6 research outputs found

    Finite time singularities of the K\"ahler-Ricci flow

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    We establish the scalar curvature and distance bounds, extending Perelman's work on the Fano K\"ahler-Ricci flow to general finite time solutions of the K\"ahler-Ricci flow. These bounds are achieved by our Li-Yau type and Harnack estimates for weighted Ricci potential functions of the K\"ahler-Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov-Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is proved for almost every fibre of collapsing solutions of the K\"ahler-Ricci flow on a Fano fibre bundle. We also apply our estimates to show that every solution of the K\"ahler-Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.Comment: All comments welcome; improved introduction and minor edit

    Geometric regularity of blow-up limits of the K\"ahler-Ricci flow

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    We establish geometric regularity for Type I blow-up limits of the K\"ahler-Ricci flow based at any sequence of Ricci vertices. As a consequence, the limiting flow is continuous in time in both Gromov-Hausdorff and Gromov-W1W_1 distance. In particular, the singular sets of each time slice and its tangent cones are close and of codimension no less than 44.Comment: All comments welcome. arXiv admin note: text overlap with arXiv:2310.0794