In this paper we prove the following results:
1
)
1)
We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures.
2
)
2)
We prove that the period map associated to any pure polarized variation of integral Hodge structures
V
\mathbb {V}
on a smooth complex quasi-projective variety
S
S
is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure.
3
)
3)
As a corollary of
2
)
2)
and of Peterzil-Starchenko’s o-minimal Chow theorem we recover that the Hodge locus of
(
S
,
V
)
(S, \mathbb {V})
is a countable union of algebraic subvarieties of
S
S
, a result originally due to Cattani-Deligne-Kaplan. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable
S
L
2
SL_2
-orbit theorem of Cattani-Kaplan-Schmid.
We extend the Ax-Schanuel theorem recently proven for Shimura varieties by Mok-Pila-Tsimerman to all varieties supporting a pure polarized integral variation of Hodge structures. The essential new ingredient is a volume bound on Griffiths transverse subvarieties of period domains.
We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period maps are quasi-projective and that the restriction of the Griffiths bundle is ample.
We systematically study the moduli theory of singular symplectic varieties which have a resolution by an irreducible symplectic manifold and prove an analog of Verbitsky's global Torelli theorem. In place of twistor lines, Verbitsky's work on ergodic complex structures provides the essential global input. On the one hand, our deformation theoretic results are a further generalization of Huybrechts' theorem on deformation equivalence of birational hyperkähler manifolds to the context of singular symplectic varieties. On the other hand, our global moduli theory provides a framework for understanding and classifying the symplectic singularities that arise from birational contractions of irreducible symplectic manifolds, and there are a number of applications to K3 [n] -type varieties.
We prove the Mercat Conjecture for rank 2 vector bundles on generic curves of every genus. For odd genus, we identify the effective divisor in Mg where the Mercat Conjecture fails and compute its slope.
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