Schedule of Follow-up Workshop to JTP "Algebraic Geometry"

Tuesday, April 24

09:30-10:30 Nicolas Addington: Zeta functions of moduli spaces of sheaves on K3 surface
10:30-11:00 Coffee Break
11:00-12:00 Chiara Camere: Twisted sheaves on K3 surfaces, Verra fourfolds and non-symplectic involutions
12:00-13:45 Lunch break
14:00-16:00 SFB working seminar for those who want to go, or free afternoon
16:00-16:30 Coffee and cake

Abstracts

Nicolas Addington: Zeta functions of moduli spaces of sheaves on K3 surface

I will report on work of my student Sarah Frei: let S be a K3 surface over a finite field F_q, let M be any (smooth, proper) moduli space of sheaves on S, and write dim M = 2n; then M has the same number of points, over any extension F_{q^m}, as the Hilbert scheme of n points on S.  If M is birational to the Hilbert scheme then then this is unsurprising, but if they're not birational then it's quite surprising.  The technique is to lift to characteristic zero and apply results of Markman on the structure of H^*(M,Q) as a monodromy representation.

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Ada Boralevi: A construction of equivariant bundles on the space of symmetric forms

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Chiara Camere: Twisted sheaves on K3 surfaces, Verra fourfolds and non-symplectic involutions

The object of this talk is the construction of a family of hyperkähler fourfolds of K3^[2]-type producing an example of the last missing case in the classification of non symplectic involutions. We will see that these varieties can be described geometrically in two different ways, as moduli spaces of twisted sheaves on K3 surfaces and as double covers of EPW quartics associated to Verra fourfolds. This is joint work with G. Kapustka, M. Kapustka and G. Mongardi.

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Dennis Eriksson: Equidistribution of electrons on Berkovich spaces

Classically, Fekete points are modeled on electrons in the plane: Given a set of k points in a compact set of the complex numbers, they are said to be a Fekete configuration if they are as far apart as possible, in the sense that their geometric mean is as big as possible. When k goes to infinity these points, and their generalizations, are expected to equidistribute towards a natural measure. This is a result in the complex setting, due to Berman-Boucksom-Witt-Nyström. In this talk, I explain how Fekete points and their equidistribution can be extended to the Berkovich setting. The construction and proof rely on a reformulation of the problem in terms of the determinant of the cohomology of a line bundle, and metrical properties of Deligne bundles. This is joint work with Sebastien Boucksom.

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Gerard Freixas i Montplet: Arithmetic intersections and the Jacquet-Langlands correspondence

I will report on joint work with S. Sankaran, that begun during our junior trimester at the HIM. We consider a compatibility between the Grothendieck-Riemann-Roch theorem in Arakelov geometry and the Jacquet-Langlands corresponence in the theory of automorphic forms. I will give a sketchy introduction to both questions, and explain how they combine to produce non-trivial relations between natural arithmetic intersection numbers on twisted Hilbert modular surfaces and Shimura curves. Given the diversity of backgrounds of the audience, the aim of the talk will mostly be an exposition of the flavour of Arakelov geometry.

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Frank Gounelas: Positivity of the cotangent bundle of a K3 surface

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Andreas Hochenegger: Formality of P-objects

We show that a P-object and simple configurations of P-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.

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Ciaran Meachan: Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational

Starting with two non-birational derived equivalent K3 surfaces, one can ask whether their Hilbert schemes of points are birational. In this talk, we will show that in some cases they are but in most cases they are not. This is join work with Giovanni Mongardi and Kota Yoshioka.

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Ernesto Carlo Mistretta: Holomorphic symmetric differentials and a birational characterisation of abelian varieties

We give a brief review of base loci and Iitaka fibrations for vector bundles, then apply this construction to give a characterisation of abelian varieties related to the vector bundle of symmetric differentials. In order to obtain a birational characterisation we can relax the assumptions on the vector bundle, but we have to deal with minimal model program and mild singularities. Part of this seminar is a joint work with S. Urbinati, and part is a work in progress.

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Giovanni Mongardi: Curve classes on irreducible holomorphic symplectic manifolds

In this talk, we prove the integral Hodge conjecture for one cycles on some (projective) IHS manifolds and discuss its consequences on Fano manifolds related to them. This is joint work with J.C. Ottem.

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Marc-Hubert Nicole: The Gross-Kohnen-Zagier theorem for (p-adic) families

Given an elliptic curve E defined over the field of rational numbers and given an imaginary quadratic field K, one may define (using the theory of complex multiplication) a K-rational point of the elliptic curve, called a Heegner point. Heegner points are crucial tools for studying the arithmetic of elliptic curves; in particular, the celebrated theorem of Gross and Zagier relates, under suitable arithmetic assumptions, the Néron-Tate height of Heegner points and the leading term of the complex L-function of E over K. The Gross-Kohnen-Zagier theorem (GKZ), complementary to the Gross-Zagier theorem mentioned above, shows that, under suitable arithmetic assumptions, the relative positions of the Heegner points, as the imaginary quadratic field varies while the elliptic curve stays fixed, are encoded by the Fourier coefficients of a certain kind of modular form called a Jacobi form. Briefly put, Heegner points are generating series for Jacobi forms. In this talk, I will explain a variant of the GKZ theorem where we make all objects involved vary in p-adic families e.g., Hida families of modular forms of varying p-adic weight, a concept we will explain from scratch. We like to view our result as the GL(2) instance of a p-adic Kudla program.

Joint work with Matteo Longo (Padova).

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Matteo Penegini: On Zariski multiplets of branch curves

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Francesco Polizzi: Monodromy representations and surfaces of general type

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Antonio Rapagnetta: Singular moduli spaces of sheaves on K3 surfaces

By the Bogomolov decomposition theorem, irreducible holomorphic symplectic manifolds play a central role in the classification of compact Kähler manifolds with numerically trivial canonical bundle. Very recently, Höring and Peternell completed the proof of the existence of a singular analogue of the Bogomolov decomposition theorem. In view of this result, singular irreducible symplectic varieties (following Greb, Kebekus and Peternell) are singular analogue of irreducible holomorphic symplectic manifolds. In a joint work with Arvid Perego, still in progress, we show that all moduli spaces of sheaves on projective K3 surfaces are singularirreducible symplectic varieties. We compute their Beauville form and the Hodge decomposition of their second integral cohomology, generalizing previous results, in the smooth case, due to Mukai, O'Grady and Yoshioka.

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Pietro Sabatino: On a generalization of the Bogomolov-Miyaoka-Yau inequality and an explicit bound for the log canonical degree of curves on open surfaces.

Let $X$, $D$ be respectively a smooth projective surface and a simple normal crossing divisor on $X$. Suppose $\kappa ( X, K_X + D)\ge 0$, given an irreducible curve $C$ on $X$ and a rational number $\alpha \in [ 0, 1 ]$, following ideas introduced by Miyaoka, we define an orbibundle $\mathcal{E}_\alpha$ as a sub vector bundle of log differentials on a suitable Galois cover of $X$ and prove a Bogomolov-Miyaoka-Yau inequality for the orbibundle and consequently for the couple $(X, D+\alpha C$. We briefly compare this construction to similar ones of e.g. Megyesi and Langer. As a consequence of this Bogovomolov-Miyaoka-Yau inequality by varying $\alpha$ we deduce, in the case $K_X+D$ big and nef and $(K_X+D)^2 > \chi ( X \setminus D)$, a bound for $(K_X+D)\cdot C)$ by an explicit function of the invariants: $(K_X+D)^2$, the topological Euler-Poncar\'e characteristic of the open surface $\chi(X\setminus D)$ and $\chi (\widetilde{C} \setminus D) $, the topological Euler-Poncar\'e characteristic of the normalization of $C$ minus the points mapping on $D$.

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