Periodic Reporting for period 1 - TameHodge (Tame geometry and transcendence in Hodge theory)
Período documentado: 2021-10-01 hasta 2023-03-31
Hodge theory, as developed by Deligne and Griffiths, is the main tool for analyzing the geometry and arithmetic of complex algebraic varieties, that is, solution sets of algebraic equations over the complex numbers. It occupies a central position in mathematics through its relations to differential geometry, algebraic geometry, differential equations and number theory. It is an essential fact that at heart, Hodge theory is NOT algebraic. On the other hand, some of the deepest conjectures in mathematics (the Hodge conjecture and the Grothendieck period conjecture) suggest that this transcendence is severely constrained.
Recent work of myself and others suggests that tame geometry, whose idea was introduced by Grothendieck in the 1980s, is the natural setting for understanding these constraints. Tame geometry, developed by model-theorist as o-minimal geometry, has for prototype real semi-algebraic geometry, but is much richer. As a spectacular application of tame geometry, Bakker, Tsimerman and I recently reproved a famous result of Cattani-Deligne-Kaplan, often considered as the most serious evidence for the Hodge conjecture: the algebraicity of Hodge loci.
I am leading a group at HU Berlin to explore this striking new connection between tame geometry and Hodge theory, with three axes: (I) attack the arithmetic of periods coming from the moduli space of abelian differentials; this opens a completely new perspective on this space cherished by dynamicists; (II) attack some fundamental questions for general variations of Hodge structures: fields of definition of Hodge loci (related to the conjecture that Hodge classes are absolute Hodge classes); atypical intersections, for instance for families of Calabi- Yau varieties; Ax-Schanuel conjecture for mixed period maps and for Hodge bundles; (III) attack Simpson’s « Standard conjecture » for local systems through the tame geometry of the non-abelian Hodge correspondence.
Recent work of myself and others suggests that tame geometry, whose idea was introduced by Grothendieck in the 1980s, is the natural setting for understanding these constraints. Tame geometry, developed by model-theorist as o-minimal geometry, has for prototype real semi-algebraic geometry, but is much richer. As a spectacular application of tame geometry, Bakker, Tsimerman and I recently reproved a famous result of Cattani-Deligne-Kaplan, often considered as the most serious evidence for the Hodge conjecture: the algebraicity of Hodge loci.
I am leading a group at HU Berlin to explore this striking new connection between tame geometry and Hodge theory, with three axes: (I) attack the arithmetic of periods coming from the moduli space of abelian differentials; this opens a completely new perspective on this space cherished by dynamicists; (II) attack some fundamental questions for general variations of Hodge structures: fields of definition of Hodge loci (related to the conjecture that Hodge classes are absolute Hodge classes); atypical intersections, for instance for families of Calabi- Yau varieties; Ax-Schanuel conjecture for mixed period maps and for Hodge bundles; (III) attack Simpson’s « Standard conjecture » for local systems through the tame geometry of the non-abelian Hodge correspondence.
After 18 months, we obtain many successes on Axis (II):
- with Bakker, Brunebarbe and Tsimerman we proved the definability of mixed period maps in an o-minimal structure.
- with Otwinowska and Urbanik we proved that the maximal irreducible components of a variation of Hodge structure defined over a number field is defined over a number field.
- with Baldi and Ullmo we proved (preprint) that as soon as the level of a variation of a Hodge structure is at least 3, then its Hodge locus of positive period dimension is an algebraic subvariety of the base (rather than a countable union of such). This dramatically improves the famous Cattani-Deligne-Kaplan result.
We also obtained major success on Axis (I):
- with Lerer we characterized geometrically the arithmetic points for the moduli of abelian differentials and started the study of their distribution (preprint).
- with Bakker, Brunebarbe and Tsimerman we proved the definability of mixed period maps in an o-minimal structure.
- with Otwinowska and Urbanik we proved that the maximal irreducible components of a variation of Hodge structure defined over a number field is defined over a number field.
- with Baldi and Ullmo we proved (preprint) that as soon as the level of a variation of a Hodge structure is at least 3, then its Hodge locus of positive period dimension is an algebraic subvariety of the base (rather than a countable union of such). This dramatically improves the famous Cattani-Deligne-Kaplan result.
We also obtained major success on Axis (I):
- with Lerer we characterized geometrically the arithmetic points for the moduli of abelian differentials and started the study of their distribution (preprint).
We proved 5 of the 13 questions of our proposal (admittedly the easiest ones). On the other hand the work with Baldi and Ullmo mentioned above is a major progress, beyond even the expectations of the proposal. We are workin hard on the remaining 8 questions...