### Instantons in complex geometry: talks and abstracts

#### 14-18 March 2011, Laboratory of Algebraic Geometry,Higher School of Economics, Moscow

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#### Vladimir Baranovsky (Irvine) Uhlenbeck compactification in arbitrary dimension

We will outline a definition of a moduli functor which compactifies the moduli of principal G-bundles on a smooth projective variety X of any dimension (for G a reductive group). Our work is based on the technique of intersection bundles due to Deligne and the semistable reduction for principal bundles due to Balaji.

#### Alexander Braverman (Brown Univ.) Towards categorical approach to instanton counting

We present a list of (partially proven) conjectures about the structure of derived categories of constructible sheaves on various instanton moduli spaces and explain how these conjectures can be applied to questions of "instanton counting" (in particular, to a version of Alday-Gaiotto-Tachikawa conjecture).

#### Vasile Brinzanescu (IMAR, Bucharest) Vector bundles on non-Kahler Calabi-Yau type 3-folds

We study moduli of relatively semi-stable vector bundles on non-Kahler principal elliptic bundles, which are Calabi-Yau type 3-folds. The main technical tools used are the twisted Fourier-Mukai transform and a spectral cover construction (joint work with A. Halanay and G. Trautmann).

#### Ugo Bruzzo (SISSA, Trieste) Uhlenbeck-Donaldson Compactification for Framed Sheaves

We introduce the moduli spaces of framed sheaves. A reason for being interested in these objects is that under suitable conditions they are nice moduli spaces (quasi-projective, smooth and fine) and provide desingularizations of moduli spaces of framed instantons. We study a "partial compactification" for these moduli spaces, called the Uhlenbeck-Donaldson compactification, and relate it to the moduli space of "ideal" framed instantons.

#### Laura Costa (Univ. de Barselona) Linear Monads on Instanton bundles on hyperquadrics

Monads appear in a wide variety of context within Algebraic Geometry. We will focus our attention on linear monads as a tool for constructing indecomposable vector bundles on hyperquadrics and on the particular case of instanton bundles on them. This is a join work with R.M. Miro-Roig.

#### Michael Finkelberg (HSE) Parabolic bundles on P^2, Quiver varieties, and Quantization

The moduli space of certain parabolic torsion free sheaves on P^2 plays an important role in the geometric representation theory (under the name of Affine Laumon space). The singularities of its affinization contain the "limit singularities" of Schubert varieties of the affine Grassmannian of SL_n. It can be realized as a quiver variety, and quantized via the quantum Hamiltonian reduction. The quantization is described in terms of affine Yangians.

#### Sergei Gorchinskiy (Steklov Institute) Polar homology

The talk is based on a joint work with A.Rosly. We discuss new complexes that compute cohomology groups of locally free sheaves on smooth algebraic varieties in characteristic zero. These complexes are algebro-geometric analogs of singular complexes for real smooth manifolds and have arised from E.Witten's program of holomorphic Chern-Simons theory.

#### Laurent Gruson (Univ. de Versailles) About the mathematical instantons with c_2 = 4

In his thesis (1996) Frederic Han gave a description of a dense open set of mathematical instantons for c_2 = 4,5 . He assumed that the second order jumping lines are in codimension 3, but there was a gap in the proof he gave of this fact. I would like to discuss the state of affairs around this in the case c_2 = 4.

#### Norbert Hoffmann (Freie Univ. Berlin) Rational families of instanton bundles on P^{2n+1}

I'll discuss the rationality problem for two kinds of moduli spaces for instanton bundles on P^{2n+1}, namely for special instantons in the sense of Spindler-Trautmann and for symplectic 't Hooft instantons in the sense of Ottaviani. The latter is joint work with Costa, Miro-Roig and Schmitt.

#### Marcos Jardim (UNICAMP) Moduli spaces of framed instanton bundles on CP^3

I will present the ADHM construction of framed instanton sheaves on CP^3, which yields a parametrization of the (fine) moduli space of framed instanton bundles in terms of matrices satisfying some quadratic equations. Using twistor theory, one can show that such moduli space coincides with the space of twistor section of the moduli space of framed bundles on CP^2. We also introduce a the notion of trisymplectic structures on a complex manifolds, and the notion of trisymplectic reduction. We show that the moduli space of framed instanton bundles can also be described in terms of a trisymplectic reduction, and conclude that it is a smooth trisymplectic manifold of the expected dimension. Joint work with Misha Verbitsky and with Amar Henni and Renato Vidal Martins.

#### Alexander Kuznetsov (Steklov Institute) Instanton bundles on Fano threefolds

I will talk about instanton bundles on Fano threefolds of index 2. In particular, jumping lines and monadic representations will be discussed.

#### Adrian Langer (University of Warsaw) Moduli spaces of framed perverse instantons on P^3

I will talk about moduli spaces of framed perverse instantons on P^3. As an open subset they contain the moduli space of framed instantons studied by I. Frenkel and M. Jardim. I will explain the connection with the moduli space of pairs on the blow up of P^3 along a line. I will also study the map between the Gieseker and Donaldson-Uhlenbeck partial compactifications of the moduli space of instantons on P^3. Finally I will construct a few counterexamples to earlier conjectures and results concerning these moduli spaces.

#### Eyal Markman (University of Massachussets) Morrison's movable cone conjecture for projective irreducible holomorphic symplectic manifolds.

We prove a version of the conjecture in the title as a consequence of the Global Torelli Theorem for irreducible holomorphic symplectic manifolds X. Let Bir(X) be the group of birational automorphisms of X. As consequence it is shown that for each non-zero integer d there are only finitely many Bir(X)-orbits of complete linear systems, which contain a reduced and irreducible divisor of Beauville-Bogomolov degree d. A variant hold for degree zero as well.

#### Rosa M. Miro-Roig (Univ. de Barcelona) Monday, Tuesday or Wednesday COHOMOLOGICAL CHARACTERIZATION OF VECTOR BUNDLES

In my talk, I will address the problem of giving a cohomological characterization of vector bundles on algebraic varieties. This is a longstanding problem in Algebraic Geometry which has its roots in an old paper by Horrocks where he gave a cohomological characterization of line bundles on projective spaces P^n.

In my talk, I will give a cohomological characterization of the bundle of p-differential forms on multiprojective spaces P^{n_1}\times ... \times P^{n_s} and a cohomological characterization of Steiner bundles on algebraic varieties. As a main tool I will use a generalized version of Beilinson's spectral sequence.

This is joint work with Costa and Soares

#### Ruxandra Moraru (Univ. Waterloo) Generalized holomorphic vector bundles and Poisson modules on Hopf surfaces

Hopf surfaces are interesting examples of compact complex surfaces that do not admit Kaehler metrics, but nonetheless admit generalized Kaehler structures. In this talk, I will describe the geometry of generalized holomorphic vector bundles on these surfaces, thus providing new non-trivial examples of generalised holomorphic vector bundles and their moduli. Time permitting, I will also describe how these bundles relate to Poisson modules on Hopf surfaces.

#### Christian Okonek (Univ. Zurich) Abelian Yang-Mills theory on a Real torus, and theta divisors of Klein surfaces

In joint work with Andrei Teleman we determine certain natural theta line bundles of Klein surfaces as elements in the Grothendieck cohomology group which classifies Real line bundles in the sense of Atiyah. These line bundles are important since they control the orientability of certain moduli spaces in Real gauge theory to a large extend.

#### Dmitri Panov (King's College London) Hyperbolic geometry and fundamental groups of complex 3-folds

We give an alternative short proof of Taubes' theorem stating that compact complex 3-folds can have arbitrary finitely presented fundamental group. This is related to a question of Gromov: Is it true that every compact manifold is homeomorphic to a quotient of the hyperbolic space H^n by an isometric (non-free) action of a discreet group? This talk is based on a joint work with Anton Petrunin.

#### Prabhakar Rao (Univ. of Missouri) ACM vector bundles on hypersurfaces

An Arithmetically Cohen Macaulay vector bundle on a hypersurface X of projective space is a bundle E for which $H^i(X,E(a))=0$ for any integer a and any i between 1 and dim(X)-1. We will discuss existence, constructions and applications of ACM bundles on smooth hypersurfaces.

#### Alexander Schmitt (Freie Univ. Berlin) Quiver Bundles

In this talk, I will first review the "classical" formalism of quiver representations, with a view on sheaves and instantons on projective manifolds. Then, I will give a survey on results on quiver bundles on projective manifolds and their moduli spaces. At the end, I will present joint work with Garcia-Prada and Heinloth on the motive of the moduli space of Higgs bundles of rank four and odd degree on a compact Riemann surface.

#### Andrei Teleman (Univ. de Provence) Holomorphic bundles and holomorphic curves on class VII surfaces. The classification problem for class VII surfaces.

The classification of complex surfaces is not finished yet. The most important gap in the Kodaira-Enriques classification table concerns the Kodaira class VII, e.g. the class of surfaces $X$ having $kod(X)=-\infty$, $b_1(X)=1$. These surfaces are interesting from a differential topological point of view, because they are non-simply connected 4-manifolds with definite intersection form. The main conjecture which (if true) would complete the classification of class VII surfaces, states that any minimal class VII surface with $b_2>0$ contains $b_2$ holomorphic curves. We explain a new approach, based on ideas from Donaldson theory, which gives existence of holomorphic curves on class VII surfaces with small $b_2$. In particular, for $b_2=1$ we obtain a proof of the conjecture, and for $b_2=2$ we prove the existence of a cycle of curves.

#### Nadezhda Timofeeva (Yaroslavl University) On Giseker-Maruyama moduli for a surface: new interpretation of classical scheme

Main (containing locally free sheaves) components of GM moduli scheme will be re alized as moduli scheme of locally free scheaves on some special class of schemes.

#### Gunther Trautmann (Univ. Kaiserslautern) Replacing the boundary by treelike vector bundles

Given a moduli space of semistable sheaves, one can try to replace its boundary of non-locally free sheaves by a variety of classes of locally free sheaves in order to eventually better understand the degenerations and the singularities of the sheaves in the boundary. This will be done for Gieseker-Maruyama moduli spaces M_S(2; P) of semistable sheaves of rank 2 with Hilbert polynomial P on a projective surface S. The replacement in this situation resembles the bubbling in Yang-Mills theory for vector bundles on 4-manifolds. A second type of replacement by vector bundles will be discussed for 1-dimensional sheaves in a Simpson moduli space M of semistable sheaves on the projective plane.

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