Algebraic Geometry Seminar
Gabriele La Nave
Univ. of Illinois, Urbana
V-soliton equations, symplectic reductions and the MMP
Abstract: It has been widely recognized by now that in order to understand the
geometry of Kaehler manifolds under the Ricci flow one needs to understand a
geometric-analytic version of Mori's Minimal Model Program. Much like in
Perelman's execution of Hamilton's program, one of the major stumbling blocks
to the development/utilization of the Ricci-flow stems from the formation of
finite time singularities.
For the Ricci flow on projective manifolds this manifests itself precisely when
the polarization determined by the moving metric hits an extremal ray so that a
birational operation is necessary.
In recent work, Tian and I proposed an approach to the description of finite
time singularities which relates the flow and its singularity formation to
variation of symplectic reductions of a Kaehler manifold endowed with a
1-dimensional (complex) Hamiltonian torus action, where the Kaehler metric in
the total space satisfies a static elliptic equation of soliton type. I will
explain how this works and how it relates to a Geometric version of the Minimal
Model program.
Wednesday April 4, 2012 at 4:00 PM in SEO 427