Meeting in November 17

  • Jeremy Daniel: Some characteristic classes of flat bundles in complex geometry
    On a compact Kähler manifold $X$, any semisimple flat bundle carries a harmonic metric. It can be used to define some characteristic classes of the flat bundle, in the cohomology of $X$. We show that these cohomology classes come from an infinite-dimensional space – constructed with loop groups – which is an analogue of the period domains used in Hodge theory.
  • Christian Urech: Simple subgroups of the Cremona group
    The Cremona group is the group of birational transformations of the complex projective plane. In 2012 Cantat and Lamy proved that the Cremona group is not simple, which answered a long standing open question. In this talk we will refine their results and show that a simple group can be embedded into the plane Cremona group if and only if it can be embedded into $\mathbb P GL(3,\mathbb C)$. Our techniques also yield new insights into the structure of subgroups consisting of elliptic elements. In particular, we will describe the structure of torsion subgroups and show that the Cremona group satisfies the Tits alternative for arbitrary subgroups; this extends a result by Cantat.
  • Michael Hoff: Moduli of lattice polarized $K3$ surfaces via relative canonical resolutions
    For a smooth canonically embedded curve $C$ of genus 9 together with a pencil $|L|$ of degree 6 , we study the relative canonical resolution of $C$ in $X$, where $X$ is the scroll swept out by the pencil $|L|$. In general, the relative canonical resolution is built with certain vector bundles on the projective line. We show that in our case the second syzygy bundle in this resolution of C is unbalanced. The proof reveals a new geometric connection between the universal Brill–Noether variety $W_{9,6}^1$ and a moduli space of lattice polarized $K3$ surfaces (for a certain rank 3 lattice). As a by-product we prove the unirationality of this moduli space. This is a joint work with Christian Bopp.
  • Matthias Wendt: Chow-Witt rings of Grassmannians and oriented Schubert calculus
    Chow-Witt rings are a recent refinement of Chow rings which take into account orientation information using quadratic forms. While Chow rings of varieties can be compared to cohomology of the complex points of the varieties, the Chow-Witt rings compare better to the cohomology of real points. In the talk I will explain the structure of the Chow-Witt rings of Grassmannian varieties over general base fields (which strongly resembles the integral cohomology of real Grassmannian manifolds). This gives rise to an “oriented” version of Schubert calculus which generalizes some results from real Schubert calculus to (almost) arbitrary fields.