#
WAGS Fall 2005: Titles and Abstracts

**Speaker:**
Vladimir Baranovsky

**Title:** Brauer Groups and Crepant Resolutions.

**Abstract:**
Recently there has been a lot of results on McKay correspondence. The
most general version starts with an
action of a finite group G on a smooth variety V and identifies the
(derived) category of G-equivariant vector
bundles on V, with the (derived) category of vector bundles on any
crepant resolution of singularities X --> V/G.
The first of these categories can be "twisted" by a projective 2-cocycle
in H^2(G, C^*); while the second may
be "twisted" by a class in the Brauer group H^2(X, O^*) using modules
over Azumaya algebras. We compute
the subgroup in H^2(G, C^*) formed by classes for which both twists
exist, and present some results related
to the conjectural equivalence of the twisted categories.

**Speaker:**
Aaron Bertram

**Title:** Thaddeus flips for K3 surfaces

**Abstract:**
The compactified Jacobian of a primitive
class of curves on a K3 surface is one of a sequence
of moduli spaces that are linked by "Mukai flops."
These moduli spaces arise naturally when one considers
stability conditions on the derived category of coherent
sheaves (following Bridgeland). There is a projective
space of extensions embedded in one of these moduli space
which exhibits Thaddeus flips as the restriction of the
Mukai flops. This is a report on joint work with Daniele Arcara.

**Speaker:**
Adrian Clingher

**Title:** On a Family of Lattice Polarized K3 Surfaces

**Abstract:**
Surfaces of type K3 with lattice polarizations of type
H+E8+E8 are classified, up to isomorphism, by two modular invariants,
in the same way elliptic curves over the field of complex numbers are
classified by the j-invariant. In this talk I will present a geometric
correspondence relating K3 surfaces in question with abelian surfaces
realized as product of two elliptic curves. This correspondence is
inspired by the F-theory/heterotic string duality. It induces an
isomorphism of Hodge structures between the two objects and it can be
used to give explicit formulas for the two modular invariants of a K3
surface. This is joint work with Charles Doran.

**Speaker:**
Sándor Kovács

**Title:** Where do nowhere vanishing 1-forms come from?

**Abstract:** A result of Carrell and Lieberman states that if a smooth
complex projective variety X admits a nowhere zero vector field then
all the characteristic numbers (intersection numbers of cohomology
classes (Chern classes) defined by the tangent bundle of X) are zero.
Carrell later asked whether the dual statement of this holds as well,
that is, whether all characteristic numbers of X are zero if X admits
a nowhere zero holomorphic 1-form.

For an example of the latter, one finds nowhere vanishing 1-forms on
products of varieties with elliptic curves, or more generally when X
admits a smooth map to an abelian variety. One can generalize this a
little further, but no essentially different examples of nowhere
vanishing 1-forms are known.

In this talk, I will discuss Carrell's question, his early results,
and other questions the original one has lead to, as well as a recent
result obtained jointly with Christopher Hacon (Utah). Finally, I will
mention a more precise conjecture that tries to answer the question in
the title.

**Speaker:**
Jun Li

**Title:** Moduli of sheaves and curves on Calabi-Yau manifolds

**Abstract:**

**Speaker:**
Thomas Nevins

**Title:** Parametrizing (complexes of) sheaves on ruled surfaces

**Abstract:**
I'll describe a not-so-obvious method to parametrize vector
bundles and more general objects on ruled surfaces. This gives a simple
conceptual description of moduli spaces of D-modules on curves, perverse
vector bundles on ruled surfaces, Hilbert schemes of points on ruled
surfaces, and phase spaces for some integrable particle systems. This is
joint work with D. Ben-Zvi and V. Ginzburg.

**Speaker:**
Chris Woodward

**Title:** Localization for moduli of G-bundles on curves

**Abstract:**
This is joint work with C. Teleman on a localization principle for the
moduli space of G-bundles which gives as a corollary the Verlinde formula
and various generalizations. An application is a proof of the
Newstead-Ramanan conjecture on the vanishing of the higher Chern classes
of the moduli space for connected G. I will talk about this and some open
problems regarding this moduli space.