We determine explicitly the Hodge ideals for the determinant hypersurface as an intersection of symbolic powers of determinantal ideals. We prove our results by studying the Hodge and weight filtrations on the mixed Hodge module O_X(*Z) of regular functions on the space X of n x n matrices, with poles along the divisor Z of singular matrices. The composition factors for the weight filtration on O_X(*Z) are pure Hodge modules with underlying D-modules given by the simple GL-equivariant D-modules on X, where GL is the natural group of symmetries, acting by row and column operations on the matrix entries. By taking advantage of the GL-equivariance and the Cohen-Macaulay property of their associated graded, we describe explicitly the possible Hodge filtrations on a simple GL-equivariant D-module, which are unique up to a shift determined by the corresponding weights. For non-square matrices, O_X(*Z) is naturally replaced by the local cohomology modules H^j_Z(X,O_X), which turn out to be pure Hodge modules. By working out explicitly the Decomposition Theorem for some natural resolutions of singularities of determinantal varieties, and using the results on square matrices, we determine the weights and the Hodge filtration for these local cohomology modules.
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.
We define defect for hypersurfaces with A-D-E singularities in complex projective normal Cohen-Macaulay fourfolds having some vanishing properties of Bott-type and prove formulae for Hodge numbers of big resolutions of such hypersurfaces. We compute Hodge numbers of Calabi-Yau manifolds obtained as small resolutions of cuspidal triple sextics and double octics with higher A_j singularities.
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 determine the second fundamental form of a variation of Hodge Structure of a smooth projective hypersurface using the classical identification of the Hodge structure and the action of the infinitesimal variation of Hodge structure with pieces of the Jacobian Ideal of the hypersurface. This simple algebraic description also implies that, in this case, the variation of Hodge structure satisfies a new set of second order partial differential equations.Comments are welcome!
In this paper we are interested in proving that the infinitesimal variations of Hodge structure of hypersurfaces of high enough degree lie in a proper subvariety of the variety of all infinitesimal variations. This is proved using a space of symmetrizers as defined by Donagi, to identify a geometric structure carried by the variety of all infinitesimal variations of Hodge structure.
We consider a hypersurface in $\mathbb{C}^n$ with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of the Jordan blocks in the Milnor monodromy.
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.
We study the Milnor monodromies of non-isolated hypersurface singularities and show that the reduced cohomology groups of the Milnor fibers are concentrated in the middle degree for some eigenvalues of the monodromies. As an application of this result, we give an explicit formula for some parts of their Jordan normal forms.
We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal $V$-filtration modulo the Jacobian ideal. Via the Tjurina subspectrum, we can compare the Hodge ideal spectrum with the Steenbrink spectrum which can be defined by the microlocal $V$-filtration. As a consequence of a formula of Mustata and Popa, these two spectra coincide in the weighted homogeneous case. We prove sufficient conditions for their coincidence and non-coincidence in some non-weighted-homogeneous cases where the defining function is semi-weigh…
We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and the author for the graded pieces with respect to the Hodge filtration of the top cohomology of the hypersurface complement in many new cases. A classical result by Severi on the position of the singularities of a nodal surface in $\mathbb{P}^3$ is improved and applications to deformation theory of nodal surfaces are given.
Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type
Mathematical School of Science, Xiamen University（厦门大学数学科学学院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
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…
Let $n$ be a positive even integer and $d$ a positive integer . To every complete family $Z$ of n dimensional degree d hypersurfaces in the projective space with isolated A-D-E singularities we construct according to an idea of Carlson-Toledo a Deligne-Mumford stack $\bar Z$ whose moduli space is $Z$ such that the monodromy representation extends. We study the corresponding periods mapping and establish an infinitesimal Torelli theorem along the isosingular strata of lZ$ under transversality assumptions. We apply this result to prove Steiness of the universal covering space of $\bar Z$.
A Hodge structure V of weight k on which a CM field acts defines, under certain conditions, a Hodge structure of weight k-1, its half twist. In this paper we consider hypersurfaces in projective space with a cyclic automorphism which defines an action of a cyclotomic field on a Hodge substructure in the cohomology. We determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces. We use our results to prove the existence of a Kuga-Satake correspondance for certain cubic 4-folds.
We calculate the fibre integrals of the hypersurface in a torus in the form of their Mellin transforms. Especially, our method works efficiently for an affine hypersurface defined by a so called simpliciable polynomial. The relations between poles of Mellin transforms of fibre integrals, the mixed Hodge structure of the cohomology of the hypersurface, the hypergeometric differential equation and the Euler characteristic of fibres are clarified.
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 …
For a smooth hypersurface of even dimension, the quadratic character of the absolute Galois group defined by the determinant of the l-adic cohomology of middle dimension is computed via the square root of the discriminant of a defining polynomial of the hypersurface.
We prove that the infinitesimal variations of Hodge structure arising in a number of geometric situations are non-generic. In particular, we consider the case of generic hypersurfaces in complete smooth projective toric varieties, generic hypersurfaces in weighted projective spaces and generic complete intersections in projective space and show that, for sufficiently high degrees, the corresponding infinitesimal variations are non-generic.
In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those varieties we calculate values of Hypergeometric series on certain $CM$ points. Our methods is based on the calculation of the Picard-Fuchs equations in higher dimensions, reducing them to the Gauss equation and then applying the Abelian Subvariety Theorem to the corresponding hypergeometric abelian varieties.
In this text we apply the methods of Hodge theory for isolated hypersurface singularities to define a signature for the Grothendieck residue pairing of these singularities.
Let $f:\mathbb{C}^{n+1} \to \mathbb{C}$ be a germ of hypersurface with isolated singularity. One can associate to $f$ a polarized variation of mixed Hodge structure $\mathcal{H}$ over the punctured disc, where the Hodge filtration is the limit Hodge filtration of W. Schmid and J. Steenbrink. By the work of M. Saito and P. Deligne the VMHS associated to cohomologies of the fibers of $f$ can be extended over the degenerate point $0$ of disc. The new fiber obtained in this way is isomorphic to the module of relative differentials of $f$ denoted $Ω_f$. A mixed Hodge structure can be defined on $Ω_…
We consider the residual B-model variation of Hodge structure of Iritani defined by a family of toric Calabi--Yau hypersurfaces over a punctured disk $D \setminus \{ 0\}$. It is naturally extended to a logarithmic variation of polarized Hodge structure of Kato--Usui on the whole disk $D$. By restricting it to the origin, we obtain a polarized logarithmic Hodge structure (PLH) on the standard log point. In this paper, we describe the PLH in terms of the integral affine structure of the dual intersection complex of the toric degeneration in the Gross--Siebert program.
We employ the inductive structure of determinantal varieties to calculate the mixed Hodge module structure of local cohomology modules with determinantal support. We show that the weight of a simple composition factor is uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariant structure of the Hodge filtration on each local cohomology module. Finally, as an application, we provide a formula for the generation level of the Hodge filtration on these modules.
The characteristic polynomial of an $r$-tuple $(A_1,..., A_r)$ of $n \times n$ matrices is the determinant $\det(x_0 I + x_1 A_1 + ... + x_r A_r)$. We show that if $r$ is at least 3 and $A = (A_1,..., A_r)$ is an $r$-tuple of matrices in general position, then up to conjugacy there are only finitely many $r$-tuples of matrices with the same characteristic polynomial as $A$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in $P^r$ is irreducible of dimension $(r-1)n^2 + 1$.
We show that for a hypersurface Batyrev's stringy E-function can be seen as a residue of the Hodge zeta function, a specialization of the motivic zeta function of Denef and Loeser. This is a nice application of inversion of adjunction. If an affine hypersurface is given by a polynomial that is non-degenerate with respect to its Newton polyhedron, then the motivic zeta function and thus the stringy E-function can be computed from this Newton polyhedron (by work of Artal, Cassou-Nogues, Luengo and Melle based on an algorithm of Denef and Hoornaert). We use this procedure to obtain an easy way to…
We prove results concerning the behavior of Hodge ideals under restriction to hypersurfaces or fibers of morphisms, and addition. The main tool is the description of restriction functors for mixed Hodge modules by means of the $V$-filtration.
We describe a relation between two invariants that measure the complexity of a hypersurface singularity. One is the Hodge spectrum which is related to the monodromy and the Hodge filtration on the cohomology of the Milnor fiber. The other is the multiplier ideal, having to do with log resolutions.
In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial $f$ in $\C^{n+1}$, where $f$ satisfies a certain regularity condition at infinity (and hence has isolated singularities). We give an algorithm which produces a basis of a localization of the Brieskorn module which is compatible with its mixed Hodge structure. As an application we show that the notion of a Hodge cycle in regular fibers of $f$ is given in terms of the vanishing of integrals of certain polynomial $n$-forms in $\C^{n+1}$ over topological $n$-cycles on the fibers of $f$. Since the $n$-th …
Mirrors $X^{\vee}$ of quasi-smooth Calabi-Yau hypersurfaces $X$ in weighted projective spaces ${\Bbb P}(w_0, \ldots, w_d)$ can be obtained as Calabi-Yau compactifications of non-degenerate affine toric hypersurfaces defined by Laurent polynomials whose Newton polytope is the lattice simplex spanned by $d+1$ lattice vectors $v_i$ satisfying the relation $\sum_i w_i v_i =0$. In this paper, we compute the stringy $E$-function of mirrors $X^\vee$ and compare it with the Vafa's orbifold $E$-function of quasi-smooth Calabi-Yau hypersurfaces $X$. As a result, we prove the equalities of Hodge numbers …
We define and study Hodge ideals associated to a coherent ideal sheaf J on a smooth complex variety, via algebraic constructions based on the already existing concept of Hodge ideals associated to Q-divisors. We also define the generic minimal exponent of J, extending the standard invariant for hypersurfaces. We relate it to Hodge ideals, and show that it is a root of the Bernstein-Sato polynomial of J.
We characterize the smallest codimension components of the Hodge locus of smooth degree $d$ hypersurfaces of the projective space $\mathbb{P}^{n+1}$ of even dimension $n$, passing through the Fermat variety (with $d\neq 3,4,6$). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension $\frac{n}{2}$. Furthermore, we prove that among all the local Hodge loci associated to a non-linear cycle passing through Fermat, the ones associated to a complete intersection cycle of type $(1,1,\ldots,1,2)$ attain the minimal possible codimension of their Zariski tangent s…
We compare the sets of Calabi-Yau threefolds with large Hodge numbers that are constructed using toric hypersurface methods with those can be constructed as elliptic fibrations using Weierstrass model techniques motivated by F-theory. There is a close correspondence between the structure of "tops" in the toric polytope construction and Tate form tunings of Weierstrass models for elliptic fibrations. We find that all of the Hodge number pairs ($h^{1, 1},h^{2, 1}$) with $h^{1,1}$ or $h^{2, 1}\geq 240$ that are associated with threefolds in the Kreuzer-Skarke database can be realized explicitly b…
The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure associated to the degeneration which is encoded as a limit mixed Hodge structure. However, this invariant is difficult to compute in practice. Using the techniques of tropical geometry we present a new formula for the motivic nearby fiber. Moreover, since there is a range of available software implementing the main algorithms in tropical geometry, our formula can be…
Given a polynomial map $f:\Bbb C^{n+1}\to\Bbb C$, one can attach to it a geometrical variation of mixed Hodge structures (MHS) which gives rise to a limit MHS. The equivariant Hodge numbers of this MHS are analytical invariants of the polynomial map and reflect its asymptotic behaviour. In this paper we compute them for a class of generic polynomials, in terms of Hodge numbers attached to isolated hypersurface singularities and Hodge numbers of cyclic coverings of projective space branched along a hypersurface.
We use methods from birational geometry to study the Hodge and weight filtrations on the localization along a hypersurface. We focus on the lowest piece of the Hodge filtration of the submodules arising from the weight filtration. This leads to a sequence of ideal sheaves called weighted multiplier ideals. The last ideal of this sequence is a multiplier ideal (and a Hodge ideal), and we prove that the first is the adjoint ideal. We also study the local and global properties of weighted multiplier ideals and their applications to singularities of hypersurfaces of smooth varieties.
We prove a formula for the Hodge numbers of square-free divisors of Calabi-Yau threefold hypersurfaces in toric varieties. Euclidean branes wrapping divisors affect the vacuum structure of Calabi-Yau compactifications of type IIB string theory, M-theory, and F-theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor $D$ requires counting fermion zero modes, which depend on the Hodge numbers $h^i({\cal{O}}_D)$. Suppose that $X$ is a smooth Calabi-Yau threefold hypersurface in a toric variety $V$, and let $D$ be the restriction to $X$ of a square-free divisor of $V$…
In this note we examine the question of expressing the determinant of the push forward of a symmetric line bundle on an abelian fibration in terms of the pull back of the relative dualizing sheaf via the zero section.
We solved the long-standing problem of describing the cohomology ring of semiample hypersurfaces in complete simplicial toric varieties. Also, the monomial-divisor mirror map is generalized to a map between the whole Picard group and the space of infinitesimal deformations for a mirror pair of Calabi-Yau hypersurfaces. This map is compatible with certain vanishing limiting products of the subrings of the chiral rings, on which the ring structure is related to a product of the roots of $A$-type Lie algebra.
