Hiroshi Sato
In this paper, we investigate when there exists a wild hypersurface bundle over a smooth proper toric variety in positive characteristic. In particular, we determine the possibilities for toric varieties with Picard number at most three or toric Fano varieties of dimension at most four. Moreover, we can construct wild hypersurface bundles over them.
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We introduce an algebraic method for describing the Hodge filtration of degenerating hypersurfaces in projective toric varieties. For this purpose, we show some fundamental properties of logarithmic differential forms on proper equivariant morphisms of toric varieties.
In this article we present a formula for the plurigenera of minimal models of nondegenerate toric hypersurfaces, which is valid in arbitrary dimension and which expresses these invariants through lattice points on the Fine interior. From this formula we derive a formula for the volume of the canonical divisor of the toric hypersurface. We also study the pluricanonical mappings and show under some restrictions how to construct a canonical model of the toric hypersurface.
In response to a question of Reid, we find all anti-canonical Calabi-Yau hypersurfaces $X$ in toric weighted projective bundles over the projective line where the general fiber is a weighted K3 hypersurface. This gives a direct generalization of Reid's discovery of the 95 families of weighted K3 hypersurfaces. We also treat the case where $X$ is fibered over the plane with general fiber a genus one curve in a weighted projective plane.
We study the geometry and cohomology of semiample hypersurfaces in toric varieties. Such hypersurfaces generalize the MPCP-desingularizations of Calabi-Yau ample hypersurfaces in the Batyrev mirror construction. We study the topological cup product on the middle cohomology of semiample hypersurfaces. In particular, we obtain a complete algebraic description of the middle cohomology of regular semiample hypersurfaces in 4-dimensional simplicial toric varieties what would be interesting for physics.
The Kodaira-Spencer map for minimal toric hypersurfaces
本文研究环面超曲面的无穷小形变，引入Kodaira-Spencer映射并计算其核，通过新Laurent多项式使计算显式化，推广了Griffiths关于射影超曲面的结果。
We give detailed descriptions of the period maps of two 2-parameter families of anti-canonical hypersurfaces in toric 3-folds. One of them is related to a Hilbert modular surface, and the other is related to the product of modular curves.
We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.
We show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This sufficient condition is always satisfied by generic K3 surfaces embedded in Fano toric 3-folds.
Given the same anti-canonical linear system on two distinct toric varieties, we provide a derived equivalence between partial crepant resolutions of the corresponding stacky hypersurfaces. The applications include: a derived unification of toric mirror constructions, calculations of Picard lattices for linear systems of quartic surfaces, and a birational reduction of Reid's list to 81 families.
Stable rationality of hypersurfaces of mock toric variety I
该研究提出作为环面簇推广的模拟环面簇，指出Del-Pezzo曲面是其非环面实例，还给出了构造这类簇中超曲面的严格环面模型的充分条件，为相关代数几何研究提供新工具。
We provide a combinatorial characterization of monomial linear systems on toric varieties whose general member is quasismooth. This is given both in terms of the Newton polytope and in terms of the matrix of exponents of a monomial basis.
We provide a construction of examples of semistable degeneration via toric geometry. The applications include a higher dimensional generalization of classical degeneration of K3 surface into 4 rational components, an algebraic geometric version of decomposing K3 as the fiber sum of two E(1)'s as well as it's higher dimensional generalizations, and many other new examples.
Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{G}_a$ be the additive group of $\mathbb{K}$. We say that an irreducible algebraic variety $X$ of dimension $n$ over the field $\mathbb{K}$ admits an additive action if there is a regular action of the group $\mathbb{G}_a^n = \mathbb{G}_a \times \ldots \times \mathbb{G}_a$ ($n$ times) on $X$ with an open orbit. In this paper we find all projective toric hypersurfaces admitting additive action.
Whitney equisingularity for families of hypersurfaces in toric varieties
本文在具有任意奇点集的环面簇框架下，建立奇点未必孤立的函数族对应超曲面族满足Whitney等奇性的条件，拓展了Eyral与Oka此前针对复空间上函数族的相关研究结果。
We provide a sufficient condition for a general hypersurface in a $\mathbb Q$-Fano toric variety to be a Calabi-Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund-Hübsch-Krawitz construction in case the ambient is a $\mathbb Q$-Fano toric variety with torsion free class group and the defining polynomial is not necessarily of Delsarte type. Finally, we introduce a duality between families of Calabi-Yau hypersurfaces which includes both Batyrev and Berglund-Hübsch-Krawitz mirror constructions. This is given in terms of a polar duality between pairs …
A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $\mathbb{Z}_2$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope $Δ$ there are families of hypersurfaces with the Newton polytopes $(λΔ)_{λ\in \mathbb{N}}$ that are asymptotically maximal when $λ$ tends to infinity. We also show that these results ge…
Stable rationality of hypersurfaces of mock toric variety II
本文用模拟环面簇替代环面簇构造退化族，借助母题方法证明了射影空间中非常一般超曲面的稳定非有理性可推出格拉斯曼簇中相应超曲面的稳定非有理性。
We build two embedded resolution procedures of a quasi-ordinary singularity of complex analytic hypersurface, by using toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection of the singularity. This result answers an open problem of Lipman in Equisingularity and simultaneous resolution of singularities, Resolution of Singularities, Progress in Mathematics No. 181, 2000, 485-503. In the first procedure the singularity is embedded as hypersurface. In the second procedure, which is inspired by a work of Goldin and Teissier for plane curves (see…
We study Gauss-Manin systems of non tame Laurent polynomial functions. We focuse on Hori-Vafa models, which are the expected mirror partners of the small quantum cohomology of smooth hypersurfaces in weighted projective spaces.
On the moduli of hypersurfaces in toric orbifolds
本文利用非约化GIT与$A$-判别式构造环面轨形超曲面模空间，给出加权射影空间中拟光滑超曲面的稳定性条件及自同构群有限性结论。
There are easy "polynomial" deformations of Calabi-Yau hypersurfaces in toric varieties performed by changing the coefficients of the defining polynomial of the hypersurface. In this paper, we explicitly constructed the ``non-polynomial'' deformations of Calabi-Yau hypersurfaces, which arise from deformations of the ambient toric variety.
We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in [3]. We also show that the Hodge Conjecture holds asymptotica…
The arithmetic volume of hypersurfaces in toric varieties and Mahler measures
该研究确定了光滑射影环面簇中超曲面的典范算术体积，并由此证明了这类超曲面上的广义Hodge指标定理，为算术代数几何领域相关问题提供了新结果。
An almost-toric hypersurface is parameterized by monomials multiplied by polynomials in one extra variable. We determine the Newton polytope of such a hypersurface, and apply this to give an algorithm for computing the implicit equation.
The most fundamental example of mirror symmetry compares the Fermat hypersurfaces in P^n and P^n/G, where G is a finite group that acts on P^n and preserves the Fermat hypersurface. We generalise this to hypersurfaces in Grassmannians, where the picture is richer and more complex. There is a finite group G that acts on the Grassmannian Gr(n,r) and preserves an appropriate Calabi-Yau hypersurface. We establish how mirror symmetry, toric degenerations, blow-ups and variation of GIT relate the Calabi-Yau hypersurfaces inside Gr(n,r) and Gr(n,r)/G. This allows us to describe a compactification of …
We study the structure of rational Picard groups of hypersurfaces of toric varieties. By using the fan structure associated to the ambient toric variety, an explicit basis of the Picard group is described by certain combinatorial data. We shall also discuss the application to Calabi-Yau spaces.
Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms between toric varieties. In particular, a complete toric description of fibers - both generic and non-generic -, image, and the flattening stratification of a toric morphism are given. Two examples are provided to illustrate the discussions. We then turn to the study of the restriction of a toric morphism to a toric hypersurface. The details of this can be und…
Toric Degenerations of Low Degree Hypersurfaces
本文证明了n维射影空间中一般d次超曲面具有环面Gröbner退化的充要条件是d≤2n-1。
We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry.…
Lefschetz section theorems for tropical hypersurfaces
本文建立环面簇热带超曲面的 Lefschetz 超平面截面定理，证明整系数热带同调群在紧致或 $\mathbb{R}^n$ 情形下无挠，并建立其与复超曲面 Hodge 数及 Hodge-Deligne 数的联系。
We identify the twisted sectors of a compact simplicial toric variety. We do the same for a generic nondegenerate Calabi-Yau hypersurface of an $n$-dimensional simplicial Fano toric variety and then explicitly compute $h^{1,1}_{orb}$ and $h^{n-2,1}_{orb}$ for the hypersurface. We give applications to the orbifold string theory conjecture and orbifold mirror symmetry.
On the arithmetic of rational hypersurfaces in toric varieties
本文研究环簇中特定多重次数超曲面的算术性质，证明其为k-有理且双有理等价于射影空间，推导其有理点分布结果，d=1时对应偶维费马三次曲面。
Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes
本文确定了由Fano多面体导出的环面$K3$超曲面的Néron-Severi格，通过引入特定椭圆纤维化并证明其生成元，从而证明了Dolgachev猜想。
This paper investigates the density of hypersurfaces in a projective normal simplicial toric variety over a finite field having a quasismooth intersection with a given quasismooth subscheme. The result generalizes the formula found by B. Poonen for smooth projective varieties. As an application, we further analyze the density of hypersurfaces with bounds on their number of singularities and on the length of their singular schemes.
We prove the Integral Hodge Conjecture for curve classes on smooth varieties of dimension at least three with nef anticanonical divisor constructed as a complete intersection of ample hypersurfaces in a smooth toric variety. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric MMP, we prove that in each case, $H_2(X,\mathbb{Z})$ is generated by classes of rational curves.
We study in detail the family of Calabi-Yau hypersurfaces defined by x_1+...+x_n+1/(x_1...x_n)=t over a finite field k. We determine its local and global monodromy and the trivial factors of its moment zeta function and Dwork's unit root zeta function.
We consider regular Calabi-Yau hypersurfaces in $N$-dimensional smooth toric varieties. On such a hypersurface in the neighborhood of the large complex structure limit point we construct a fibration over a sphere $S^{N-1}$ whose generic fibers are tori $T^{N-1}$. Also for certain one-parameter families of such hypersurfaces we show that the monodromy transformation is induced by a translation of the $T^{N-1}$ fibration by a section. Finally we construct a dual fibration and provide some evidence that it describes the mirror family.
We prove the structure theorem of the intersection complexes of toric varieties in the category of mixed Hodge modules. This theorem is due to Bernstein, Khovanskii and MacPherson for the underlying complexes with rational coefficients. As a corollary the Euler characteristic Hodge numbers of non-degenerate toric hypersurface can be determined by the Euler characteristic subtotal Hodge numbers together with combinatorial data of Newton polyhedra. This is used implicitly in an explicit formula by Batyrev--Borisov. Note that a formula for the Euler characteristic subtotal Hodge numbers in terms …
We study elliptically fibered K3 surfaces, with sections, in toric Fano threefolds which satisfy certain combinatorial properties relevant to F-theory/Heterotic duality. We show that some of these conditions are equivalent to the existence of an appropriate notion of a Weierstrass model adapted to the toric context. Moreover, we show that if in addition other conditions are satisfied, there exists a toric semistable degeneration of the elliptic K3 surface which is compatible with the elliptic fibration and F-theory/Heterotic duality.
Tropical Periods for Calabi-Yau Hypersurfaces in non--Fano Toric Varieties
本文利用多面体描述非-Fano环面簇及其反典范Calabi-Yau超曲面，通过广义Duistermaat-Heckman测度从镜像流形周期证明$\widehatΓ$-猜想适用于非-Fano环境空间，扩展了早期工作。
This paper is a continuation of our paper math.AG/0205321 where we have built a combinatorial model for the torus fibrations of Calabi-Yau toric hypersurfaces. This part addresses the connection between the model torus fibration and the complex and Kähler geometry of the hypersurfaces.
We prove the metric version of the SYZ conjecture for a class of Calabi-Yau hypersurfaces inside toric Fano manifolds, by solving a variational problem whose minimizer may be interpreted as a global solution of the real Monge-Ampère equation on certain polytopes. This does not rely on discrete symmetry.
We study the B-model chiral ring of Calabi-Yau hypersurfaces in Batyrev's mirror construction. The main result is an explicit description of a subring of the chiral ring of semiample regular (transversal to torus orbits) Calabi-Yau hypersurfaces. This subring includes the marginal operators and contains all information about the correlation functions used by physicists. Computation of the chiral ring passes through a description of cohomology of semiample hypersurfaces. Here, we develop the techniques for calculating of the cohomology of resolutions.
We use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds. For a mirror pair of toric branes on a mirror pair of toric hypersurfaces we derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs. We define a linear sigma model for the brane geometry and describe a correspondence between dual toric polyhedra and toric brane geometries. The method is applied to study examples with obstructed and classically unobstructed brane moduli at various p…
Toric polar maps and characteristic classes
该研究证明复射影空间中超曲面的环面极映射多重度与特定开集的Chern-Schwartz-MacPherson类系数符号一致，给出平面曲线映射次数的局部不变量公式，并构造了任意维数的双有理环面极映射的不可约超曲面族。
We construct classes in the motivic cohomology of certain 1-parameter families of Calabi-Yau hypersurfaces in toric Fano n-folds, with applications to local mirror symmetry (growth of genus 0 instanton numbers) and inhomogeneous Picard-Fuchs equations. In the case where the family is classically modular the classes are related to Belinson's Eisenstein symbol; the Abel-Jacobi map (or rational regulator) is computed in this paper for both kinds of cycles. For the "modular toric" families where the cycles essentially coincide, we obtain a motivic (and computationally effective) explanation of a p…
Hybrid toric varieties and the non-archimedean SYZ fibration on Calabi-Yau hypersurfaces
本文利用热带收缩构造了一类极大退化 Calabi-Yau 超曲面上的非阿基米德 SYZ 纤维化，并证明在离散对称性条件下其度量势沿纤维为常数。
In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in toric variety. We will also be able to give precise formulation of SYZ mirror conjecture in general (including singular locus and duality of singular fibres).
We describe in purely combinatorial terms dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces. The same structures arise on the base of a special Lagrangian torus fibration in the Strominger-Yau-Zaslow conjecture. We study the topological torus fibration in the large complex structure limit and show that it coincides with our combinatorial model.
Quasi algebraically closed fields, or $C_1$ fields, are defined in terms of a low degree condition. Namely, the field $K$ is $C_1$ if every degree $d$ hypersurface of the projective space $\mathbb{P}_K^n$ contains a $K$-point as soon as $d\leq n$. In this article we define a notion of low toric degree generalizing this condition for hypersurfaces of simplicial projective split toric varieties. This allows us to prove a particular case of the $C_1$ conjecture of Kollár, Lang and Manin : any smooth separably rationally connected variety that can be embedded as such a hypersurface over a $C_1$ fi…
We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times\mathbb{C}$ along $H\times 0$, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to $H$. The main applications concern affine hypersurfaces of general type, for which…
Framed duality and mirror symmetry for toric complete intersections
本文将完全环面簇中的超曲面族$f$-镜像对称系统扩展至完全交子簇族，基于分次对偶发展了相关理论，并探讨了广义Landau-Ginzburg/完全交对应。
For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous combinatorial formula holds in the arithmetic setting: the global height of a 1-codimensional cycle with respect to a toric divisor equipped with a semipositive toric metric can be expressed in terms of mixed integrals of the $v$-adic roof functions associated to the metric and the Legendre-Fenchel dual of the $v$-adic Ronkin function of the Laurent polynomial o…
We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether-Lefschetz divisor in the sense of Hassett.
We find through a systematic analysis that all but 29,223 of the 473.8 million 4D reflexive polytopes found by Kreuzer and Skarke have a 2D reflexive subpolytope. Such a subpolytope is generally associated with the presence of an elliptic or genus one fibration in the corresponding birational equivalence class of Calabi-Yau threefolds. This extends the growing body of evidence that most Calabi-Yau threefolds have an elliptically fibered phase.
We give a method to estimate Seshadri constants on toric varieties at any point. By using the estimations and toric degenerations, we can obtain some new computations or estimations of Seshadri constants on non-toric varieties. In particular, we investigate Seshadri constants on hypersurfaces in projective spaces and Fano 3-folds with Picard number one in detail.
In the spirit of a theorem of Wood, we give necessary and sufficient conditions for a family of germs of analytic hypersurfaces in a smooth projective toric variety X to be interpolated by an algebraic hypersurface with a fixed class in the Picard group of X.
The study of the geometry of Calabi-Yau fourfolds is relevant for compactifications of string theory, M-theory, and F-theory to various dimensions. This work introduces the mathematical machinery to derive the complete moduli dependence of the periods of non-trivial three-forms for fourfolds realized as hypersurfaces in toric ambient spaces. It sets the stage to determine Picard-Fuchs-type differential equations and integral expressions for these forms. The key tool is the observation that non-trivial three-forms on hypersurfaces in toric ambient spaces always stem from divisors that are build…
We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with $h^{1, 1} \geq 140$ or $h^{2, 1} \geq 140$ that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number is 228 or greater. We find that for small $h^{1, 1}$ the fraction of polytopes in the KS database that do not have a genu…
On the Jacobian syzygies for generic toric models
本文针对仿射环面中的一般超曲面，在射影空间关联超曲面配置，并计算了相应雅可比代数的最小分次分解。
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