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Abstract We prove a functional transcendence theorem for the integrals of algebraic forms in families of algebraic varieties. This allows us to prove a geometric version of André’s generalization of the Grothendieck period conjecture, which we state using the formalism of Nori motives. More precisely, we prove a version of the Ax–Schanuel conjecture for the comparison between the flat and algebraic coordinates of an arbitrary admissible graded polarizable variation of integral mixed Hodge structures. This can be seen as a generalization of the recent Ax–Schanuel theorems of [13, 18] for mixed period maps.
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Abstract We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasi-projective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period map to the image of the period map of the associated graded. On the one hand, we show in a precise manner that the parts of this map parametrizing extension data of non-adjacent-weight pure Hodge structures are quasi-affine. On the other hand, extensions of adjacent-weight pure polarized Hodge structures are parametrized by a compact complex torus (the intermediate Jacobian) equipped with a natural theta bundle which is ample in Griffiths transverse directions.Our proof makes heavy use of o-minimality, and recent work with B. Klingler associating an {\mathbb{R}_{\mathrm{an},\exp}}-definable structure to mixed period domains and admissible mixed period maps.more » « less
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Free, publicly-accessible full text available December 1, 2026
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Abstract A flat vector bundle on an algebraic variety supports two natural definable structures given by the flat and algebraic coordinates. In this note, we show these two structures are compatible, subject to a condition on the local monodromy at infinity that is satisfied for all flat bundles underlying variations of Hodge structures.more » « less
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We generalize the finiteness theorem for the locus of Hodge classes withfixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodgeclasses to self-dual classes. The proof uses the definability of periodmappings in the o-minimal structure $$\mathbb{R}_{\mathrm{an},\exp}$$.more » « less
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Abstract We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky’s global Torelli theorem in the smooth case (assuming b 2 ≥ 5 {b_{2}\geq 5} ) which does not use the existence of a hyperkähler metric or twistor deformations.more » « less
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Abstract We extend the decomposition theorem for numerically K -trivial varieties with log terminal singularities to the Kähler setting. Along the way we prove that all such varieties admit a strong locally trivial algebraic approximation, thus completing the numerically K -trivial case of a conjecture of Campana and Peternell.more » « less
