Thomas Agugliaro (U. Strasbourg) Hodge standard conjecture for powers
Abstract: The Hodge standard conjecture predicts positivity of intersection forms on algebraic cycles. It was formulated by Grothendieck in the Sixties, modeled on the Hodge-Riemann relations. Only recently some progress has been made, based on p-adic Hodge theory. As most conjectures on algebraic cycles, it behaves badly under powers. In this talk, we will investigate this question and prove the conjecture for powers of simple abelian varieties of prime dimension.
Joseph Ayoub (U. Zurich): Classicality of the motivic Galois group
Abstract: By construction, the motivic Galois group is a derived affine group scheme over $\Q$ acting on the singular cohomology of algebraic varieties. It follows from the standard motivic conjectures that this group is classical, i.e., an affine group scheme in the usual sense. Concretely, this means that there are no nontrivial operations on the singular cohomology of algebraic varieties of degree $\geq 1$. We will discuss a proof of this conjecture.
Dennis Eriksson (Gothenburg/Chalmers UT): On smooth fillings of Calabi-Yau families
Abstract: For a family of projective manifolds over a punctured disc, it is not always possible to extend it smoothly. One obstruction to these extensions is given by monodromy. Famous good reductions theorems state that for abelian varieties trivial monodromy implies the possibility to fill smoothly. For general Calabi-Yau families there are however examples of families of trivial monodromy with no smooth fillings. Motivated by our work in mirror symmetry we introduce an invariant that give obstructions to the existence of smooth fillings in these cases, even after finite base extensions and birational equivalence. In this talk I will overview known results in the area, and explain how the new invariant generalises previously known cases. This is joint work with Gerard Freixas.
Gerard Freixas i Montplet (CNRS/École Polytechnique): Riemann-Roch isomorphism and Griffiths bundles
Abstract: The application of the Grothendieck-Riemann-Roch formula (GRR) on universal families over some moduli spaces yields remarkable identities for Hodge-type bundles, which can sometimes be lifted to canonical isomorphisms between natural line bundles. Examples of such are the Mumford isomorphism on the moduli space of curves, or the key-formula of Moret-Bailly on the moduli space of Abelian varieties. These can actually be seen as particular instances of a program suggested by Deligne, consisting in lifting GRR in codimension 1 to a functorial isomorphism of line bundles. In recent work with Dennis Eriksson, we have completed Deligne’s program, and we have considered consequences for combinations of Griffiths bundles, in turn motivated by genus one mirror symmetry. In this talk, I will provide an overview of this work.
Bruno Klingler (Humboldt U. Berlin) Special loci for algebraic varieties.
Abstract: I will discuss various special loci for algebraic varieties, their properties, and their (conjectural) relations.
Colleen Robles (Duke U.) Completions of two-parameter period maps by nilpotent orbits
Abstract: Hodge theory has been a powerful tool in the study of moduli of curves, principally polarized abelian varieties, and K3 surfaces. Examples include Mumford’s proof that the moduli space of principally polarized abelian varieties of genus at least 7 is of general type, and Friedman’s proof of global Torelli for K3 surfaces. Both these arguments utilized Mumford’s compactification of the associated period space. The compactification exists only when the period space is locally Hermitian symmetric (which roughly corresponds to curves, ppav and K3 surfaces). It is a long-standing problem, going back to questions raised by Griffiths in the 1960s and 1970s, to generalize Mumford’s compactification to arbitrary period spaces. It turns out that the correct generalization is to construct a completion of the period map for the given family/moduli space. I will explain recent results that do this for two-parameter period maps, and discuss some applications to the study of Calabi-Yau 3-folds (a la Friedman’s result).
Sara Torelli (U. Torino) Infinitesimal rigidity of modular morphisms via the second Fujita decomposition
Abstract: Recently, new results of global rigidity of modular morphisms coming from geometric variations of Hodge structures associated to families of curves, namely the Torelli and Prym morphisms, have given new insights on the classical rigidity results from the sixties. In the same spirit, we consider the infinitesimal rigidity of the same kind of morphisms, and we provide a proof based on the second Fujita decomposition of the Hodge bundle of a family of curves. This is joint work with Giulio Codogni and Víctor González-Alonso.
David Urbanik (IHES): Degrees of Hodge Loci
Abstract: Hodge and Noether-Lefschetz loci of families of algebraic varieties parameterize fibres of the family with extra algebraic structure. They are algebraic varieties inside a base variety, and one can associate to them a degree. We explain how to prove asymptotic upper and lower bounds on this degree in terms of the Hodge norm of the Hodge vectors which define the associated components. Our methods reduce the problem to counting rational points in Siegel set orbits and an explicit analysis of such orbits.
9:30-10:30: Sara Torelli
10:30-11:00: Coffee break
11:00-12:00: Bruno Klingler
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12:00-14:00 Lunch break
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Thursday, May 15 Afternoon in Albano House 1, Floor 2, Room 7:
14:00-15:00 Gerard Freixas i Montplet
15:00-15:30 Coffee break
15:30-16:30 Dennis Eriksson
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Thursday evening: Workshop Dinner
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Friday, May 16 Morning in Albano House 1, Floor 2, Room 4:
9:30-10:30 David Urbanik
10:30-11:00 Coffee break
11:00-12:00: Joseph Ayoub
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12:00-14:00 Lunch break
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Friday, May 16 Afternoon in Albano House 1, Floor 2, Room 5:
14:00-15:00 Thomas Agugliaro
15:00-15:30 Coffee break
15:30-16:30 Colleen Robles
I thank the following for their support:
The Knut & Alice Wallenbeg Foundation
The Swedish Research Council