Schedule

All lectures in Room 2.049 in the IRIS Haus

Thursday [30.05.2024]

10:00 - 11:00 D. Oprea
11:15 - 12:15 R. Laza

Lunch

14:00 - 15:00 F. Moretti
15:15 - 16:15 M. Lelli Chiesa
16:30 - 17:30 D. Ranganathan

Workshop Dinner

Friday [31.05.2024]

10:00 - 11:00 O. de Gaay Fortman
11:15 - 12:15 T. Krämer

Lunch

14:00 - 15:00 A. Iribar López
15:15 - 16:15 R. Pandharipande

Titles and Abstracts

Olivier de Gaay Fortman (Hannover)

Powers of an abelian variety isogenous to a Jacobian

In this talk, I present a joint project with Stefan Schreieder concerning isogenies between powers of abelian
varieties and Jacobians. We show that no power of a very general principally polarized abelian variety is isogenous
to a Jacobian. The analogous statement holds true for powers of the intermediate Jacobian of a very general cubic
threefold. We also classify all ways in which a power of the Jacobian of a very general hyperelliptic curve can be
isogenous to another Jacobian. These results have applications regarding the Coleman-Oort conjecture, and the integral
Hodge conjecture for abelian varieties.

Aitor Iribar López (ETHZ)

Geometry of Noether-Lefschetz loci in the moduli of abelian varieties I

The general abelian variety does not have any abelian subvarieties. The locus of abelian varieties that do have
such a subvariety is part of the Noether-Lefschetz locus inside the moduli of p.p.a.v, and using an old
construction by Debarre, we can parametrize its irreducible components.

I will explain how to calculate their tautological projection of the associated cycles (recently developed by
Canning, Molcho, Opera and Pandharipande), showing that they are the Fourier coefficients of a modular form, as
conjectured by Greer and Lian. Then, I will explain the connection between certain Gromov-Witten invariants of
a moving elliptic curve and the following question: Is the tautological projection a ring homomorphism?

Thomas Krämer (HU Berlin)

All exceptional Tannaka groups on abelian varieties arise from cubic threefolds

To any subvariety of an abelian variety one may attach the reductive group which is the Tannaka group of the
corresponding perverse intersection complex. These groups play a fundamental role in recent work on the topology
and arithmetic of irregular varieties. While we usually want them to be big, there are exceptional examples: The
Fano surface of lines on a smooth cubic threefold has the Tannaka group E6.

In the talk I, will show that these are the only exceptional cases: Under mild dimension assumptions, any smooth
subvariety of an abelian variety with an exceptional simple Tannaka group is isomorphic to the Fano surface of lines
on a smooth cubic threefold. This significantly enlarges the scope of our work on the Shafarevich conjecture with
Javanpeykar, Lehn and Maculan. The key idea is to control the Hodge decomposition on cohomology by a cocharacter of
the Tannaka group of Hodge modules, and to compare this group-theoretic data with Hodge number estimates by
Lazarsfeld-Popa and Lombardi. This is work in progress with Christian Lehn and Marco Maculan.

Margherita Lelli Chiesa (Rome III)

Gaussian maps for curves on Abelian surfaces

By a result of Arbarello, Bruno and Sernesi, Brill-Noether general curves lying on K3 surfaces are characterized by
the non-surjectivity of their Gaussian map. Concerning curves on Abelian surfaces, a conjecture by Colombo, Frediani,
Pareschi predicts their characterization in terms of the corank of their second Gaussian map . In a different spirit,
I will show that the Prym-Gauss map of a Prym curve on an abelian surface is never surjective. I will then talk about
a joint work in progress with Arbarello and Bruno aimed to show that a general Prym canonical curve with non-surjective
Prym-Gauss map always lies on a surface in P^{g-2}.

Radu Laza (Stony Brook)

The Core of Calabi-Yau degenerations

It is problem of high interest to construct meaningful compactifications for moduli spaces of algebraic varieties.
For varieties of general type and Fano type the situation is fairly well understood via KSBA theory and K-stability
respectively. In the remaining K-trivial case, one understands the “classical cases” (abelian varieties, K3 surfaces,
and hyper-Kaehler manifolds) via period maps and Hodge theory. Thus, the only remaining primitive case
for constructing compact moduli spaces is that of strict Calabi-Yau’s of dimension at least 3.

In this talk, I conjecture the existence of a minimal compactification for the moduli of Calabi-Yau varieties, an
analogue of the Baily-Borel compactification in the classical case. I will then explain a construction that produces
such a compactification at least at the set-theoretic level. The key concept here is the Type and Core of a Calabi-Yau
degeneration. The Type is an integer invariant that measures the depth of the degeneration, while the Core is a
pure Hodge structure of Calabi-Yau type, which should be understood as the Hodge theoretic piece that survives
in any reasonable model of the degeneration.

Federico Moretti (HU Berlin)

Families of Jacobians with isotrivial factors

Let Y be a variety equipped with a top form. I will present a simple genus bound for a family of subvarieties covering
Y in term of the dimension of the family. As an application I will prove a genus bound for families of Jacobians with a
given isotrivial factor. If the dimension of the family is big enough, the bound is sharp and realized only by the family
of degree 2 covers of a given curve. This is based on joint work with Josh Lam and Giovanni Passeri.

Dragos Oprea (UC San Diego)

Tautological projection for cycles on the moduli space of abelian varieties

The tautological ring of the moduli space of principally polarized abelian varieties was introduced and computed
by van der Geer in the 1990s. I will show that every cycle class on the moduli space of principally polarized abelian
varieties can be decomposed canonically into a tautological and a non-tautological part. Furthermore, I will
explain that the tautological components of all cycles parametrizing product abelian varieties can be expressed in
terms of Schur determinants. This is based on joint work with Samir Canning, Sam Molcho and Rahul Pandharipande.

Rahul Pandharipande (ETHZ)

Geometry of Noether-Lefschetz loci in the moduli of abelian varieties II

I will speculate about the connection of projections of Noether-Lefschetz loci to integrals in Gromov-Witten theory.
The lecture is related to results of S. Canning, F. Greer, A. Iribar López, C. Lian, S. Molcho, D. Oprea, A. Pixton, and H.-H. Tseng.

Dhruv Ranganathan (Cambridge)

Gromov-Witten theory and the tautological ring

Gromov-Witten theory is a natural source of cohomology classes on the moduli space of curves. I will explain a
speculation, going back to Levine and Pandharipande, that the cohomology classes produced by Gromov-Witten
theory lie in a surprisingly small part of cohomology: the tautological ring. I will explain how logarithmic
Gromov-Witten theory can be used to approach this speculation, and try to share some of the subtleties involved
in this approach. Based on recent and ongoing work with Davesh Maulik.