from the space of admissible double covers of stable curves to different toroidal compactifications of the moduli space of principally polarized abelian varieties. By separating the combinatorial problems from the geometric aspects we can reduce this to the computation of certain monodromy cones. In this way we not only shed new light on the extension results of Alexeev, Birkenhake, Hulek, and Vologodsky for the second Voronoi toroidal compactification, but we also apply this to ot…

Co-abelian toroidal compactifications of torsion free ball quotients

Let X' be the toroidal compactification of the quotient of the complex 2-ball by a torsion free lattice G of SU(2,1). We say that X'is co-abelian if there is an abelian surface, birational to X'. The present work can be viewed as an illustration for the presence of a plenty of non-compact co-abelian torsion free toroidal compactifications. More precisely, it shows that all the admissible values for the volume of a torsion free quotient of the complex 2-ball are attained by co-abelian Picard modular ones over Eisenstein numbers. The article provides three types of infinite series of finite unra…

Functorial resolution except for toroidal locus. Toroidal compactification

Let $X$ be any variety in characteristic zero. Let $V \subset X$ be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of $X$ except for V. It is a morphism $f: Y \to X$ , which does not modify the subset $ V $ and transforms $X$ into a toroidal embedding $Y$, with singularities extending those on $V$. Moreover, the exceptional divisor has simple normal crossings on $Y$. The theorem naturally generalizes the Hironaka canonical desingularization. It does not modify the nonsingular locus $V$ and transforms $X$ into a nonsingular variety $Y$. Th…

The nef cone of toroidal compactifications of ${\cal A}_4$

We determine the cone of nef divisors on the Igusa and Voronoi compactifications of the moduli space of principally polarised abelian 4-folds. We also show that the canonical bundle on the Igusa compactification of A_4(n) is ample for n at least 3. Analogous results for A_2 and A_3 were earlier obtained by the first author but the case of A_4 is far less simple.

The fundamental group of a toroidal compactification of a Hermitian locally symmetric space

The present work obtains the fundamental group of a toroidal compactification X' of a non-compact quotient X of a Hermitian symmetric space D of non-compact type by a lattice L in the isometry group G of D. As a consequence it derives the equality of the ranks of the first homology groups of X' and X with integral coefficients. The paper provides also a sufficient condition on a torsion free non-uniform lattice L, under which the fundamental group of X' is residually finite. Articles of Hummel-Schroeder, Hummel and Di Cerbo imply that the toroidal compactifications X' of generic non-compact to…

Jet differentials on toroidal compactifications of ball quotients

We give explicit estimates for the volume of the Green-Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green-Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one.

Compactifications of moduli of elliptic K3 surfaces: stable pair and toroidal

We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space, one for the ramification divisor and another for the rational curve divisor. In the course of the proof, we further develop the theory of integral affine spheres with 24 singularities. We also construct moduli of rational (generalized) elliptic stable slc surfaces of types ${\bf A_n}$ ($n\ge1$), ${\bf C_n}$ ($n\ge0$) and …

Boundary branch divisor of toroidal compactifications

We prove that any toroidal compactification of arithmetic quotient of Hermitian symmetric domain has no boundary branch divisor, in the setting where the algebraic group is of adjoint type.

Special Cycles on Toroidal Compactifications of Orthogonal Shimura Varieties

We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.

Mixed sheaves on Shimura varieties and their higher direct images in toroidal compactifications

Let (P,X) be Shimura data, M=M(P,X,K) the Shimura variety of level K. To an algebraic representation of P, one can associate a mixed sheaf (variation of Hodge structure, l-adic sheaf) on M. In the paper, we study the degeneration of such sheaves along strata in toroidal compactifications of M. The main result (2.8 in the Hodge setting, 3.9 in the l-adic setting) gives a formula for this degeneration in terms of Hochschild cohomology of certain unipotent subgroups of P. The new version differs from the earlier one in that the proof of 2.8 was rewritten. In particular, the effect of Saito's spec…

拟射影簇上主丛的构造

Principal bundles on toroidal compactifications of integral canonical models of abelian-type Shimura varieties

本文研究了在积分典范模型的托里奥利紧致化上，主$\mathcal{G}^c$-和$M^c$-丛的典范扩展构造方法。

Finite volume complex-hyperbolic surfaces, their toroidal compactifications, and geometric applications

We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Moreover, several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riemannian metrics of nonpositive curvature, but which do not admit Kähler metrics of nonpositive curvature. An infinite class of such examples arise as smooth toroidal compactifications of ball quotients.

Semi-toric and toroidal compactifications as log minimal models, and applications to weak K-moduli

We give a characterization of toroidal (resp., semi-toric) compactifications due to Ash-Mumford-Rapoport-Tai (resp., Looijenga) as log minimal models and apply it to study weak K-moduli compactifications, giving a different proof to a theorem of Alexeev-Engel. We also discuss towards further generalization, in particular revisit Shah-Sterk compactification of moduli of polarized Enriques surfaces to show compatibility with log K-stability.

Intersection Theory on Tropicalizations of Toroidal Embeddings

We show how to equip the cone complexes of toroidal embeddings with additional structure that allows to define a balancing condition for weighted subcomplexes. We then proceed to develop the foundations of an intersection theory on cone complexes including push-forwards, intersections with tropical divisors, and rational equivalence. These constructions are shown to have an algebraic interpretation: Ulirsch's tropicalizations of subvarieties of toroidal embeddings carry natural multiplicities making them tropical cycles, and the induced tropicalization map for cycles respects push-forwards, in…

The intersection cohomology of the Satake compactification of ${\mathcal A}_g$ for $g\le 4$

We completely determine the intersection cohomology of the Satake compactifications of the moduli space of principally polarized abelian varieties in genera 2,3,4, except for the degree 10 intersection cohomology in genus 4. We also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the Satake compactification in these genera. As a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved.

On the classification of toroidal compactifications with $3\bar{c}_{2}=\bar{c}^{2}_{1}$ and $\bar{c}_{2}=1$

We classify the smallest finite volume complex hyperbolic surfaces with cusps which admit smooth toroidal compactifications and which are not birational to a bi-elliptic surface. Remarkably, there is only one such surface which appears to be the compactification of a Picard modular surface.

Classification and arithmeticity of toroidal compactifications with $3\bar{c}_{2}=\bar{c}^{2}_{1}=3$

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally discovered by Hirzebruch, is the blowup of an Abelian surface at one point. The others are bielliptic surfaces blown up at one point. These appear to be the first known examples of smooth toroidal compa…

Toroidal compactifications of integral models of Shimura varieties of Hodge type

We construct projective toroidal compactifications for integral models of Shimura varieties of Hodge type. We also construct integral models of the minimal (Satake-Baily-Borel) compactification. Our results essentially reduce the problem to understanding the integral models themselves. As such, they cover all previously known cases of PEL type, as well as all cases of Hodge type involving parahoric level structures. At primes where the level is hyperspecial, we show that our compactifications are canonical in a precise sense. We also provide a new proof of Y. Morita's conjecture on the everywh…

Irrational periods of Hilbert Eisenstein series via toroidal compactification

We show that the periods of the holomorphic Eisenstein series of level 1, parallel weight 2, on a Hilbert modular surface are not rational, even up to scaling. This is deduced from a study of the mixed Hodge structure on the cohomology of the Hilbert modular surface, where we find extension classes related to the units of the real quadratic field. We prove similar results for Hilbert modular varieties of all dimensions, and produce extensions of Galois representations in étale cohomology. The key point is to study the restriction of the canonical extension of the Hodge bundle to the boundary o…

Toroidal compactifications of torsion free local complex hyperbolic spaces

Let B be the complex n-dimensional ball and X' be the toroidal compactification of a quotient B/G by a torsion free lattice G of SU(n,1). For an arbitrary G-rational boundary point p, denote by U(p) the commutant of the unipotent radical of the stabilizer of p in SU(n,1) and put G(U) for the subgroup of G, generated by the intersections of G with U(p) for all G-rational boundary points p. The present note establishes that the fundamental group of X' is isomorphic to the quotient G / G(U). As a consequence, the first integral homology group of X' turns to be a quotient of the first integral hom…

On Calabi-Yau compactifications of toric Landau-Ginzburg models for Fano complete intersections

Toric Landau--Ginzburg models of Givental's type for Fano complete intersections are known to have Calabi--Yau compactifications. We give an alternative proof of this fact. As an output of our proof we get a description of fibers over infinity for compactified toric Landau--Ginzburg models.

Extending Torelli map to toroidal compactifications of Siegel space

It has been known since the 1970s that the Torelli map $M_g \to A_g$, associating to a smooth curve its jacobian, extends to a regular map from the Deligne-Mumford compactification $\bar{M}_g$ to the 2nd Voronoi compactification $\bar{A}_g^{vor}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $\bar{A}_g^{perf}$ is also regular, and moreover $\bar{A}_g^{vor}$ and $\bar{A}_g^{perf}$ share a common Zariski open neighborhood of the image of $\bar{M}_g$. We also show that the map to the Igusa monoidal transform (central cone compactification) is NOT regul…

Shimura 簇环面紧化上 $p$-adic shtukas 的典范扩张

Canonical extensions of $p$-adic shtukas on toroidal compactifications of Shimura varieties

本文在 Abel 型 Shimura 簇的环面紧化的整模型上，通过研究边界处 shtuka 的“退化”，构造了具有拟抛物水平结构的 $p$-adic shtuka 的典范扩张，并定义了 Pappas-Rapoport 意义下的典范整模型。

球商的环面紧化的曲率性质与Shafarevich猜想

Curvature properties and Shafarevich conjecture for toroidal compactifications of ball quotients

本文研究有限体积复双曲流形的环面紧化，探讨了其上满足非正曲率性质的Kähler度量的存在性，并通过对足够深的非均匀算术格群商证明万有覆盖为Stein，验证了Shafarevich猜想。

The Geometry on Smooth Toroidal Compactifications of Siegel varieties

This is a part of our joint program. The purpose of this paper is to study smooth toroidal compactifications of Siegel varieties and their applications, we also try to understand the Kähler-Einstein metrics on Siegel varieties through the compactifications. Let $A_{g,Γ}:=H_g/Γ$ be a Siegel variety, where $H_g$ is the genus-$g$ Siegel space and $Γ$ is an arithmetic subgroup in $Aut(H_g)$. There are four aspects of this paper : 1.There is a correspondence between the category of degenerations of Abelian varieties and the category of limits of weight one Hodge structures. We show that any cusp of…

Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

For any degenerating Calabi-Yau family, we introduce new limit space which we call galaxy, whose dense subspace is the disjoint union of countably infinite open Calabi-Yau varieties, parametrized by the rational points of the Kontsevich-Soibelman's essential skeleton, while dominated by the Huber adification over the Puiseux series field. Other topics include: projective limits of toroidal compactifications, locally modelled on what we call the limit toric varieties, the way to attach tropicalized family to given Calabi-Yau family, which are weakly related to each other.

On the canonical divisor of smooth toroidal compactifications

In this paper, we show that the canonical divisor of a smooth toroidal compactification of a complex hyperbolic manifold must be nef if the dimension is greater or equal to three. Moreover, if $n\geq 3$ we show that the numerical dimension of the canonical divisor of a smooth $n$-dimensional compactification is always bigger or equal to $n-1$. We also show that up to a finite étale cover all such compactifications have ample canonical class, therefore refining a classical theorem of Mumford and Tai. Finally, we improve in all dimensions $n\geq 3$ the cusp count for finite volume complex hyperb…

皮卡模簇的环面紧化上的有理曲线

Rational curves on toroidal compactifications of Picard modular varieties

该研究改进了皮卡模簇环面紧化典范丛丰富性的维数界，构造了含特定有理曲线的无穷多不可公度非紧皮卡模簇，证明相关曲线在任意高维均存在。

具有高阶非辛自同构的K3曲面模空间的紧化

Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism

该研究针对具有3阶和4阶非辛自同构（不动点含亏格≥2曲线）的K3曲面模空间，描述其Baily-Borel、环面及KSBA几何紧化，对3阶处理所有最大维族，证明环面与KSBA紧化可通过ADE根格简单描述。

Intersection cohomology of Drinfeld's compactifications

Let $X$ be a smooth complete curve, $G$ be a reductive group and $P\subset G$ a parabolic. Following Drinfeld, one defines a compactification $\widetilde{\on{Bun}}_P$ of the moduli stack of $P$-bundles on $X$. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of $\widetilde{\on{Bun}}_P$. The description is given in terms of the combinatorics of the Langlands dual Lie algebra $\check{\mathfrak g}$.

紧化 $A_g$ 的低度余弦学

The low degree cohomology of compactifications of $A_g$

本文研究了紧化 $A_g$ 的低度余弦学，证明了其上同调中仅有限个不可约伽罗瓦表示可能出现，并给出了相关应用。

Multiple realizations of varieties as ball quotient compactifications

We study the number of distinct ways in which a smooth projective surface $X$ can be realized as a smooth toroidal compactification of a ball quotient. It follows from work of Hirzebruch that there are infinitely many distinct ball quotients with birational smooth toroidal compactifications. We take this to its natural extreme by constructing arbitrarily large families of distinct ball quotients with biholomorphic smooth toroidal compactifications.

Toroidal b-divisors and Monge-Ampère measures

We generalize the intersection theory of nef toric (Weil) b-divisors on smooth and complete toric varieties to the case of smooth and complete toroidal embeddings. As a key ingredient we show the existence of a limit measure, supported on the weakly embedded rational conical polyhedral space attached to the toroidal embedding, which arises as a limit of discrete measures defined via tropical intersection theory on the polyhedral space. We prove that the intersection theory of nef toroidal Cartier b-divisors can be extended continuously to nef toroidal Weil b-divisors and that their degree can …

阿贝尔型模形式的整数模型的算术紧化

Arithmetic compactifications of integral models of Shimura varieties of abelian type

本文研究了阿贝尔型模形式整数模型的算术紧化，通过构造合适的尖点标记和锥分解，扩展了扭曲构造，并验证了Pink公式。

Extended period domains, algebraic groups, and higher Albanese manifolds

For a linear algebraic group G over the field of rational numbers, we consider the period domains D classifying G-mixed Hodge structures, and construct the extended period domains as toroidal partial compactifications. We give an interpretation of higher Albanese manifolds by Hain and Zucker by using the above D for some G, and extend them via the toroidal partial compactifications.

环形紧化与Mumford-Tate域的堆叠上同调

Toroidal Compactifications and Stacky Cohomology of Mumford--Tate Domains

本文构造并研究了Mumford-Tate域的log-toric Hodge堆栈，通过将Kato-Usui部分紧化由neat算术群商得，实现了Hodge理论与堆栈边界数据的分离。

Complexes d'intersection des compactifications de Baily-Borel. Le cas des groupes unitaires sur Q

In this work, we calculate the trace of a power of the Frobenius endomorphism on the fibers of the intersection complex of the Baily-Borel compactification of a Shimura variety associated to a unitary group over Q. Our main tool is Pink's theorem about the restriction to the strata of the Baily-Borel compactification of the direct image of a local system on the Shimura variety. To use this theorem, we give a new construction of the intermediate extension of a pure perverse sheaf as a weight truncated of the full direct image. More generally, we are able to define analogs in positive characteri…

Intersection theory of the stable pair compactification of the moduli space of six lines in the plane

We describe sequences of blowups of $\overline{M}_{0,5} \times \overline{M}_{0,5}$ and $\mathbf{P}^2 \times \mathbf{P}^2$ yielding a small resolution of the stable pair compactification $\overline{M}(3,6)$ of the moduli space $M(3,6)$ of six lines in $\mathbf{P}^2$. These blowup sequences can be viewed, respectively, as generalizations of Keel's and Kapranov's constructions of $\overline{M}_{0,n}$. We use these blowup sequences to describe the intersection theory of $\overline{M}(3,6)$. In particular, we show that the Chow ring of any small resolution of $\overline{M}(3,6)$ has a presentation …

Non-affine Landau-Ginzburg models and intersection cohomology

We construct Landau-Ginzburg models for numerically effective complete intersections in toric manifolds as partial compactifications of families of Laurent polynomials. We show a mirror statement saying that the quantum D-module of the ambient part of the cohomology of the submanifold is isomorphic to an intersection cohomology D-module defined from this partial compactification and we deduce Hodge properties of these differential systems.

热带紧化中的退化与$\overline{M}_{0,n}$的热带交理论

Degenerations in tropical compactifications and tropical intersection theory of $\overline{M}_{0,n}$

本文提出热带紧化中子簇极限环的公式，扩展了Dickenstein-Feichtner-Sturmfels和Katz的结果，并应用于$\overline{M}_{0,n}$的热带化研究，引入了新的热带$ψ$-超曲面。

Intersection cohomology of the Uhlenbeck compactification of the Calogero-Moser space

We study the natural Gieseker and Uhlenbeck compactifications of the rational Calogero-Moser phase space. The Gieseker compactification is smooth and provides a small resolution of the Uhlenbeck compactification. This allows computing the IC stalks of the Uhlenbeck compactification.

A desingularization of Kontsevich's compactification of twisted cubics in $V_5$

By definition, the del Pezzo $3$-fold $V_5$ is the intersection of $\mathrm{Gr}(2,5)$ with three hyperplanes in $\mathbb{P}^9$ under the Plücker embedding. Rational curves in $V_5$ have been studied in various contents of Fano geometry. In this paper, we propose an explicit birational relation of the Kontsevich and Simpson compactifications of twisted cubic curves in $V_5$. As a direct corollary, we obtain a desingularized model of Kontsevich compactification which induces the intersection cohomology group of Kontsevich's space.

On the cone of effective surfaces on $\overline{\mathcal A}_3$

We determine five extremal effective rays of the four-dimensional cone of effective surfaces on the toroidal compactification $\overline{\mathcal A}_3$ of the moduli space ${\mathcal A}_3$ of complex principally polarized abelian threefolds, and we conjecture that the cone of effective surfaces is generated by these surfaces. As the surfaces we define can be defined in any genus $g\ge 3$, we further conjecture that they generate the cone of effective surfaces on the perfect cone toroidal compactification of ${\mathcal A}_g$ for any $g\ge 3$.

The perfect cone compactification of quotients of type IV domains

The perfect cone compactification is a toroidal compactification which can be defined for locally symmetric varieties. Let $\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}$ be the perfect cone compactification of the quotient of the type IV domain $D_{L}$ associated to an even lattice $L$. In our main theorem we show that the pair ${ (\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}, Δ/2) }$ has klt singularities, where $Δ$ is the closure of the branch divisor of ${ D_{L}/\widetilde{O}^{+}(L) }$. In particular this applies to the perfect cone compactification of the moduli space of $2d$-polarised $K3$ surf…

Nisnevich local Good compactifications

For a local complete intersection morphism, we establish fiberwise denseness in the $n$-dimensional irreducible components of the compactification Nisnevich locally.

Applications of principal isogenies to constructions of ball quotient surfaces

Let $(({\mathbb B} / Γ_1)', T(1))$ be á torsion free toroidal compactification with abelian minimal model $(A_1, D(1))$. An arbitrary principal isogeny $μ_a : A_2 \rightarrow A_1$, $a \in {\mathbb C}$ pulls-back $(A_1, D(1))$ to the abelian minimal model $(A_2, D(2))$ of a torsion free toroidal compactification $(({\mathbb B} / Γ_2)', D(2))$. The present work makes use of the isogeny pull-backs of abelian ball quotient models, towards the construction of infinite series of mutually non-birational co-abelian torsion free Galois covers $(({\mathbb B} / Γ_n)', T(n))$ of a ball quotient compactifi…

Hodge Bundles on Smooth Compactifications of Siegel Varieties and Applications

We study Hodge bundles on Siegel varieties and their various extensions to smooth toroidal compactifications. Precisely, we construct a canonical Hodge bundle on an arbitrary Siegel variety so that the holomorphic tangent bundle can be embedded into the Hodge bundle, and we observe that the Bergman metric on the Siegel variety is compatible with the induced Hodge metric. Therefore we obtain the asymptotic estimate of the Bergman metric explicitly. Depending on these properties and the uniformitarian of Kähler-Einstein manifold, we study extensions of the tangent bundle over any smooth toroidal…

Complexes d'intersection des compactifications de Baily-Borel : Le cas des variétés de Siegel

In this work, we calculate the trace of a Hecke correspondance composed with a power of the Frobenius endomorphism on the fibre of the intersection complexes of the Baily-Borel compactification of a Siegel modular variety. Our main tool is Pink's theorem about the restriction to the strata of the Baily-Borel compactification of the direct image of a local system on the Shimura variety. To use this theorem, we give a new construction of the intermediate extension of a pure perverse sheaf as a weight truncation of the full direct image. More generally, we are able to define analogs in positive c…

A resolution of singularities for Drinfeld's compactification by stable maps

Drinfeld's relative compactification plays a basic role in the theory of automorphic sheaves, and its singularities encode representation-theoretic information in the form of intersection cohomology. We introduce a resolution of singularities consisting of stable maps from nodal deformations of the curve into twisted flag varieties. As an application, we prove that the twisted intersection cohomology sheaf on Drinfeld's compactification is universally locally acyclic over the moduli stack of $G$-bundles at points sufficiently antidominant relative to their defect.

On the GHKS compactification of the moduli space of K3 surfaces of degree two

We investigate a toroidal compactification of the moduli space of K3 surfaces of degree $2$ originating from the program formulated by Gross-Hacking-Keel-Siebert. This construction uses Dolgachev's formulation of mirror symmetry and the birational geometry of the mirror family. Our main result in an analysis of the toric fan. For this we use the methods developed by two of us in a previous paper.

Effective results for complex hyperbolic manifolds

The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We estimate the number of ends of such manifolds in terms of their volume. We give effective bounds on the number of complex hyperbolic manifolds with given upper bounds on the volume. Moreover, we give two sided bounds on their Picard numbers in terms of the volume and the number of…