The Conical Kahler-Ricci Flow and the Analytic Log Minimal Model ProgramPublic Deposited
We prove a uniform scalar curvature bound for solutions of the conical Kahler-Ricci flow when the twisted canonical bundle is semiample and the cone divisor is obtained from the associated Iitaka-Kodaira fibration. In the course of the proof we establish uniform bounds for the potential of the metric and its time derivative. Next, we classify the behavior of the conical Kahler-Ricci flow on Hirzebruch surfaces under the Calabi ansatz when the cone divisor is the exceptional curve. We show that the flow always reaches a singularity in finite time and converges in the Gromov-Hausdorff topology to either a two dimensional projective orbifold, the Riemann sphere, or a single point. The limiting behavior depends only on numerical properties of the initial Kahler class and the degree of the Hirzebruch surface. In particular, this establishes the first example of the flow contracting the cone divisor itself at the singular time, and shows that the conical flow may contract curves of self-intersection less than (-1), as opposed to the behavior of the smooth Kahler-Ricci flow.\ Finally, we outline some conjectures regarding the structure of finite time non-collapsing singularities of the conical Kahler-Ricci flow, construct geometric examples where we expect such behavior to occur, and formulate some new conjectures about the singularity formulation in the Gromov-Hausdorff limit when the twisted canonical bundle is big and nef.