## June Conference - Titles and abstracts

Michel Brion

Title : Homogeneous bundles over abelian varieties

Abstract : A vector bundle on an abelian variety is called homogeneous if it is isomorphic to all of its pull-backs by translations; there is a similar notion for principal bundles. Homogeneous vector bundles over abelian varieties were classified by Mukai in algebro-geometric terms, via the Fourier-Mukai transform. We will present a classification of homogeneous principal bundles, based on the structure and representations of algebraic groups; this yields an alternative proof of Mukai's result.

Patrick Brosnan

Title : Zero loci of admissible normal functions

Abstract : Admissible normal functions are certain a priori analytic objects attached to variations of Hodge structure. To be a little more precise, they are certain sections of bundles of intermediate Jacobians. They can arise from a family of algebraic cycles or more generally from a Hodge class. Since they are analytic in nature, it is not clear that, over an algebraic base, their zero loci are algebraic. I will report on joint work with Pearlstein showing that, in fact, they are.

Lucia Caporaso

Title : Tropical and algebraic curves: comparing their moduli spaces.

Abstract : The talk will describe the construction and main properties of moduli spaces of tropical curves, together with their compactification. Some interesting analogies with moduli spaces of Deligne-Mumford stable curves will be illustrated.

Frédéric Campana

Title: Birational stability of the cotangent bundle, and orbifold rational curves.

Abstract: For any complex projective manifold X, we introduce a birational invariant k_{++}(X)>= k(X), the difference measuring the birational unstability of its cotangent bundle, and controlling k(Y) from above, for any Y dominated by X. Conjecturally, k_{++}(X)=k(R(X)), with R(X) being the rational quotient' of X. For example: k_{++}(X)=-\infty if and only if X is rationally connected

We intend to explain why and how to extend these notions to the category of smooth orbifolds', and how to reduce the above conjecture to 3 other conjectures, 2 of which are standard in the LMMP, the last one an extension of Miyaoka's generic semi-positivity to lc pairs with c_1=0.

As an application outside of the birational classification, we will mention the isotriviality conjecture for families of canonically polarised manifolds parametrised by a special' quasi-projective manifold.

Details in arXiv: 1001.3763

Fabrizio Catanese

Title: A characterization of varieties whose universal cover is a polydisk or a bounded symmetric domain of tube type.

Abstract : I will report on joint work with M. Franciosi, and the characterizations we gave of surfaces whose universal cover is a product of curves. And especially on joint work with A. Di Scala, giving the characterization of varieties whose universal cover is a polydisk, respectively a bounded symmetric domain of tube type. One defines, for a projective manifold of dimension n, a special tensor as a (non zero) section of the n-th symmetric power of the cotangent bundle twisted by the anticanonical divisor. And a semispecial tensor as a (non zero) section of the n-th symmetric power of the cotangent bundle twisted by the anticanonical divisor and by a 2-torsion line bundle.

Theorem 1. The universal cover of X is the polydisk iff 1) or, equivalently , 2) holds.

1) X has ample canonical bundle and admits a semispecial tensor such that at some point p the corresponding tangential hypersurface splits as a product of linear forms.

2) X has ample canonical bundle and admits a semispecial tensor such that at some point p the corresponding tangential hypersurface is reduced.

Theorem 2. If X has ample canonical bundle and admits a semispecial tensor, then the universal cover of X is a bounded symmetric domain D of tube type. The domain D is determined by the multiplicities of the irreducible components of the corresponding tangential hypersurface.

We have then a corollary which extends previous results by Kazhdan.

Corollary. Assume that the universal covering of X is a bounded symmetric domain D of tube type. Let X^s be a Galois conjugate of X : then also the universal cover of X^s is biholomorphic to D.

I will also give a general introduction concerning the current knowledge on projective varieties whose universal cover is a bounded domain in C^n.

Pierre-Emmanuel Chaput

Title : Littlewood-Richardson rule for minuscule Schubert calculus.

Abstract : Given a homogeneous space G/P, we will give a combinatorial formula for the intersection number of three so-called "minuscule" Schubert cells in G/P. The combinatorics are incoded in Schutzenberger's jeu de taquin, performed on a quiver which depends on G/P and the three involved Schubert cells.

Jean-Pierre Demailly

Title : Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture

Abstract : We will show that a combination of holomorphic Morse inequalities and of a probabilistic estimate of curvature of k-jets as k tends to infinity is sufficient to imply the existence of global jet differentials on every variety of general type. This yields the algebro-differential degeneration of entire curves on such varieties, or more generally on "directed varieties" of general type. Our hope is that this result can be used to prove the generalized Green-Griffiths-Lang conjecture by an induction process.

Tommaso de Fernex

Title : The valuation space of an isolated normal singularity.

Abstract : Which normal isolated singularities admit a finite endomorphism of degree at least two? The origin of this question comes from an old paper of Wahl, where he gives an answer in dimension two. By looking at the Zariski decomposition of the log canonical divisor on a given resolution, Wahl defines the "characteristic number" of the surface singularity (an invariant that vanishes if and only if the singularity is log canonical) and proves that if there is such an endomorphism, then this invariant is zero. In this talk I will discuss how this property is in fact a quite general fact, not limited to dimension two. In higher dimensions, the question is related to the analogous question on the existence of finite polarized endomorphisms of projective varieties. With this in mind, we work in full generality, without assuming the singularity to be Q-Gorenstein. This will lead us to consider all resolutions, which will be contextualized by working directly on the Zariski--Riemann space and, concurrently, on the full space of valuations of rank one centered at the singular point. The talk is based on work in progress with Charles Favre and Sebastien Boucksom.

Brendan Hasset

Title: Constructing rational curves on K3 surfaces

Abstract : Thirty years ago, Mori and Mukai proved that most' complex K3 surfaces admit infinitely many rational curves. Showing that a particular K3 surface has this property remains a challenge, especially when the surface has Picard rank one. We survey techniques applicable to these surfaces, especially reducing mod p and lifting the resulting configurations of rational curves to characteristic zero. This allows us to show that degree two K3 surfaces of Picard rank one have infinitely many rational curves. (joint with Bogomolov and Tschinkel)

Jun-Muk Hwang

Title : Deformation of the space of lines on the 5-dimensional hyperquadric

Abstract : Let F be the space of lines on the 5-dimensional hyperquadric. F is a 7-dimensional homogeneous projective manifold. We show that a projective manifold which arises as a deformation of F is biholomorphic to either F itself or the G_2-horospherical variety X^5 studied by Pasquier and Perrin. The key point of the proof is to show that a 7-dimensional uniruled projective manifold of Picard number 1 with the variety of minimal rational tangents isomorphic to a certain Hirzebruch surface is biholomorphic to X^5. A main new ingredient in the proof is the Cartanian geometry of the geometric structure determined by such a Hirzebruch surface: the construction of a Cartan connection and the investigation of its curvature. This geometric structure is associated to a non-reductive graded Lie algebra and has not been studied classically.

János Kollár

Title : A local version of the Kawamata-Viehweg vanishing theorem

Abstract : We give a criterion for a divisorial sheaf on a log terminal variety to be Cohen-Macaulay. The log canonical case and applications to moduli problems are also considered.

Title: Notes on the compactification of the moduli space of polarized K3 surfaces

Abstract: An important and challenging problem in algebraic geometry is to give a geometric compactification for the moduli space of polarized K3 surfaces. Ideas based on the minimal model program give a rather canonical solution to the compactification problem for K3s. However, in practical terms, very little is known about the resulting compactification (e.g. its structure, and the boundary objects parameterized by it). In this talk, I will discuss two complementary approaches, based on GIT and arithmetics respectively, to the MMP inspired compactification for K3 surfaces.

Robert Lazarsfeld

Title: Positivity of cycles on abelian varieties

Abstract: The cones of divisors and curves defined by various positivity conditions on a smooth projective variety have been the subject of a great deal of work in algebraic geometry, and by now they are quite well understood. However the analogous cones for cycles of higher codimension have started to come into focus only recently. I will discuss a couple of computations on abelian varieties where one can work out the picture fairly completely -- already here one sees some non-classical phenomena. I will also discuss some of the many open problems that present themselves around this circle of ideas. (This is joint work with Olivier Debarre, Lawrence Ein and Claire Voisin.)

Davesh Maulik

Title: Quantum cohomology of framed sheaves

Abstract: We will discuss some geometric constructions on the moduli space of torsion-free sheaves on the $\mathbb{C}2$, framed at infinity, and how they can be used to understand its quantum cohomology. As time permits, we will discuss applications to other symplectic resolutions and Donaldson-Thomas theory in higher rank. (joint with A. Okounkov)

Rahul Pandharipande

Title: Algebraic cobordism of varieties and bundles.

Abstract: I will discuss an approach to algebraic cobordism for varieties via double point degenerations (joint work with M. Levine). Recent applications to cobordisms of pairs [X,E] where E is a bundle on X will be treated (joint work with Y.-P. Lee). In both cases, the simplest geometric relations lead to well-behaved theories.

Christian Pauly

Title : On the monodromy of the Hitchin connection.

Abstract : In this talk I will show that the monodromy representation of the projective Hitchin connection on the sheaf of generalized theta functions on the moduli space of vector bundles over a curve has an element of infinite order in its image. I will explain the link with conformal blocks.

Mihai Paun

Title : Skoda division theorem and metrics with minimal singularities

Abstract : We will discuss some properties of positively curved metrics on adjoint bundles of big and klt divisors, together with a few applications/speculations.

Christian Peskine

Title : The k-secant lemma, a vanishing theorem, and related conjectures

Nicolas Perrin

Title : On the quantum K-theory of homogeneous spaces

Abstract : (joint work with A. Buch, P.-E. Chaput and L. Mihalcea)

The quantum K-theory of a smooth projective variety X was defined by Givental and Lee as a generalisation of the quantum cohomology. One of the new aspects of this theory is the fact that the structure constants of the product are, a priori, only defined on the tensor product of the cohomology H^*(X,Z) with the completed ring Z[[q1,...,qn]] of quantum parameters. If the structure constants are defined over Z[q1,...,qn] we say that the product is finite.

We shall explain in this talk that for homogeneous spaces, the quantum K-theoretic product is finite provided that some Gromov-Witten varieties are rationnaly connected. We shall then explain that these rationally connectedness results hold true for some homogeneous spaces.

Yongbin Ruan

Title: Landau-Ginzburg/Calabi-Yau Correspondence

Abstract: A far reaching correspondence from physics suggests that the Gromov-Witten theory of a Calabi-Yau hypersurface of weight projective space (more generally a toric variety) can be computed by the singularity theory of its de ning polynomial. In this talk, I will present some of works (jointly with Alessandro Chiodo) towards establishing this correspondence mathematically as well as some of surprises and speculations.

Jason Starr

Title : Families of abelian varieties over a higher-dimensional base

Abstract : Given a family of complex Abelian varieties over a (quasi-projective) base variety, Tom Graber and I prove that sections are "witnessed" by every sufficiently general curve in the base: the restriction map from sections of the family over the entire base to sections over the curve is an isomorphism. One can of course attack this by variations of Hodge structures, as I will explain. But our proof uses no Hodge theory; it is a simple application of "N\'eron models" applied to the original family. In particular, this leads to a strategy for proving the analogue in positive characteristic, where Hodge-theoretic methods do not apply. No background in N\'eron models will be assumed.

Ravi Vakil

Title : The ring of invariants of n points on the projective line.

Abstract : The GIT quotient of a small number of points on the projective line has long been known to have beautiful geometry. For example, the case of six points is intimately connected to the outer automorphism of S_6. We extend this picture to an arbitrary number of points, with arbitrary weights, completely describing the equations of the moduli space. (In some sense there is only one equation.) The geometry of the key case of eight points is new, but has quite a classical flavor. This is joint work with Ben Howard, John Millson, and Andrew Snowden.