Topology of Algebraic Varieties (TAV)

On some questions about minimal log discrepancies

Mircea Mustata
University of Michigan
March 3, 2015
The minimal log discrepancy is a measure of singularities of pairs. While akin to the log canonical threshold, it turns out to be much more difficult to study, with many questions still open. I will discuss a question about the boundedness of divisors that compute the minimal log discrepancy on a fixed germ. This is joint work (in progress) with Yusuke Nakamura.

Automorphisms of smooth canonically polarised surfaces in characteristic 2

Nikolaos Tziolas
University of Cyprus
February 25, 2015
Let $X$ be a smooth canonically polarised surface defined over an algebraically closed field of characteristic 2. In this talk I will present some results about the geometry of $X$ in the case when the automorphism scheme $\mathrm{Aut}(X)$ of $X$ is not smooth, or equivalently $X$ has nontrivial global vector fields.

Projectivity of the moduli space of KSBA stable pairs and applications

Zsolt Patakfalvi
Princeton University
February 24, 2015
KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs are higher dimensional generalizations of (weighted) stable pointed curves. I will present a joint work in progress with Sándor Kovács on proving the projectivity of this moduli space, by showing that certain Hodge-type bundles are ample on it. I will also mention applications to the subadditivity of logarithmic Kodaira dimension, and to the ampleness of the CM line bundle.

The cohomology groups of Hilbert schemes and compactified Jacobians of planar curves

Luca Migliorini
University of Bologna; Member, School of Mathematics
February 18, 2015
I will first discuss a relation between the cohomology groups (with rational coefficients) of the compactified Jacobian and those of the Hilbert schemes of a projective irreducible curve $C$ with planar singularities, which extends the classical Macdonald formula, relating the cohomology groups of the symmetric product of a nonsingular curve to those of its Jacobian. The result follows from a "Support theorem" for the relative Hilbert scheme family associated with a versal deformation of the curve $C$.

Proper base change for zero cycles

Moritz Kerz
University of Regensburg; Member, School of Mathematics
February 17, 2015
We study the restriction map to the closed fiber for the Chow group of zero-cycles over a complete discrete valuation ring. It turns out that, for proper families of varieties and for certain finite coefficients, the restriction map is an isomorphism. One can also ask whether for other motivic cohomology groups with finite coefficients one gets a restriction isomorphism.

Moduli of degree 4 K3 surfaces revisited

Radu Laza
Stony Brook University; von Neumann Fellow, School of Mathematics
February 3, 2015
For low degree K3 surfaces there are several way of constructing and compactifying the moduli space (via period maps, via GIT, or via KSBA). In the case of degree 2 K3 surface, the relationship between various compactifications is well understood by work of Shah, Looijenga, and others. I will report on work in progress with K. O’Grady which aims to give similar complete description for degree 4 K3s.

A birational model of the Cartwright-Steger surface

Igor Dolgachev
University of Michigan
January 21, 2015
A Cartwright-Steger surface is a complex ball quotient by a certain arithmetic cocompact group associated to the cyclotomic field $Q(e^{2\pi i/12})$, its numerical invariants are with $c_1^2 = 3c_2 = 9, p_g = q = 1$. It is a cyclic degree 3 cover of a simply connected surface of general type with $c_1^2 = 2, p_g = 1$. A similar construction in the case of the cyclotomic fields $Q(e^{2\pi i/5})$ (resp. $Q(e^{2\pi i/7})$) leads to the beautiful geometry of a del Pezzo surface of degree 5 and its K3 double cover branched along the union of lines (resp.

On descending cohomology geometrically

Sebastian Casalaina-Martin
University of Colorado at Boulder
January 20, 2015
In this talk I will present some joint work with Jeff Achter concerning the problem of determining when the cohomology of a smooth projective variety over the rational numbers can be modeled by an abelian variety. The primary motivation is a problem posed by Barry Mazur. We provide an answer to Mazur's question in two situations. First, we show that the third cohomology group can be modeled by the cohomology of an abelian variety over the rationals provided the Chow group of points is supported on a curve.