Talks and Schedule

Monday

9:30-10:00 Registration

10:30-11:30 Arbarello

Ribbons

Abstract: I will report on joint work in progress with A. Bruno and M. Lelli-Chiesa on Prym-canonical curves, Gaussian maps, and ribbons.

Coffee break

12:00-13:00 Voisin

Universally defined cycles

Abstract: I prove that universally defined cycles on smooth varieties of dimension d  are uniquely given by polynomials in Chern classes. I prove a similar result for universally defined cycles on products of smooth varieties of dimension d_1,...,d_r.  I will also discuss a  related statement concerning universally defined cycles on powers of smooth varieties of a given dimension, which  is still open and work in progress.

16:30-17:30 Meazzini

Koszul-Tate resolutions and cotangent cohomology

Abstract: We construct a Koszul-Tate resolution for an affine scheme X defined by an ideal in a polynomial ring. This allows to prove formulas for the cotangent cohomology T* of X, which turn out to be particularly interesting in the case of monomial ideals, for which there is a Z^n-grading on each T^i. In particular, we extend some results previously obtained by Altmann-Christophersen for Stanley-Reisner schemes. This is a joint work in progress with Nathan Ilten and Andrea Petracci.

Coffee break

18:00-19:00 Benoist

The period-index problem for Stein surfaces

Abstract: De Jong's period-index theorem states the period and the index of a class in the Brauer group of the function field of a complex algebraic surface coincide. We will prove a complex-analytic analogue of this statement: the period and the index of a class in the Brauer group of the field of meromorphic functions of a Stein surface coincide.

Tuesday

9:00-10:00 Laza

Isotrivial Lagrangian fibrations of compact hyper-kähler manifolds

Abstract: I will discuss some basic results on the structure of isotrivial Lagrangian fibrations for hyper-Kaehler manifolds. First, we establish a basic dichotomy: Type A (Kummer type) and Type B (Deligne-Mostow type). Then, in both cases, we prove that the smooth fibers of the fibration are isogeneous to powers of elliptic curves. We classify the Type A case and show that only the K3^n and Kum_n can occur.  Finally, I will discuss some partial results towards the classification of hyper-Kaehler manifolds of Kummer type, that is birational to a quotient of an abelian variety. This is joint work with Yoon-Joo Kim and Olivier Martin.

10:30-11:30 Fu

Motive of moduli spaces of vector bundles on curves

Abstract: I will report some recent progress on the study of motives of various moduli spaces of vector bundles on curves with additional structures (e.g. Higgs field, parabolic structure etc.). Precise formulas of motives of such moduli spaces in rank 2 and 3 are obtained. The talk is based on series of work in collaboration with Victoria Hoskins and Simon Pepin Lehalleur.

Coffee break

12:00-13:00 M. Lehn

Symplectic hypersurfaces

Abstract: If one tries to classify symplectic singularities by their codimension rather than the dimension, already the first case of

codimension 1 leads to interesting questions. I will explain the geometry of the known examples. This is a report on joint work with Namikawa, Sorger and van Straten.

16:30-17:30 Huybrechts

(Ramified) Brauer classes on projective varieties and hyperkähler manifolds

Abstract: I will discuss recent results on unramified Brauer classes on projective varieties and discuss a version of the period-index conjecture for hyperkähler manifolds.

Coffee break

18:00-19:00 Poster session and scientific discussion

Wednesday

9:00-10:00 Markman

Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds

Abstract: Fix a positive integer d. Let K be the imaginary quadratic number field Q(sqrt{-d}). Let C be a curve of genus 3, let J be its Jacobian, and let AJ(C) be the image of C in J via the Abel-Jacobi map. Let E' be the ideal sheaf of d+1 disjoint translates of AJ(C) in J tensored with the theta line bundle. Define E" similarly replacing AJ(C) by -AJ(C). The main results are:

1) Consider Orlov's derived equivalence F:D^b(JxJ)->D^b(A), where A=JxPic^0(J). Let DMon(J)=Spin(6,6,Z) be the derived monodromy group of J. Let L be the line in the total cohomology ring H(J) spanned by ch(E') and ch(E")^*. Let G be the subgroup of DMon(J) fixing all points of L.  The isomorphism H(JxJ) -> H(A) induced by F conjugates the diagonal action of G to the special Mumford-Tate group of a generic deformation (A',e') of (A,e),  where e:K ->End_Q(A) is a complex-multiplication determined by the line L.

2) Orlov's derived equivalence F maps the outer tensor product of E' and E" to a complex derived-dual to a rank 8d reflexive sheaf R on A.

3) The triple (A,e,R) deforms to a triple (A',e',R'), where R' is a twisted reflexive sheaf over an abelian variety of Weil type (A',e') deformation equivalent to (A,e) in an open subset of the 9-dimensional moduli of such pairs (A',e'). The algebraicity of the Hodge-Weil classes in the middle-cohomology of A' follows.

The result (1) is generalized replacing J by an abelian n-fold X, n>1, and replacing the line L by a line secant to the spinor variety in the even cohomology of X.

The result (3) is conditional on the following. 

Conjecture: The unobstructedness theorem of Buchweitz-Flenner for deformations of semiregular coherent sheaves generalizes to semiregular twisted reflexive sheaves.

10:30-11:30 Diverio

A birational version of Gromov's Kähler hyperbolicity

Abstract: In his wonderful book "Shafarevich maps and automorphic forms", J. Kollár asked almost 30 years ago for a "good" birational version of the notion of Kähler hyperbolicity introduced by M. Gromov in the early '90s. Kähler hyperbolic manifolds are compact Kähler manifold admitting a Kähler form whose pull-back to the universal cover becomes d-exact and moreover with a bounded primitive. Such manifolds are of general type (more than this: with ample canonical bundle), Kobayashi hyperbolic, and with large fundamental group.  We shall report on the recently intruduced class of weakly Kähler hyperbolic manifolds which indeed provides such a birational generalization. Among other things, we shall explain that these manifolds are of general type, satisfy a precise quantitative version of the Green-Griffiths conjecture and have generically arbitrarily large fundamental group. Moreover, their properties permit to verify Lang's conjecture for Kähler hyperbolic manifolds. This is a joint work with F. Bei, B. Claudon, P. Eyssidieux, and S. Trapani.

Coffee break

12:00-13:00 Manetti

Some remarks on deformations of (twisted) Higgs bundles

Abstract: We apply some recent algebraic results by Bandiera, Lepri, Manetti and Meazzini to deformation theory of Higgs bundles (both classical and twisted). In particular we introduce a semiregularity-type morphism annihilating obstructions and we prove a formality result in the classical polystable case.

Thursday

9:00-10:00 Bakker

Twistor geometry of local systems and the Shafarevich conjecture

Abstract: Shafarevich asked whether the universal cover of a smooth projective variety X is always holomorphically convex, meaning it admits a proper map to a Stein space.  This was proven in the linear case---namely when X admits an almost faithful representation of its fundamental group---by Eyssidieux--Katzarkov--Pantev--Ramachandran using techniques from non-abelian Hodge theory.  In joint work with Y. Brunebarbe and J. Tsimerman, we prove a version of the linear Shafarevich conjecture for quasiprojective varieties.  The twistor geometry of the stack of local systems is an essential part of the proof, and we will focus on its role in the story.

10:30-11:30 Saccà

Compactification of Lagrangian fibrations

Abstract: Lagrangian fibered Hyper-K\"ahler manifolds are the natural generalization of elliptic K3 surfaces and have been used to study and construct examples of compact Hyper-K\"ahler manifolds and (possibly singular) symplectic varieties. In this talk I will talk about some compactification techniques for quasi-projective Lagrangian fibrations, with applications to the study of Prym, Intermediate Jacobian, Albanese, dual fibrations etc.

Coffee break

12:00-13:00 Langer

Projective contact log varieties

Abstract: Contact manifolds are odd-dimensional analogues of hyper-Kähler manifolds. The main aim of the talk is to survey some results on contact structures on smooth complex projective log varieties. I will show how to study Mori type log contractions of such contact log varieties using generalizations of some standard results on the loci of rational curves. To do so I also need to study more general contact structures on some special Lie algebroids.

16:30-17:30 Pacienza

On the movable and the nef cone conjectures in the relative case

Abstract: Very recently Li-Zhao related the relative Kawamata-Morrison cone conjecture to the existence of Shokurov’s polytopes and Gachet-Lin-Stenger-Wang established, in the absolute case, the equivalence between the movable and the nef cone Kawamata-Morrison conjectures (under standard MMP conjectures). Building upon these works, together with Andreas Hoering and Zhixin Xie we generalize Gachet-Lin-Stenger-Wang to the relative case. In particular we obtain that the relative Kawamata-Morrison nef cone conjecture holds for varieties fibered in K3 surfaces.

Coffee break

18:00-19:00 Yin

A Beauville decomposition for the Hitchin system

Abstract: Beauville, and later Deninger-Murre, applied the theory of Fourier transforms and obtained a natural decomposition of the Chow ring of an abelian scheme. I will present an extension of this theory to the abelian fibration given by the Hitchin system, and discuss some consequences. Extensions of the Beauville decomposition to abelian fibrations are typically met with two difficulties: 1) extending the Poincaré line bundle; 2) the problem of the “smaller supports”. I will explain how to address these issues in this particular case. Joint work with Davesh Maulik and Junliang Shen.

Friday

9:00-10:00 Fatighenti

Examples of modular vector bundles with and without moduli, I

Abstract: We exhibit examples of slope-stable modular non-rigid vector bundles on a hyperkähler manifold of K3^[2]-type. These are obtained by performing standard linear algebra constructions on the examples studied by O’Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic 4-fold and the Debarre-Voisin hyperkähler. This is a joint work with Claudio Onorati.  In this first half of the talk we will concentrare on introducing the main definitions and properties, working out some explicit examples and stating the main result.

10:30-11:30 Onorati

Examples of modular vector bundles with and without moduli, II

Abstract: We exhibit examples of slope-stable modular non-rigid vector bundles on a hyperkähler manifold of K3^[2]-type. These are obtained by performing standard linear algebra constructions on the examples studied by O’Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic 4-fold and the Debarre-Voisin hyperkähler. This is a joint work with Enrico Fatighenti.  In this second half of the talk we will focus on the proofs and we will see that these bundles are atomic if and only if they are rigid.

Coffee break

12:00-13:00 Debarre

Quadrics on smooth Gushel-Mukai varieties

Abstract: We study Hilbert schemes of quadrics on smooth Gushel-Mukai varieties. This is joint work with Alexander Kuznetsov.