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Based on the compactness of the moduli of non-collapsed Calabi–Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we show the convergence of the Kähler Ricci flow in an appropriate topology and prove the partial-$C^0$-conjecture.
We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
We prove that on Fano manifolds, the Kähler–Ricci flow produces a “most destabilising” degeneration, with respect to a new stability notion related to the $H$-functional. This answers questions of Chen–Sun–Wang and He.
We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman’s $\mu$‑functional on Fano manifolds, resolving a conjecture of Tian–Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kähler–Ricci soliton, then the Kähler–Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian–Zhu and Tian–Zhang–Zhang–Zhu, where either the manifold was assumed to admit a Kähler–Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.
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