We establish a polynomial recursion formula for linear Hodge integrals. It is obtained as the Laplace transform of the cut-and-join equation for the simple Hurwitz numbers. We show that the recursion recovers the Witten-Kontsevich theorem when restricted to the top degree terms, and also the combinatorial factor of the lambda_g formula as the lowest degree terms.
Using Atiyah-Bott localization on the space of stable maps to the stack quotient $[\mathbb{P}^1/\mathbb{Z}_2]$, we find recursions that determine all Hodge integrals with descendent insertions at one marked point on the hyperelliptic locus $\overline{\mathcal{H}}_{g, 2g + 2} \subseteq \overline{\mathcal{M}}_{g, 2g + 2}$. The initial conditions required for our recursions are gravitational descendents at one marked point, which are known to be $\frac{1}{2}$. We discover a new structure concerning these intersection numbers: for a fixed monomial of $λ$-classes, the resulting family of hyperellip…
Despite the failure of the integral Hodge conjecture, we show that the rational Hodge conjecture implies an integral version (modulo torsion) of the absolute Hodge conjecture.
In this paper we discuss an obstruction to the integral Hodge conjecture, which arises from certain behavior of vanishing cycles. This allows us to construct new counter-examples to the integral Hodge conjecture. One typical such counter-example is the product of a very general hypersurface of odd dimension and an Enriques surface. Our approach generalizes the degeneration argument of Benoist-Ottem.
We define a map $\mathcal{P}_M$ attached to any polarized Hodge module $M$ such that the restriction of $\mathcal{P}_M$ to a locus on which $M$ is a variation of Hodge structures induces the usual period integral pairing for this variation of Hodge structures. In the case that $M$ is the minimal extension of a simple polarized variation of Hodge structures $V$, we show that the homotopy image of $\mathcal{P}_M$ is the minimal extension of the graph morphism of the usual period integral map for $V$.
This paper initiates a study of Hodge integrals on moduli spaces of pseudostable curves. We prove an explicit comparison formula that allows one to effectively compute any pseudostable Hodge integral in terms of intersection numbers on moduli spaces of stable curves, and we use this comparison to prove that pseudostable Hodge integrals are equal to their stable counterparts when they are linear in lambda classes, but not when they are nonlinear. This suggests that pseudostable Gromov-Witten invariants are equal to usual Gromov-Witten invariants for target curves, but not for higher-dimensional…
We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, any product of Jacobians of smooth projective curves over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a proper curve of compact type over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture …
We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.
A conjectural relationship between the GUE partition function with even couplings and certain special cubic Hodge integrals over the moduli spaces of stable algebraic curves is under consideration.
As the first step of proving the Hodge-FVH correspondence recently proposed in [19], we derive the Virasoro constraints and the Dubrovin--Zhang loop equation for special cubic Hodge integrals. We show that this loop equation has a unique solution, and provide a new algorithm for the computation of these Hodge integrals. We also observe the gap phenomenon for certain special cubic Hodge free energies.
We provide a closed form expression for linear Hodge integrals on the hyperelliptic locus. Specifically, we find a succinct combinatorial formula for all intersection numbers on the hyperelliptic locus with one $λ$-class, and powers of a $ψ$-class pulled back along the branch map. This is achieved by using Atiyah-Bott localization on a stack of stable maps into the orbifold $[\mathbb{P}^1/\mathbb{Z}_2]$.
We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi-Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of int…
We establish the real integral Hodge conjecture for 1-cycles on various classes of uniruled threefolds (conic bundles, Fano threefolds with no real point, some del Pezzo fibrations) and on conic bundles over higher-dimensional bases which themselves satisfy the real integral Hodge conjecture for 1-cycles. In addition, we show that rationally connected threefolds over non-archimedean real closed fields do not satisfy the real integral Hodge conjecture in general and that over such fields, Bröcker's EPT theorem remains true for simply connected surfaces of geometric genus zero but fails for some…
A conjecture on the relation between the cubic Hodge integrals and the topological vertex in topological string theory is resolved. A central role is played by the notion of generalized shift symmetries in a fermionic realization of the two-dimensional quantum torus algebra. These algebraic relations of operators in the fermionic Fock space are used to convert generating functions of the cubic Hodge integrals and the topological vertex to each other. As a byproduct, the generating function of the cubic Hodge integrals at special values of the parameters therein is shown to be a tau function of…
We prove that there exists a pencil of Enriques surfaces defined over $\mathbb{Q}$ with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question of Murre on the universality of the Abel-Jacobi maps in codimension three.
We give a simple proof that Kottwitz's PEL type integral models of Shimura varieties admit closed embeddings into Siegel integral models. We also show that Rapoport's and Kottwitz's integral models agree with Kisin's integral models for relevant Shimura data.
We present a detailed overview of the construction of the A_inf-cohomology theory from the preprint "Integral p-adic Hodge theory", joint with B. Bhatt and P. Scholze. We focus particularly on the p-adic analogue of the Cartier isomorphism via relative de Rham--Witt complexes.
We use the universal generation of algebraic cycles to relate (stable) rationality to the integral Hodge conjecture. We show that the Chow group of 1-cycles on a cubic hypersurface is universally generated by lines. Applications are mainly in cubic hypersurfaces of low dimensions. For example, we show that if a generic cubic fourfold is stably rational then the Beauville--Bogomolov form on its variety of lines, viewed as an integral Hodge class on the self product of its variety of lines, is algebraic. In dimension $3$ and $5$, we relate stable rationality with the geometry of the associated i…
We study higher genus Fan--Jarvis--Ruan--Witten theory of any chain polynomial with any group of symmetries. Precisely, we give an explicit way to compute the cup product of Polishchuk and Vaintrob's virtual class with the top Chern class of the Hodge bundle. Our formula for this product holds in any genus and without any assumption on the semi-simplicity of the underlying cohomological field theory.
We study the integral Hodge conjecture in complex codimension $2$ and $3$ for approximations to the classifying space of groups of type A. In degree two, we prove a conjecture of Ben Antieau, extending his two counterexamples to a general family of groups. In particular, when reduced to a finite field, these spaces extend Antieau's counterexamples to the integral $\ell$-adic Tate conjecture from the cases $\ell = 2, 3$ to all primes $\ell$.
We discuss the space of sections and certain bisections on a quadric surfaces bundle $X$ over a smooth curve. The Abel-Jacobi from these spaces to the intermediate Jacobian will be shown to be dominant with rationally connected fibers. As an application, we prove that the integral Hodge conjecture holds for degree four integral Hodge classes of fourfolds fibered by quadric bundles over a smooth curve. This gives an alternative proof of a result of Colliot-Th{é}l{è}ne and Voisin.
Given an integral p-adic variety, we observe that if the integral Hodge--de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral p-adic Hodge theory, we show that such a nice behavior is guaranteed if the p-adic variety can be lifted to an analogue of second Witt vectors and satisfies some bound on dimension and ramification index. This is a (ramified) mixed characteristic analogue of results due to Deligne--Illusie and Fontaine--Messing. Lastly, we discuss an example illustrating the necessity of t…
We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds. Our algorithm is based on computation of the list of elementary divisors of both the lattice of linear algebraic cycles, and the lattice of Hodge cycles written in terms of vanishing cycles, and observing that these two lists are the same.
In a previous paper we proved that after a simple transformation the generating series of the linear Hodge integrals on the moduli space of stable curves satisfies the hierarchy of the Intermediate Long Wave equation. In this paper we present a much shorter proof of this fact. Our new proof is based on an explicit formula for the one-point linear Hodge integrals that was found independently by Faber, Pandharipande and Ekedahl, Lando, Shapiro, Vainshtein.
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…
