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

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.

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