Abstracts

Arend Bayer

Title: PT/DT-correspondence for orbifolds

Abstract: The PT/DT-correspondence relates counting invariants of stable pairs
with Donaldson-Thomas invariants (counting invariants of subschemes).
We will show how to use wall-crossing to extend the
PT/DT-correspondence to smooth 3-dimensional
orbifolds that are Calabi-Yau satisfying one additional condition
("Hard Lefschetz condition"). The formulas become particular explicit
in the case of transversal An-singularities,
where the orbifold structure is locally given by C x [C/Zn].

Kai Behrend

Title: On the calculus underlying Donaldson-Thomas theory

Abstract: On a manifold there is the graded algebra of polyvector fields
with its Lie-Schouten bracket, and the module of de Rham differentials with
exteriour differentiation. This package is called a "calculus". The moduli
space of sheaves (or derived category objects) on a Calabi-Yau threefold has
a kind of "virtual calculus" on it, at least conjecturally. In particular,
this moduli space has virtual de Rham cohomology groups, which categorify
Donaldson-Thomas invariants, at least conjecturally. We describe some
attempts at constructing such a virtual calculus. This is work in progress.

Tom Bridgeland

Title: Hall algebras and curve-counting invariants

Abstract: Joyce, Toda and others have shown how Hall algebras of coherent
sheaves can be used to prove results about Donaldson-Thomas-type curve-counting
invariants. In particular, the DT/PT correspondence and a weak rationality
statement can be proved in this way. In this talk I will explain some of the
Hall algebra technology and outline one approach to these results.

Jim Bryan

Title: Motivic Donaldson-Thomas invariants and the virtual motive of the Hilbert
scheme of points

Abstract: Let X[n] be the Hilbert scheme of n points on X. When X is a curve or a
surface, X[n] is smooth there are beautiful formulas for their Euler characteristics,
Betti numbers, and in fact, their motives. When X is a threefold, X[n] is in general
singular, reducible, and not of the expected dimension. However, when X is a
Calabi-Yau threefold, X[n] is virtually smooth and has a natural virtual motive. The
Euler characteristics of these virtual motives are the degree zero Donaldson-Thomas
invariants of X and so they can be regarded as motivic Donaldson-Thomas invariants of
X. We give a formula for these motives which naturally generalizes the corresponding
formulas when dimension of X is 1 or 2. This is joint work with Kai Behrend and Balazs
Szendroi.

Igor Burban

Title: Hall algebra of coherent sheaves on curve orbifolds

Abstract: This talk is based on a joint project with Olivier Schiffmann.
Let X be a projective curve over a finite field k and G be a finite
group of automorphisms such that X/G = P1. We study properties
of the Hall algebra of the category of G-equivariant coherent sheaves
on X (what can be interpreted as the category of coherent sheaves
on the orbifold [X/G] or a “weighted projective line”).
This study is motivated by numerous applications to the
structure theory of quantum groups. It turns out that the
Hall algebra of the category of coherent sheaves
on [P1/G] is closely related with the quantum affine
algebra of the Dynkin type determined by G. The case of
elliptic curves leads to certain quantum toroidal algebras.
I am going to show how various properties of quantum algebras
can be categorified in this way, using the technique of derived
categories, stability conditions etc.

Michele Cirafici

Title: Cohomological gauge theory and Donaldson-Thomas invariants

Abstract: I will discuss the relation between Donaldson-Thomas theory on a
local Calabi-Yau threefold and a six dimensional topological gauge theory.
Donaldson-Thomas invariants are believed to count certain bound states of
D-branes in string theory. The topological gauge theory arises as an effective
description of the D-brane system and Donaldson-Thomas type invariants can be
seen as generalized instanton configurations. One can associate to these
generalized instantons a ADHM-like formalism which is based on a certain
quiver.

Arijit Dey

Title: Irreducibility of parabolic moduli over algebraic surface.

Abstract: We will sketch a proof of irreducibility of moduli space of
parabolic rank 2 bundles over an algebraic surface for c2 » 0
and with an irreducible parabolic divisor D of X.

Henrique Sá Earp

Abstract: The notion of asymptotic stability of holomorphic bundles emerges as
boundary condition in geometric analysis (Hermite-Yang-Mills problem) over
certain noncompact Calabi-Yau 3-folds. It amounts to the stability of the
bundle restricted to the "divisor at infinity", hence it is a purely
topological property.

In the context of Fano 3-folds with a K3 divisor at infinity, I will show
how examples of asymptotically stable bundles are obtained as cohomology of
linear monads. This context is noteworthy since such spaces are building
blocks in the construction of real 7-dimensional manifolds with holonomy
group G2, which appeal to some physicists studying String/M-theory.

This is joint work with Prof. Marcos Jardim (Unicamp, SP, Brazil).

Mario García Fernández

Title: Coupled equations for Kähler metrics and Yang-Mills connections

Abstract: We introduce a system of partial differential equations coupling a Kähler metric
on a compact complex manifold X and a connection on a principal bundle over X.
These equations intertwine two well studied quantities, the first being the
curvature of a Hermite Yang-Mills connection (HYM) and the second being the scalar
curvature of a Kähler metric. They depend on a positive real parameter and have
an interpretation in terms of a moment map µ on an infinite-dimensional
symplectic manifold, where the group of symmetries G is an extension of the gauge
group of the bundle that moves the base X. The moduli space of solutions is the
symplectic quotient
M = µ-1 (0)/G,
that inherits a Kähler structure. The problem considered merges the well-studied
theories of Hermitian Yang-Mills connections (obtained for > 0) and constant
scalar curvature Kähler metrics (which correspond to = 0) into a unique theory.
We use the moment map interpretation of the coupled equations to give necessary
and sufficient conditions for the existence of solutions. Building on the work of
A. Futaki, we provide an obstruction using an adapted version of the Futaki
invariant for the coupled equations. We give a sufficient condition, obtained via
a deformation argument, that is satisfied in a large family of examples. Relying
on previous work of S. K. Donaldson, we define an algebraic (poly)stability
condition for a pair consisting of a polarized variety and a bundle, and
conjecture that the existence of solutions implies the polystability of the pair.
(Joint work with Luis Álvarez-Cónsul and Oscar García-Prada)

Mark Gross

Title: Smoothing surface singularities via mirror symmetry

Abstract: I will talk about recent work with Paul Hacking and
Sean Keel, which constructs a mirror family to any rational
surface Y along with an anti-canonical cycle of rational curves
D. In the particular case that D is contractible to a cusp
singularity, the mirror family is a smoothing of the so-called
"dual" cusp singularity, thus proving an old conjecture of
Looijenga on the smoothability of cusp singularities.

The mirror family is constructed explicitly from relative
Gromov-Witten invariants of Y, using ideas derived from
the tropical vertex of myself, Pandharipande, and Siebert

Paul Hacking

Title: Exceptional vector bundles associated to degenerations of surfaces.

Abstract: In the 1980s J. Wahl described degenerations of surfaces for
which there are no vanishing cycles. In this setting, we construct a
rigid holomorphic bundle on the general fiber which is analogous to a
vanishing cycle. This provides a geometric interpretation of
exceptional bundles on rational surfaces and suggests a way to
understand the boundary of the moduli space of surfaces of general
type via the classification of rigid bundles on such surfaces.

Oleksandr Iena

Title: Modification of the Simpson moduli space $M_{3m+1}(\mathbb P_2)$ by vector bundles

Abstract: We consider the moduli space of stable vector bundles on curves embedded in
$\mathbb P_2$ with Hilbert polynomial 3m+1 and construct a compactification
of this space by vector bundles that resemble the bundles considered by
The result $\tilde M$ is a blow up of the Simpson moduli space
$M_{3m+1}(\mathbb P_2)$. Joint work with Günther Trautmann.

Dominic Joyce

Abstract: Calabi-Yau manifolds are compact Kähler manifolds with trivial canonical bundle.
They have a rich geometric structure. Calabi-Yau 3-folds are of interest in String Theory and are
the subject of "Mirror Symmetry" — a family of conjectures relating the complex geometry of one
Calabi-Yau 3-fold X with the symplectic geometry of a different Calabi-Yau 3-fold X*. The
"Donaldson-Thomas invariants" of a Calabi-Yau 3-fold X, introduced by Richard Thomas in 1998,
are integers which "count" Gieseker (semi)stable coherent sheaves on X in a fixed topological
class (Chern character). Coherent sheaves are algebro-geometric objects which generalize
holomorphic vector bundles. Donaldson-Thomas invariants have the nice property that they are
unchanged under deformations of the complex structure of X. They have attracted attention recently
through the "MNOP Conjecture", which relates Donaldson-Thomas invariants counting rank 1 sheaves
(ideal sheaves) to Gromov-Witten invariants counting curves on X. Thomas' original definition worked
only for topological classes in which there are no strictly semistable sheaves. Also, the dependence of
the invariants on the stability condition (polarization / Kähler class) was not understood. We will
describe a new generalization of Donaldson-Thomas invariants, with the following properties:

• they take values in the rationals;
• they are defined for all topological classes, and are equal to Donaldson-Thomas invariants when these are defined;
• they are unchanged under deformations of the complex structure of X; and
• they transform according to a known wall-crossing formula under change of stability condition.

This is related to the 2008 paper by Kontsevich and Soibelman. Joint work with Yinan Song.

Sean Lawton

Abstract:
Let X be the moduli of SL(n,C), SU(n), GL(n,C), or U(n) valued
representations of a rank r free group. We compute the fundamental group of
X and show that these four moduli otherwise have identical higher homotopy
groups. We then classify the singular stratification of X. This comes down
to showing that the singular locus corresponds exactly to reducible
representations if there exist singularities at all. Lastly, we show that
the moduli X are generally not topological manifolds, except for a few
examples we explicitly describe. This is joint work with C. Florentino.

Sukhendu Mehrotra

Title: Moduli spaces of simple sheaves on K3 surfaces and deformations

Abstract: We study integral transforms between a K3 surface X and moduli
spaces of simple sheaves on X. The transform whose kernel is the (pseudo)
universal sheaf of the moduli problem is shown to be generically faithful.
To conclude, applications of this result to "non-commutative" deformations of X
are discussed. This is work in progress, joint with Eyal Markman.

Vikram Metha

Title: The fundamental group scheme of a family of smooth projective
varieties over W

Abstract: We construct the fundamental group scheme for a family over W, using
the theory of Tannaka Lattices of Wedhorn. We prove the Kunneth formula as well
as a version of base-change (joint work with S. Subramanian).

André Oliveira

Abstract:
We consider holomorphic quadratic pairs of type (2,d) over a smooth
projective curve X. The stability condition for these objects depends
on a real parameter $\alpha$, and our aim is to study the connectedness
of the corresponding moduli spaces $\mathcal{N}_{\alpha}(2,d)$. This is
done through the analysis of the changes in the spaces
$\mathcal{N}_{\alpha}(2,d)$ when the parameter $\alpha$ varies, and also
through a detailed study, using an analogue of the Hitchin map, of a
particular space $\mathcal{N}_{\alpha_0}(2,d)$.
We shall also mention the relation between these spaces and the moduli
space of certain representations of $\pi_1X$.

Rahul Pandharipande

Title: I. Introduction to curve counting on 3-folds

Abstract: I will discuss the Donaldson-Thomas counting theory for
curves on a 3-fold X. The relationship to the theory of stable
quotients (concerning rank 1 sheaves on curves C in X
with a section) will be treated. Aspects of
the study of toric 3-folds via the vertex will be presented.

Title: II. Local curves

Abstract: A curve C sitting in a rank 2 bundle N -> C as the 0 section is the
simplest model for a curve in a 3-fold. I will discuss
the well-developed theory of local curves including the relationship
to the quantum cohomology of the Hilbert scheme of points and
more recent understanding of the descendent theory.

Markus Reineke

Title: Quiver moduli and integrality of DT-type invariants

Abstract: Moduli spaces of representations of quivers allow to model
wall-crossing formulas for the DT-type invariants of M. Kontsevich and Y.
Soibelman. We dicuss how integrality properties of these invariants can be
derived from functional equations characterizing generating series of Euler
characteristics of quiver moduli.

Bernd Siebert

Title: A tropical view on Landau-Ginzburg models.

Abstract: Mirror symmetry has been suggested to associate to a variety with
effective anticanonical bundle a so-called Landau-Ginzburg model. Mathematically
this is simply a variety with a holomorphic function, the superpotential. In the
talk I will report on work in progress with Michael Carl and Max Pumperla
(Hamburg) how this often mysterious process fits very naturally into my joint
program with Mark Gross on an explanation of mirror symmetry via toroidal
degenerations and tropical geometry. One feature is that properness of the
superpotential is equivalent to the irreducibility of the anticanonical divisor
on the mirror side. Moreover, the Gromov-Witten invariants of the local
(non-compact) Calabi-Yau geometries associated to the Fano side naturally come up
via our tropical scattering process on the Landau-Ginzburg side.

Balázs Szendröi

Title: The Hilbert space of four points on affine three-space and its refined DT invariant

Abstract: Let C3[n] denote the Hilbert scheme of n points on affine three-space;
this is an example of a moduli space of
sheaves on a Calabi-Yau threefold, and of a singular scheme cut out inside
a smooth embedding space by a global
superpotential. In the simplest nontrivial case n=4, the scheme C3[4] and
its embedding can be described very
explicitly in terms of the Plucker embedding of a Grassmannian. This leads
to an explicit computation of the cohomology
of the sheaf of vanishing cycles of the superpotential, which is a
refinement of the Donaldson-Thomas invariant attached
to C3[4]. This is joint work with Alexandru Dimca, and is also related to
joint work with Jim Bryan and Kai Behrend which
will be reported on in Jim Bryan's talk.

Yukinobu Toda

Title: Moduli stacks of stable quotients and the wall-crossing

Abstract: The notion of stable quotients is introduced by
Marian-Oprea-Pandharipande, to give a compactification of
the moduli space of maps from Riemann surfaces to the
Grassmannian, which is different from stable map compactification.
In this talk I will give a generalized notion of stable quotients
which depends on a certain stability parameter, and show that
stable quotients and stable maps are related by wall-crossing
phenomena.

Graeme Wilkin

Title: Morse theory and stable pairs

Abstract: In the early 1980s Atiyah and Bott described a new approach to studying the
cohomology of the moduli space of stable bundles: the equivariant Morse
theory of the Yang-Mills functional. There are many other interesting moduli
spaces that fit into a similar framework, however the catch is that the
total space is singular, and it is not obvious how to construct the Morse
theory of the appropriate functional. In this talk I will describe how to
get around these difficulties for the moduli space of stable pairs, for
which we prove a Kirwan surjectivity theorem and give a Morse-theoretic
interpretation of the change in cohomology due to a flip. This builds upon
earlier work with George Daskalopoulos, Jonathan Weitsman and Richard
Wentworth for rank 2 Higgs bundles.
This is joint work with Richard Wentworth.