The Ricci curvature of symplectic quotients of Fano manifolds

AbstractThis work is a continuation of the author’s previous paper [Greatest lower bounds on the Ricci curvature of toric Fano manifolds, Adv. Math. 226 (2011), 4921–4932]. On any toric Fano manifold, we discuss the behavior of the limit metric of a sequence of metrics which are solutions to a continuity family of complex Monge–Ampère equations in the Kähler–Einstein problem. We show that the limit metric satisfies a singular complex Monge–Ampère equation. This gives a conic-type singularity for the limit metric. Information on conic-type singularities can be read off from the geometry of the moment polytope.

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Ricci curvature in Kähler geometry

International Journal of Mathematics ◽

10.1142/s0129167x17400092 ◽

2017 ◽

Vol 28 (09) ◽

pp. 1740009

Author(s):

Song Sun

Keyword(s):

Ricci Curvature ◽

Einstein Metrics ◽

Complex Analysis ◽

Einstein Metric ◽

Tangent Cones ◽

Kahler Geometry ◽

Fano Manifolds ◽

Winter School ◽

Kähler Einstein Metrics ◽

Hausdorff Limits

These are the notes for lectures given at the Sanya winter school in complex analysis and geometry in January 2016. In Sec. 1, we review the meaning of Ricci curvature of Kähler metrics and introduce the problem of finding Kähler–Einstein metrics. In Sec. 2, we describe the formal picture that leads to the notion of K-stability of Fano manifolds, which is an algebro-geometric criterion for the existence of a Kähler–Einstein metric, by the recent result of Chen–Donaldson–Sun. In Sec. 3, we discuss algebraic structure on Gromov–Hausdorff limits, which is a key ingredient in the proof of the Kähler–Einstein result. In Sec. 4, we give a brief survey of the more recent work on tangent cones of singular Kähler–Einstein metrics arising from Gromov–Hausdorff limits, and the connections with algebraic geometry.

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Bergman kernel along the Kähler–Ricci flow and Tian’s conjecture

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2014-0015 ◽

2016 ◽

Vol 2016 (717) ◽

Cited By ~ 2

Author(s):

Wenshuai Jiang

Keyword(s):

Recent Work ◽

Ricci Curvature ◽

Ricci Flow ◽

Bergman Kernel ◽

Fano Manifolds ◽

Bergman Kernels ◽

Complex Dimension ◽

Short Time

AbstractIn this paper, we study the behavior of Bergman kernels along the Kähler–Ricci flow on Fano manifolds. We show that the Bergman kernels are equivalent along the Kähler–Ricci flow for short time under certain condition on the Ricci curvature of the initial metric. Then, using a recent work of Tian and Zhang, we can solve a conjecture of Tian for Fano manifolds of complex dimension at most 3.

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Greatest lower bounds on the Ricci curvature of Fano manifolds

Compositio Mathematica ◽

10.1112/s0010437x10004938 ◽

2010 ◽

Vol 147 (1) ◽

pp. 319-331 ◽

Cited By ~ 32

Author(s):

Gábor Székelyhidi

Keyword(s):

Ricci Curvature ◽

Lower Bounds ◽

Einstein Metrics ◽

Upper Bounds ◽

Fano Manifold ◽

Fano Manifolds ◽

Existence Time ◽

Kähler Einstein Metrics ◽

Bounded Below ◽

Kahler Metric

AbstractOn a Fano manifoldMwe study the supremum of the possibletsuch that there is a Kähler metricω∈c1(M) with Ricci curvature bounded below byt. This is shown to be the same as the maximum existence time of Aubin’s continuity path for finding Kähler–Einstein metrics. We show that onP2blown up in one point this supremum is 6/7, and we give upper bounds for other manifolds.

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