The partial C 0 -estimate along a general continuity path and applications
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https://doi.org/10.1007/s11425-019-1656-2
The partial C 0-estimate along a general continuity path and applications Ke Feng1 & Liangming Shen2,∗ 1School
of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China; 2School of Mathematical Sciences, Beihang University, Beijing 100191, China Email: [email protected], [email protected] Received September 18, 2019; accepted February 23, 2020
Abstract
We establish a new partial C 0 -estimate along a continuity path mixed with conic singularities along
a simple normal crossing divisor and a positive twisted (1, 1)-form on Fano manifolds. As an application, this estimate enables us to show the reductivity of the automorphism group of the limit space, which leads to a new proof of the Yau-Tian-Donaldson conjecture. Keywords MSC(2010)
partial C 0 -estimate, K¨ ahler-Einstein metric, conic K¨ ahler metric 53C44
Citation: Feng K, Shen L M. The partial C 0 -estimate along a general continuity path and applications. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-019-1656-2
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Introduction
Finding canonical metrics on K¨ahler manifolds is a central problem in K¨ahler geometry. In 1970s in the celebrated work [41], Yau solved Calabi’s conjecture and established the existence of Ricci flat K¨ahler metrics on K¨ahler manifolds with c1 (M ) = 0. Aubin [2] and Yau also established the existence of K¨ahlerEinstein metrics with negative Ricci curvature on K¨ahler manifolds with c1 (M ) < 0. The main idea is to establish a priori estimates for the solutions to the family of complex Monge-Amp`ere equations along a continuity path. The remaining problem is the Fano case, i.e., c1 (M ) > 0. Unlike the two cases above, Matsushima [24] and Futaki [16] showed that there are obstructions to the existence of K¨ahler-Einstein metrics on Fano manifolds. Thus there are Fano manifolds which do not admit K¨ahler-Einstein metrics. To solve the K¨ahler-Einstein problem on Fano manifolds, Tian made crucial progress in [30] which first introduced the partial C 0 -estimate. Let us recall the basic settings of this problem: Let (M, ω0 ) √ ¯ be a Fano manifold with a K¨ahler metric ω0 ∈ [2πc1 (M )], which satisfies that Ric(ω0 ) = ω0 + −1∂ ∂h √ ¯ where h is a smooth ω0 -PSH (pluri-subharmonic) function on M. Suppose ω = ω0 + −1∂ ∂φ is the K¨ahler-Einstein metric on M . Then φ satisfies the following complex Monge-Amp`ere equation: √ ¯ n = eh−φ ω n , (ω0 + −1∂ ∂φ) 0 * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Feng K et al.
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∫ where φ satisfies that M eh−φ ω0n = V. To solve this equation, a standard way as [2, 41] is to establish the solution to the following continuous family of Monge-Amp`ere equations (ω0 +
√ ¯ t )n = eh−tφt ω n −1∂ ∂φ 0
(1.1)
∫ for t ∈ [0, 1] with M eh−tφt ω0n = V. Tian realized that it was impossible to derive a priori C 0 -estimate of (1.1) directly as [41]. Instead, he noted that differ
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