DGA-Models of variations of mixed Hodge structures

We define objects over Morgan's mixed Hodge diagrams which will be algebraic models of unipotent variations of mixed hodge structures over K\"ahler manifolds. As an analogue of Hain-Zucker's equivalence between unipotent variations of mixed Hodge structures and mixed Hodge representations of the fundamental group with Hain's mixed hodge structure, we give an equivalence between the category of our VMHS-like objects and the category of mixed Hodge representations of the dual Lie algebra of Sullivan's minimal model with Morgan's mixed Hodge structure. By this result, we can put various (tannakian theoretical) non-abelian mixed Hodge structures on the category of our new objects like the taking fibers of variations of mixed Hodge structures at points. By certain modifications of the result, we also give models of non-unipotent variations of mixed Hodge structures.Comment: 30 page

Hodge loci and absolute Hodge classes

We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are defined over number fields (assuming the initial family is), as expected from the Hodge conjecture. We also show that the Hodge conjecture for (weakly) absolute Hodge classes reduces to the Hodge conjecture for (weakly) absolute Hodge classes on varieties defined over number fields

On integral Hodge classes on uniruled or Calabi-Yau threefolds

If X is a smooth projective complex threefold, the Hodge conjecture holds for degree 4 rational Hodge classes on X. Kollar gave examples where it does not hold for integral Hodge classes of degree 4, that is integral Hodge classes need not be classes of algebraic 1-cycles with integral coefficients. We show however that the Hodge conjecture holds for integral Hodge classes of degree 4 on a uniruled or a Calabi-Yau threefold

Log Hodge groups on a toric Calabi-Yau degeneration

We give a spectral sequence to compute the logarithmic Hodge groups on a hypersurface type toric log Calabi-Yau space, compute its E_1 term explicitly in terms of tropical degeneration data and Jacobian rings and prove its degeneration at E_2 under mild assumptions. We prove the basechange of the affine Hodge groups and deduce it for the logarithmic Hodge groups in low dimensions. As an application, we prove a mirror symmetry duality in dimension two and four involving the usual Hodge numbers, the stringy Hodge numbers and the affine Hodge numbers.Comment: 49 pages, 3 figure

The Hodge ring of Kaehler manifolds

We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and those of arbitrary Kaehler manifolds. The consideration of certain natural ideals in the Hodge ring allows us to determine exactly which linear combinations of Hodge numbers are birationally invariant, and which are topological invariants. Combining the Hodge and unitary bordism rings, we are also able to treat linear combinations of Hodge and Chern numbers. In particular, this leads to a complete solution of a classical problem of Hirzebruch's.Comment: Dedicated to the memory of F. Hirzebruch. To appear in Compositio Mat

On the Hodge-Newton filtration for p-divisible groups of Hodge type

A p-divisible group, or more generally an F-crystal, is said to be Hodge-Newton reducible if its Hodge polygon passes through a break point of its Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a canonical filtration called the Hodge-Newton filtration. The notion of Hodge-Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge-Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz's result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre-Tate deformation theory for local Shimura data of Hodge type. We also apply our deformation theory to study some congruence relations on Shimura varieties of Hodge type.Comment: 31 page