**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 A_{n}-singularities,

where the orbifold structure is locally given by C x [C/Z_{n}].

**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.

**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.

**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.

**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 = P^{1}. 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 [P^{1}/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.

**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.

**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 c_{2} » 0

and with an irreducible parabolic divisor D of X.

**Title:** Asymptotically stable bundles

**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).

**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)

**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

**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.

**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

Nagaraj and Seshadri.

The result $\tilde M$ is a blow up of the Simpson moduli space

$M_{3m+1}(\mathbb P_2)$. Joint work with Günther Trautmann.

**Title:** A theory of generalized Donaldson-Thomas invariants

**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.

**Title:** Singularities of free group character varieties

**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.

**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.

**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).

**Title:** Quadratic pairs and the Hitchin map

**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.

**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.

**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.

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

**Abstract:** Let C^{3}[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 C^{3}[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 C^{3}[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.

**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.

**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.