Kähler-Ricci flow on stable Fano manifolds

Tosatti, Valentino

doi:10.1515/crelle.2010.019

Cited by 33 publications

(31 citation statements)

References 33 publications

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“…It was shown in [16,19,30] that on a manifold with positive first Chern class the flow converges to a Kähler-Einstein metric when one exists. Convergence is also known under certain weaker assumptions (see [6,14,17,18,27,31,37] and the discussions therein).…”

Section: Introductionmentioning

confidence: 99%

Contracting exceptional divisors by the Kähler-Ricci flow II

Song

Weinkove

2013

Proceedings of the London Mathematical Society

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We investigate the case of the Kähler-Ricci flow blowing down disjoint exceptional divisors with normal bundle O(−k) to orbifold points. We prove smooth convergence outside the exceptional divisors and global Gromov-Hausdorff convergence. Moreover, we prove a conjecture of the authors by establishing the result that the Gromov-Hausdorff limit coincides with the metric completion of the limiting metric under the flow. This improves and extends the previous work of the authors. We apply this to P 1 -bundles which are higher-dimensional analogues of the Hirzebruch surfaces. We also consider the case of a minimal surface of general type with only distinct irreducible (−2)-curves and show that solutions to the normalized Kähler-Ricci flow converge in the Gromov-Hausdorff sense to a Kähler-Einstein orbifold.

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Section: Introductionmentioning

confidence: 99%

Contracting exceptional divisors by the Kähler-Ricci flow II

Song

Weinkove

2013

Proceedings of the London Mathematical Society

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“…By a conjecture of Yau [40], a necessary and su‰cient condition for M to admit a Kähler-Einstein metric is that M be 'stable in the sense of geometric invariant theory'. Indeed, the problem of using stabil-ity conditions to prove convergence properties of the Kähler-Ricci flow is an area of considerable current interest and we refer the reader to [21], [19], [22], [23], [25], [31], [24], [36] and [5] for some recent advances (however, this list of references is far from complete). One might expect that the su‰ciency part of the Yau-Tian-Donaldson conjecture can be proved via the flow (1.1).…”

Section: Introductionmentioning

confidence: 99%

The Kähler–Ricci flow on Hirzebruch surfaces

Song

Weinkove

2011

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We investigate the metric behavior of the Kähler-Ricci flow on the Hirzebruch surfaces, assuming the initial metric is invariant under a maximal compact subgroup of the automorphism group. We show that, in the sense of Gromov-Hausdor¤, the flow either shrinks to a point, collapses to P 1 or contracts an exceptional divisor, confirming a conjecture of Feldman-Ilmanen-Knopf. We also show that similar behavior holds on higher-dimensional analogues of the Hirzebruch surfaces.

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“…If we assume one of the Gromov-Hausdorff limit is smooth, then the uniqueness statement follows from the main result of [31] (based on the Lojasiewicz-Simon technique). The part on the relation with K-stability is new even if we assume the curvature of ω(t) is uniformly bounded, under which Székelyhidi [33] and Tosatti [39] have obtained some partial results with extra assumptions. Theorem 1.2 follows from a finite dimensional result that we will elaborate in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Kähler–Ricci flow, Kähler–Einstein metric, and K–stability

2018

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We prove the existence of Kähler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kähler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kähler-Ricci flow on Fano manifolds. This is in turn based on a general finite dimensional discussion, which is interesting in its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kähler manifold to K-stability assuming bounds on geometry.

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Kähler-Ricci flow on stable Fano manifolds

Abstract: Abstract. We study the Kähler-Ricci flow on Fano manifolds. We show that if the curvature is bounded along the flow and if the manifold is K-polystable and asymptotically Chow semistable, then the flow converges exponentially fast to a Kähler-Einstein metric.

Cited by 33 publications

References 33 publications

Contracting exceptional divisors by the Kähler-Ricci flow II

Contracting exceptional divisors by the Kähler-Ricci flow II

The Kähler–Ricci flow on Hirzebruch surfaces

Kähler–Ricci flow, Kähler–Einstein metric, and K–stability

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