# V-soliton equations, symplectic reductions and Kahler-Ricci flow

# V-soliton equations, symplectic reductions and Kahler-Ricci flow

It has been widely recognized by now that in order to understand the geometry of Kahler manifolds under the Ricci flow one needs to understand a geometric-analytic version of Mori's Minimal Model Program. One of the major stumbling blocks is due to formation of finite time singularities. In recent work, Tian and I proposed an approach to the description of finite time singulariteis 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 (called V-soliton). I will explain how this works and how it relates to a Geometric version of the Minimal Model program. I will then describe the nature of the (degenerate) elliptic equations one needs to solve and the regularity of solutions.