Vector Bundles on Algebraic Curves

Derived Categories and the Langlands Programme

Abstracts

    V Baranovsky

    Uhlenbeck compactification as a stack

    We explain how a concept of a "quasi-map" into the moduli space of bundles on a curve gives rise to a "functor of families" description for the Uhlenbeck compactification of moduli of vector bundles on a smooth projective surface. Our work generalizes an earlier construction of Gaitsgory-Braverman-Finkelberg-Kuzntesov for bundles on a projective plane. We also establish a connection with earlier related work by Gomez, Schmitt and Balaji.

    M Bolognesi

    A structure theorem for SUC(2) and the moduli of pointed rational curves

    In this talk I will describe a construction recently given with A.Alzati (Milan). Let C be a genus g ≥ 2 curve and SUC(2) the moduli space of semistable rank 2 vector bundles on C with trivial determinant. A few low genus cases are well known: if g = 2 then SUC(2) ≅ IP3 and if g = 3 it is the Coble quartic in IP7. Via some remarks about classification maps of extensions, I will show that for any genus, SUC(2) is birational to a fibration with rational fibers over a projective space IPg. Moreover I will show how the fibers can be birationally identified to M2g0, the moduli space of rational 2g-pointed curves. If g = 3 this gives a birational isomorphism between the Coble quartic and a fibration in Segre cubics over a IP3.

    I Burban

    Tilting on rational projective curves

    My talk is based on a joint work in progress with Yuriy Drozd. Our main result is the following. Let X be a reduced rational projective curve of any arithmetic genus, having only nodes or cusps as singularities. Then there exists a finite-dimensional algebra A of global dimension two and a fully faithful exact functor D¯(Coh(X))  D¯(A-mod) having a left adjoint functor. This technique allows to show the dimension of the bounded derived category of coherent sheaves on X is at most two, confirming a conjecture posed by R. Rouquier. In the case of degenerations of elliptic curves, this leads to a particularly nice class of algebras called "gentle". From the one side, our approach brings a new light on the representation theory of gentle algebras, on the other it suggests a new approach to some open problems about derived categories of coherent sheaves on degenerations of elliptic curves.

    P-H Chaudouard

    On the truncated Hitchin fibration

    The talk will be based on a joint work with Gérard Laumon. Ngô Bao Châu has proved the Langlands-Shelstad fundamental lemma thanks to a deep cohomological study of the elliptic part of the Hitchin fibration. Outside the elliptic open set, the Hitchin fibration is no more proper. I shall introduce in this talk a better open subset of the Hitchin fibration. I shall show how one can extend Ngô's cohomological theorems to this situation. As a consequence, we prove the weighted fundamental lemma, which is a generalization of the fundamental lemma due to Arthur and which is needed in the stabilization of the Arthur-Selberg trace formula.

    D Diaconescu

    Chamber structure and wallcrossing for ADHM invariants

    A construction of equivariant counting invariants of ADHM sheaves on curves will be presented. This construction depends on a real stability parameter, resulting in a chamber structure in the ADHM theory of curves. A wallcrossing result will be proven using the theory of Behrend functions for Artin stacks developed by Joyce and Song. Some wallcrossing conjectures as well as applications to local stable pair theory and BPS invariants will be briefly discussed.

    D Eisenbud

    Cohomology tables of vector bundles on IPn

    I will speak about joint work with Frank Schreyer that describes the possible cohomology tables of vector bundles, and to a certain extent all coherent sheaves, on IPn, up to rational multiples. Although the characterization is very concrete and direct, the heart of the argument seems to be a derived category connection with graded modulesover the polynomial ring, clearly related to but apparently different than the usual BGG correspondence. I will describe this work, and some recent extensions of it, and try to make clear where the mystery lies.

    G Hein

    Fourier-Mukai transforms and minimal bundles

    We call a vector bundle E on a algebraic curve X minimal, when h0(LE) > 0 for all line bundles L of degree zero, and for all proper subsheaves E'  E this fails. Using the theory of the Fourier-Mukai transforms we show that minimal vector bundles on X define ample divisors on the Picard torus Pic0(X). As a corollary we obtain that the moduli space of rank two bundles with odd degree together with its Poincaré bundle is a Quot scheme of an appropriate vector bundle.

    J Heinloth

    Remarks on a conjecture of Hausel and Rodriguez-Villegas.

    Hausel and Rodriguez-Villegas made a very intriguing series of conjectures on the cohomology of the moduli space of Higgs-Bundles on a curve (if rank and degree are coprime). In particular their conjectures imply that the intersection form on the cohomology should vanish if the genus of the curve is at least 2. After recalling these conjectures we would like to report on some recent supporting evidence for some of these conjectures.

    D Hernández Serrano

    Equations of the moduli of Higgs pairs

    The moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and characterize its image. This characterization provides us with a method for explicitely computing KP-type equations that describe the moduli space of Higgs pairs. Finally, for the case where the spectral cover is totally ramified at a fixed point of the curve, these equations can be given in terms of the characteristic coefficients of the Higgs field. A connection between these techniques and the Langlands program will be also discussed.

    D Huybrechts

    Chow groups and derived categories of K3 surfaces

    In this talk I will survey what is known about the group of autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. I will mention results of Mukai and Orlov and a conjecture of Bridgeland. Every autoequivalence acts on the cohomology and on the Chow ring of the K3 surface. In a joint work with Macri and Stellari we recently completed the description of the action on cohomology. The Chow ring, which is huge due to Mumford's classical result, admits a natural subring studied by Beauville and Voisin. We show that this subring is preserved under derived equivalences.

    A King

    Dimers and Calabi-Yaus

    I will give an introduction to the dimer model construction of non-commutative toric 3-Calabi-Yau algebras which resolve toric 3-Calabi-Yau singularities, e.g. they are derived equivalent to the usual (commutative) crepant resolution.

    B Kreussler

    1. On derived categories of coherent sheaves

    Derived categories of coherent sheaves and their auto-equivalences have been used to study moduli spaces of vector bundles, to study the birational geometry of projective varieties and to define new invariants of the underlying variety. After giving a brief overview over these topics, I shall focus on the case of irreducible singular cubic curves and present results jointly obtained with Igor Burban.

    2. Vector bundles on cubic curves and Yang-Baxter equations

    This talk is based on joint work with Igor Burban. Starting with a flat family E  T of irreducible projective curves having trivial dualising sheaf and with a fine relative moduli space M  T of vector bundles on the given family, we construct an isomorphism between two vector bundles on an open set of M xT M xT E xT E, which gives us a family of solutions of the associative Yang-Baxter equation. This generalises a construction of Polishchuk for a single curve. The construction provides an example of the philosophy underlying homological mirror symmetry that structural properties of derived categories should lead to solutions of equations from mathematical physics.

    E Macri

    Categorical invariants for cubics

    It is a famous theorem of Clemens-Griffiths and Tyurin that a cubic threefold is determined by its intermediate Jacobian. We discuss a categorical version of this theorem, namely, that a cubic threefold is determined by a semi-orthogonal component of its derived category. Our approach is to construct an interesting Bridgeland stability condition on this component and recover the Fano surface of lines as a moduli space of stable objects. If time permits, we will also discuss some generalization to the case of cubic fourfolds. This is joint work with P. Stellari, and partly with M. Bernardara and S. Mehrotra.

    J Martens

    Moduli of parabolic Higgs bundles and Atiyah algebroids

    We will discuss the holomorphic Poisson geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points in terms of Lie algebroids. Using the Hitchin system in this setting and we will show how this provides a global analogue of the Grothendieck-Springer resolution. This is joint work with M. Logares.

    T Ngo Dac

    Fundamental lemma and Hitchin fibration after Ngo Bao Chau

    In this talk, I will report the recent proof of Ngo Bao Chau of the fundamental lemma for Lie algebras. This conjecture can be resumed to be a family of equalities of local orbital integrals. Surprisingly, it turns out that it has been proved via the study of a well-studied geometric object - the Hitchin fibration. I will explain the connection between the orbital integrals and the Hitchin fibration and emphasize on the geometry of the latter. This talk serves as a preparation for the talk of P-H. Chaudouard.

    Y Pandey

    Prym varieties and abelianisation of G-bundles

    We describe some recent contributions to the abelianisation program via the calculation of polarisations on Prym-varieties. Consider a Galois cover π: ZX of smooth projective curves with Galois group a Weyl group W of a simple Lie algebra G. For a dominant weight λ, we consider the intermediate curve Yλ = Z/Stab(λ). One can realise a Prym-variety Pλ  Jac(Yλ) and we denote by φλ the restriction of the principal polarisation of Jac(Yλ) upon Pλ. For two dominant weights λ and μ, we construct a correspondence Δλ,μ on Yλ x Yμ and calculate the pull-back of φμ by Δλ,μ in terms of φλ. We also describe a result for N-bundles wher N is any extension of a finite group W by an abelian group T. These results originate from the speaker's thesis supervised by Christian Pauly and subsequent work.

    K-G Schlesinger

    S-duality and the geometric Langlands program: A short overview

    We start from supersymmetric (N=4) gauge theory in four dimensions and the S-duality (the generalization of the well known electric-magnetic duality of the Maxwell equations to the case of a nonabelian gauge theory) conjecture. Applying two steps - a topological twist and a reduction to two dimensions - we arrive at mirror symmetry for a very special two dimensional field theory from which one rediscovers the Langlands duality for an algebraic curve and a given gauge group (the nonabelian gauge group we started from). Finally, we will explain how recent results of Braverman and Finkelberg on the searched for geometric Langlands duality for algebraic surfaces look like from a physics perspective: Here, one has to invoke S-duality of type IIB string theory (or, alternatively, an M-theory perspective), instead of S-duality of the four dimensional gauge theory, which gives a deep relationship between Langlands duality and the famous dualities of string-/M-theory.

    U Stuhler

    Hecke algebras and motivic integration

    In the last years there have been various attempts to define Hecke algebras in the context of motivic integration. Some of these attempts will be discussed. There will be outlined analogies with the classical case of the Hecke algebras related to locally compact reductive groups. Additionally applications towards the homological algebra of these rings will be given as well as relations to the geometric local Langlands program.

    Y Varshavsky

    1. On the Langlands correspondence over function fields

    Let K be a field of rational functions of a curve over a finite field, and let A be the ring of adeles of K. Then the Langlands conjecture for GL(r) (proved by Drinfeld for r=2 and by Lafforgue in general) asserts that there is a canonical bijection between (ℓ-adic) r-dimensional irreducible representations of the Galois group of K and certain infinite-dimensional representations of the group GL(r,A). The aim of my talk is to formulate the result, and outline the strategy of the proof. If time permits I will also speak about Langlands conjecture for arbitrary reductive groups.

    2. From automorphic sheaves to the cohomology of the moduli spaces of F-bundles

    Drinfeld has initiated (in the case of GL(2)) two geometric approaches to global Langlands conjectures over function fields: one by constructing of so-called "automorphic sheaves" and another by studying the cohomology of the moduli spaces of so-called F-sheaves (which are also known as "Drinfeld's shtukas"). The purpose of this lecture is to connect these two approaches (for any reductive group G). This is a joint work with Alexander Braverman.


Berlin, June 15-19, 2009

An activity of
SFB 647 “Space-Time-Matter“

www.raumzeitmaterie.de

Venue

Konrad-Zuse-Zentrum
für Informationstechnik Berlin
Takustrasse 7
D-14195 Berlin-Dahlem
Germany

www.zib.de