Topology of Algebraic Varieties (TAV)

Universal Chow group of zero-cycles on cubic hypersurfaces

Claire Voisin
Centre national de la recherche scientifique; Distinguished Visiting Professor, School of Mathematics
November 12, 2014
We discuss the universal triviality of the \(\mathrm{CH}_0\)-group of cubic hypersurfaces, or equivalently the existence of a Chow-theoretic decomposition of their diagonal. The motivation is the study of stable irrationality for these varieties. Our main result is that this decomposition exists if and only if it exists on the cohomological level.

Zarhin's trick and geometric boundedness results for K3 surfaces

François Charles
Université Paris-Sud
November 11, 2014
Tate's conjecture for divisors on algebraic varieties can be rephrased as a finiteness statement for certain families of polarized varieties with unbounded degrees. In the case of abelian varieties, the geometric part of these finiteness statements is contained in Zarhin's trick. We will discuss such geometric boundedness statements for K3 surfaces over arbitrary fields and holomorphic symplectic varieties, with application to direct proofs of the Tate conjecture for K3 surfaces that do not involve the Kuga-Satake correspondence.

Elliptic genera of Pfaffian-Grassmannian double mirrors

Lev Borisov
Rutgers University
November 5, 2014
For an odd integer \(n > 3\) the data of generic n-dimensional subspace of the space of skew bilinear forms on an n-dimensional vector space define two different Calabi-Yau varieties of dimension \(n-4\). Specifically, one is a complete intersection of n hyperplanes in the Grassmannian \(G(2,n)\) and the other is a complete intersection of \(n(n-3)/2\) hyperplanes in the Pfaffian variety of degenerate skew forms. In \(n=7\) case, these have been investigated by Rodland and were (heuristically) found to have the same mirror family.

Beauville's splitting principle for Chow rings of projective hyperkaehler manifolds

Lie Fu
Member, School of Mathematics
November 4, 2014
Being the natural generalization of K3 surfaces, hyperkaehler varieties, also known as irreducible holomorphic symplectic varieties, are one of the building blocks of smooth projective varieties with trivial canonical bundle. One of the guiding conjectures in the study of algebraic cycles of such varieties is Beauville's splitting principle. Concerning the weak form of the splitting principle, I want to report some progress on the closely related Beauville-Voisin conjecture.

Mirror symmetry & Looijenga's conjecture

Philip Engel
Columbia University
October 29, 2014
A cusp singularity is an isolated surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In the 1980's Looijenga conjectured that a cusp singularity is smoothable if and only if the minimal resolution of the dual cusp is the anticanonical divisor of some rational surface. This conjecture can be related to the existence of certain integral affine-linear structures on a sphere.

Singular moduli spaces and Nakajima quiver varieties

Giulia Saccà
Member, School of Mathematics
October 28, 2014
The aim of this talk is to study a class of singularities of moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations.

Extending differential forms and the Lipman-Zariski conjecture

Sándor Kovács
University of Washington; Member, School of Mathematics
October 22, 2014
The Lipman-Zariski conjecture states that if the tangent sheaf of a complex variety is locally free then the variety is smooth. In joint work with Patrick Graf we prove that this holds whenever an extension theorem for differential 1-forms holds, in particular if the variety in question has log canonical singularities.

The structure of instability in moduli theory

Daniel Halpern-Leistner
Member, School of Mathematics
October 21, 2014
In many examples of moduli stacks which come equipped with a notion of stable points, one tests stability by considering "iso-trivial one parameter degenerations" of a point in the stack. To such a degeneration one can often associate a real number which measures "how destabilizing" it is, and in these situations one can ask the question of whether there is a "maximal destabilizing" or "canonically destabilizing" degeneration of a given unstable point.

Positive cones of higher (co)dimensional numerical cycle classes

Mihai Fulger
Princeton University
October 21, 2014
It is classical to study the geometry of projective varieties over algebraically closed fields through the properties of various positive cones of divisors or curves. Several counterexamples have shifted attention from the higher (co)dimensional case. They show that the analogous definitions do not lead to analogous positivity properties. To correct the negative outlook, we look at stronger positivity conditions on numerical classes. A sample result is that the pseudoeffective cone, the closure of the cone of effective \(k\)-dimensional cycle classes is pointed.