Mini-courses

Charles Doran: Fibration and degeneration in Calabi-Yau geometry

1. Elliptic curves, K3 surfaces, and Calabi-Yau threefolds.
2. Variations of Hodge structure and Picard-Fuchs ODE.
3. Modular parametrization for lattice-polarized K3 surfaces.
4. Calabi-Yau fibrations and degenerations.
5. Mirror symmetry: old and new.

As a preparation for this series of lectures, participants might want to have a look at Chapter 7 in Hodge theory, edited by Cattani, Zein, Griffiths, Trang, which summarises the fundamentals of the theory for variations of Hodge structures.

You can find here the slides for Charles' second talk (given on Wednesday).

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Arno Kret: Automorphic representations and Galois representations

1. Automorphic forms.
2. The paper of Buzzard-Gee.
3. Shimura varieties.
4. Construction of Galois representions using the above; (part of) final lecture will be about my work with Shin on $GSp$ and $GSO$.

If you are thinking of preparing for this mini-course before the school, you could look at Taylor’s ICM paper on Galois representations or at the paper by Buzzard and Gee, The conjectural connections between automorphic representations and Galois representations.

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Colleen Robles: The Clemens-Schmid Exact Sequence

1. Review of monodromy and period maps.
2. Schmid’s nilpotent orbit theorem.
3. Mixed Hodge structures.
4. The Clemens-Schmid exact sequence.

The material is covered in these notes .

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Abstracts

Philip Engel: Local systems underlying variation of Hodge structure

Generalizing finiteness theorems of Parshin, Arakelov, and Faltings, Deligne proved in 1987 that only finitely many $\mathbb Z$-local systems of a fixed rank underlie a polarized variation of Hodge structure, over a fixed quasiprojective variety. Deligne conjectured that an appropriate form of this finiteness also holds in families of quasiprojective varieties. In the 1990's, Simpson's refined this conjecture in the following form: the nonabelian Hodge locus is algebraic. I will discuss joint work with Salim Tayou that these conjectures are true when the algebraic monodromy group is cocompact.

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Paola Frediani: Asymptotic directions in the moduli space of curves.

I will report on a joint work with E. Colombo and G.P. Pirola, where we study asymptotic directions in the tangent bundle of the moduli space of curves of genus $g$, namely those tangent directions that are annihilated by the second fundamental form of the Torelli map. I will give examples of asymptotic directions for any $g$ at least $4$. I will show that if the rank $d$ of a tangent direction at $[C]$ (with respect to the infinitesimal deformation map) is less than the Clifford index of the curve $C$, then the tangent direction is not asymptotic. If the rank of a tangent direction $v$ is equal to the Clifford index of the curve, I will give sufficient conditions ensuring that the infinitesimal deformation $v$ is not asymptotic. Finally I will determine all asymptotic directions of rank $1$ and give an almost complete description of asymptotic directions of rank $2$.

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Ben Moonen: Tautological rings of Shimura varieties

On a Shimura variety, we have a collection of natural vector bundles, called automorphic bundles, which are constructed from group representations. The tautological ring of the Shimura variety is defined to be the ring generated by all Chern classes of these automorphic bundles; this may be considered on the level of cohomology, but also on more refined levels, such as the Chow ring. The construction extends to (sufficiently nice) toroidal compactifications of Shimura varieties. In my talk, I will explain that there is a very beautiful conjectural description of this tautological ring. I will present several examples and will explain what is the problem that remains open. If time permits, I will also explain why we know more in characteristic $p>0$; this is based on the theory of $G$-zips (about which I will not assume any knowledge).

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Emre Sertöz: Computing the transcendence and linear relations of 1-periods

I will sketch a modestly practical algorithm to compute all linear relations with algebraic coefficients between any given finite set of 1-periods. As a special case, we can algorithmically decide transcendence of 1-periods. This is based on the "qualitative description" of these relations by Huber and Wüstholz. We combine their result with the recent work on computing the endomorphism ring of abelian varieties. This is a work in progress with Jöel Ouaknine (MPI SWS) and James Worrell (Oxford).

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Orsola Tommasi: Geometry of fine compactified Jacobians

The degree d universal Jacobian parametrizes degree d line bundles on smooth curves. There are several approaches on how to extend it to a proper family over the moduli space of stable curves. In this talk, we introduce a simple definition of a fine compactified Jacobian, both for a single nodal curve and over the moduli space. We discuss their combinatorial characterization and their geometry and explain how this leads to new examples already in the case of curves of genus 1. This is joint work with Nicola Pagani.

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Sara Torelli: On the difference map to the Jacobian of curves in K3 surfaces

For a general smooth projective curve $C$ of genus $g\geq 3$, the image of the difference map $C\times C\to JC$ does not intersect the torsion locus of $JC$ of any order $n$. In this talk we consider the problem for pairs of points on curves arising by normalization from the Severi variety of $2$-pointed $\delta$-nodal curves in the linear system of a general primitively polarized K3 surface $(X, H)$. We prove that for any non negative integer $n$, the locus in such Severi variety with the property that the difference of the two marked points is $n$-torsion in the Jacobian of the normalization of the $\delta$-nodal curve is not empty of the expected dimension $2$ if and only if a certain Brill-Noether number is non negative. We then discuss applications. This is a work in progress with Andreas Leopold Knutsen.

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