Abstracts - Spring 2012

Speaker: Mark Gross

Title: Mirror symmetry for rational surfaces and smoothing surface singularities

Abstract: I will talk about recent work with Sean Keel and Paul Hacking on a general mirror construction for pairs (Y,D), where Y is a rational surface and D is an anti-canonical cycle of rational curves. The construction is controlled by certain Gromov-Witten invariants of the pair (Y,D). I will discuss a number of applications, including resolving a conjecture of Looijenga on smoothability of cusp singularities and the existence of canonical sections of line bundles of Y, called theta functions.

Speaker: Sasha Merkurjev

Title: Algebraic cycles and norm varieties 

Abstract: Let p be a prime integer and F a field of characteristic 0.Let X be the norm variety of a symbol in K_n(F)/p, constructed in the proof of the Bloch-Kato conjecture.  We prove that the function field F(X) has the following property: for any equidimensional variety Y, the change of field homomorphism CH(Y) ---> CH(Y_{F(X)}) of Chow groups with coefficients in integers localized at p is surjective in codimensions < (dim X)/(p-1).  This is joint work with N. Karpenko.

Speaker: Burt Totaro

Title:  Symmetric differentials and the fundamental group  

Abstract:  Consider a smooth complex projective variety X.  Hodge theory shows that sections of exterior

powers of the cotangent bundle are related to the topology of X.  What about symmetric powers of the cotangent bundle? We discuss the relation between the topology of X and its "symmetric differentials". One interest of these results is that symmetric differentials give information in the direction of "Kobayashi hyperbolicity"; for example, they limit how many rational curves X can contain.

Speaker: Kirsten Wickelgren

Title: 2-Nilpotent real section conjecture

Abstract: Grothendieck's section conjecture predicts that the rational points of a hyperbolic curve over a number field are determined by the \'etale fundamental group of the curve. This talk is about a 2-nilpotent section conjecture over R: for a geometrically connected curve X over R such that the normalization of each component has real points, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action. It follows as a corollary that for X smooth and proper, the set of real points of X equipped with a real tangent direction is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with its Gal(C/R) action.