arXiv subjects

Jian Song

Publications and source records attributed to Jian Song.

239 records · Page 14Linked to original sources

arXiv2006

The Kähler-Ricci flow on surfaces of positive Kodaira dimension

The existence of Kähler-Einstein metrics on a compact Kähler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has become one of the most powerful tools in geometric analysis. We study the Kähler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical is a generalized Kähler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric classification for Käher surfaces with a numerical effective canonical line bundle by the Kähler-Ricci flow. In general, we propose a program of finding canonical metrics on canonical models of projective varieties of positive Kodaira dimension.

math.DG↗

arXiv2004

On the convergence and singularities of the J-flow with applications to the Mabuchi energy

The J-flow of S. K. Donaldson and X. X. Chen is a parabolic flow on Kahler manifolds with two Kahler metrics. It is the gradient flow of the J-functional which appears in Chen's formula for the Mabuchi energy. We find a positivity condition in terms of the two metrics which is both necessary and sufficient for the convergence of the J-flow to a critical metric. We use this result to show that on manifolds with ample canonical bundle, the Mabuchi energy is proper on all Kahler classes in an open neighborhood of the canonical class defined by a positivity condition. This improves previous results of Chen and of the second author. We discuss the implications of this for the problem of the existence of constant scalar curvature Kahler metrics. We also study the singularities of the J-flow and, under certain conditions (which always hold for dimension two) derive estimates away from a subvariety. In the case of Kahler surfaces we use these estimates to confirm, at least in a certain sense, a conjectural remark of Donaldson that if the J-flow does not converge then it should blow up over some curves of negative self-intersection.

math.DG↗

arXiv2004

The SzegÖ kernel on an orbifold circle bundle

The analysis of holomorphic sections of high powers $L^N$ of holomorphic ample line bundles $L\to M$ over compact Kähler manifolds has been widely applied in complex geometry and mathematical physics. The Tian-Yau-Zelditch's asymptotic expansion of the Szegö kernel of a circle bundle plays an important role in Kähler-Einstein geometry. We generalize the expansion for compact Kähler orbifolds with only finite isolated singularities and analyze the behavior of the expansion near the singularities.

math.DG↗

arXiv2003

The α-Invariant on Toric Fano Manifolds

The global holomorphic α-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate plurisubharmonic functions and compute the α-invariant of toric Fano manifolds.

math.DG↗

arXiv2002

The $α$-Invariant on $CP^2#2\bar{CP^2}$

In this paper, we apply the Tian-Yau-Zelditch expansion of the Bergman kernel on polarized Kähler metrics to approximate plurisubharmonic functions and compute the $α$-invariant of $CP^2#2\bar{CP^2}$, which is exactly 1/3. In addition we prove Tian's conjecture on the generalized Moser-Trudinger inequality in a special case prescribed by the $α$-invariant.

math.DG↗