219,976 research outputs found
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
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