Geometry of Kähler metrics and foliations by holomorphic discs.

*(English)*Zbl 1182.32009A Kähler metric is called extremal if the complex gradient vector field of its scalar curvature is holomorphic. Whenever this vector field vanishes, one obtains a so called constant scalar curvature Kähler metric. In the canonical Kähler class, such a metric must be Kähler Einstein.

A main result of this paper proves that, in each Kähler class of a compact Kähler manifold, there exists a unique (up to holomorphic transformations) extremal Kähler metric.

The authors establish a partial regularity theory for homogeneous complex Monge-Ampère equations (HCMA) via the study of foliations by holomorphic curves and their relations to HCMA equations. To this end the authors introduce notions of smoothness of different “degree”. These different layers of smoothness are in one-to-one correspondence to various notions of smoothness of the moduli space of holomorphic discs.

They show transversality results. In particular, the set of boundary values such that the corresponding moduli space \(\mathcal M\) induces an almost super regular foliation is generically open and closed. Let \( K\) denote the energy function [T. Mabuchi, Osaka J. Math. 24, 227–252 (1987; Zbl 0645.53038)]. Then \(K\) is subharmonic when restricted to a disk family of almost smooth solutions, which implies that the \(K\) energy is always bounded from below. To do this, the authors introduce a notion of weak Kähler Ricci flow.

As an application result, the authors obtain that if \((M,L)\) is a polarized algebraic manifold, admitting a constant scalar curvature metric with Kähler class equal to \(c_1(L)\), then \((M,L)\) is asymptotically \(K\)-semistable or CM-stable in the sense of [G. Tian, Invent. Math. 130, No. 1, 1–37 (1997; Zbl 0892.53027)].

A main result of this paper proves that, in each Kähler class of a compact Kähler manifold, there exists a unique (up to holomorphic transformations) extremal Kähler metric.

The authors establish a partial regularity theory for homogeneous complex Monge-Ampère equations (HCMA) via the study of foliations by holomorphic curves and their relations to HCMA equations. To this end the authors introduce notions of smoothness of different “degree”. These different layers of smoothness are in one-to-one correspondence to various notions of smoothness of the moduli space of holomorphic discs.

They show transversality results. In particular, the set of boundary values such that the corresponding moduli space \(\mathcal M\) induces an almost super regular foliation is generically open and closed. Let \( K\) denote the energy function [T. Mabuchi, Osaka J. Math. 24, 227–252 (1987; Zbl 0645.53038)]. Then \(K\) is subharmonic when restricted to a disk family of almost smooth solutions, which implies that the \(K\) energy is always bounded from below. To do this, the authors introduce a notion of weak Kähler Ricci flow.

As an application result, the authors obtain that if \((M,L)\) is a polarized algebraic manifold, admitting a constant scalar curvature metric with Kähler class equal to \(c_1(L)\), then \((M,L)\) is asymptotically \(K\)-semistable or CM-stable in the sense of [G. Tian, Invent. Math. 130, No. 1, 1–37 (1997; Zbl 0892.53027)].

Reviewer: Gabriela Paola Ovando (Freiburg)

##### MSC:

32Q20 | Kähler-Einstein manifolds |

32W20 | Complex Monge-Ampère operators |

32Q15 | Kähler manifolds |

53C12 | Foliations (differential geometric aspects) |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

##### Keywords:

extremal Kähler metrics; homogeneous complex Monge-Ampere equation; foliations; holomorphic discs
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\textit{X. Chen} and \textit{G. Tian}, Publ. Math., Inst. Hautes Étud. Sci. 107, 1--107 (2008; Zbl 1182.32009)

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