### Instantons in complex geometry: talks and abstracts

Home | Venue | Schedule | Program | Poster | PDF
####
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.

Home | Venue | Schedule | Program | Poster | PDF