Ricci flow on compact Kahler manifolds

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The PI proposes to elucidate the behavior of the Ricci flow on an arbitrary compact Kahler manifold. This is a major question in the field of complex differential geometry, which has received much attention in recent years, thanks also to Perelman's landmark work on the Ricci flow on compact real 3-manifolds. In the Kahler case, it is known that the Ricci flow preserves the Kahler condition, and it is thus also known as the Kahler-Ricci flow (KRF). Its maximal existence time can be computed cohomologically in terms of the initial metric and of the complex structure of the manifold, and in particular it is infinite if and only if the canonical bundle of the manifold is numerically effective. An influential program of Song-Tian aims to understand the behavior of the flow, at least when the Kahler manifold is projective algebraic, by relating it to the Minimal Model Program in algebraic geometry. The first main goal of the proposal will be to study singularities which form in finite time. Earlier work of the PI and Collins proved that in this case the flow forms singularities along an analytic subvariety V, proving a conjecture of Feldman-Ilmanen-Knopf. The PI proposes to go substantially further: first, he proposes to show that the diameter of the evolving metrics remains uniformly bounded above at the singular time. Second, to show that the total volume goes to zero at the singularity if and only if the manifold admits a Fano fibration structure (and the limiting cohomology class is pulled back from the base of this fibration). The PI proved this earlier with Zhang when the complex dimension is at most 3. And third, he proposes to show that if the total volume does not go to zero, then the subvariety V above can be contracted complex analytically and the flow can restart on a new compact analytic space, as suggested by Song-Tian. This process is expected to terminate in finitely many steps, either with a Fano fibration or with a solution that exists for all positive time. The second main goal of the proposal will be to understand the long-time behavior of solutions that exist for all positive time. As mentioned above, these exist precisely when the canonical bundle is numerically effective. A long-standing conjecture in algebraic geometry (abundance) predicts that in this case the canonical bundle is semiample, so the manifold is fibered by Calabi-Yau manifolds over a lower-dimensional base, in general with singular fibers. In this case, the PI proposes to identify the global Gromov-Hausdorff limit of the (normalized) flow, as conjectured by Song-Tian, and to prove higher regularity away from the singular fibers. Lastly, without assuming that the canonical bundle is semiample, the PI proposes to show that the KRF converges weakly to a closed positive current in the canonical class (which corresponds to a singular Hermitian metric on the canonical bundle), which is independent of the initial metric of the flow, and has minimal singularities.