scholarly journals Kähler-Einstein metrics: Old and New

2017 ◽  
Vol 4 (1) ◽  
pp. 200-244
Author(s):  
Daniele Angella ◽  
Cristiano Spotti
Keyword(s):  
Einstein Metrics ◽  
Complex Manifolds ◽  
Kähler Einstein Metrics ◽  
Compact Complex Manifolds

AbstractWe present classical and recent results on Kähler-Einstein metrics on compact complex manifolds, focusing on existence, obstructions and relations to algebraic geometric notions of stability (K-stability). These are the notes for the SMI course "Kähler-Einstein metrics" given by C.S. in Cortona (Italy), May 2017. The material is not intended to be original.

ASYMPTOTIC CHOW SEMI-STABILITY AND INTEGRAL INVARIANTS

2004 ◽  
Vol 15 (09) ◽  
pp. 967-979 ◽  
Author(s):  
AKITO FUTAKI
Keyword(s):  
Einstein Metrics ◽  
Automorphism Groups ◽  
Complex Manifolds ◽  
Invariant Polynomials ◽  
Integral Invariants ◽  
Natural Extensions ◽  
Kähler Einstein Metrics ◽  
Compact Complex Manifolds

We define a family of integral invariants containing those which are closely related to asymptotic Chow semi-stability of polarized manifolds. It also contains an obstruction to the existence of Kähler–Einstein metrics and its natural extensions by the author, Calabi and Bando as Kählerian invariants and by Morita and the author as invariant polynomials of the automorphism groups of compact complex manifolds.

Affine compact almost-homogeneous manifolds of cohomogeneity one

2009 ◽  
Vol 7 (1) ◽  
Author(s):  
Daniel Guan
Keyword(s):  
Einstein Metrics ◽  
Compact Manifolds ◽  
Complex Manifolds ◽  
Homogeneous Manifolds ◽  
Einstein Manifolds ◽  
Extremal Metrics ◽  
Cohomogeneity One ◽  
Fano Manifolds ◽  
Kähler Einstein Metrics

AbstractThis paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.

Existence of coupled Kähler–Einstein metrics using the continuity method

2018 ◽  
Vol 29 (05) ◽  
pp. 1850041 ◽  
Author(s):  
Vamsi Pritham Pingali
Keyword(s):  
Calculus Of Variations ◽  
Einstein Metrics ◽  
Canonical Bundle ◽  
Complex Manifolds ◽  
Continuity Method ◽  
Kähler Einstein Metrics

In this paper, we prove the existence of coupled Kähler–Einstein metrics on complex manifolds whose canonical bundle is ample. These metrics were introduced and their existence in the said case was proven by Hultgren and Nyström using calculus of variations. We prove the result using the method of continuity. In the process of proving estimates, akin to the usual Kähler–Einstein metrics, we reduce existence in the Fano case to a [Formula: see text] estimate.

scholarly journals STABILITY AND HERMITIAN–EINSTEIN METRICS FOR VECTOR BUNDLES ON FRAMED MANIFOLDS

2012 ◽  
Vol 23 (09) ◽  
pp. 1250091
Author(s):  
MATTHIAS STEMMLER
Keyword(s):  
Vector Bundles ◽  
Einstein Metrics ◽  
Coherent Sheaf ◽  
Einstein Metric ◽  
Complex Manifolds ◽  
Holomorphic Vector Bundles ◽  
Compact Complex Manifolds ◽  
Torsion Free

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford–Takemoto and Hermitian–Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that KX ⊗ [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization KX ⊗ [D] coincides with the degree with respect to the complete Kähler–Einstein metric gX\D on X\D. For stable holomorphic vector bundles, we prove the existence of a Hermitian–Einstein metric with respect to gX\D and also the uniqueness in an adapted sense.

scholarly journals Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Martin de Borbon
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Projective Line ◽  
Tangent Cones ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Complex Curves ◽  
Complex Dimensions ◽  
Irregular Case ◽  
Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

scholarly journals Schottky groups acting on homogeneous rational manifolds

2019 ◽  
Vol 2019 (753) ◽  
pp. 23-56 ◽  
Author(s):  
Christian Miebach ◽  
Karl Oeljeklaus
Keyword(s):  
Deformation Theory ◽  
Group Actions ◽  
Picard Group ◽  
Schottky Group ◽  
Fundamental Groups ◽  
Complex Manifolds ◽  
Schottky Groups ◽  
Algebraic Dimension ◽  
Geometric Invariants ◽  
Compact Complex Manifolds

AbstractWe systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of {{\rm{SL}}(2,\mathbb{C})/\Gamma} for Γ a discrete free loxodromic subgroup of {{\rm{SL}}(2,\mathbb{C})}, previously obtained by A. Guillot.

scholarly journals Higher-page Bott–Chern and Aeppli cohomologies and applications

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dan Popovici ◽  
Jonas Stelzig ◽  
Luis Ugarte
Keyword(s):  
Positive Integer ◽  
Spectral Sequence ◽  
Complex Manifolds ◽  
Serre Duality ◽  
Frölicher Spectral Sequence ◽  
Compact Complex Manifolds

Abstract For every positive integer r, we introduce two new cohomologies, that we call E r {E_{r}} -Bott–Chern and E r {E_{r}} -Aeppli, on compact complex manifolds. When r = 1 {r\kern-1.0pt=\kern-1.0pt1} , they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when r ≥ 2 {r\geq 2} . They provide analogues in the Bott–Chern–Aeppli context of the E r {E_{r}} -cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page- ( r - 1 ) {(r-1)} - ∂ ⁡ ∂ ¯ {\partial\bar{\partial}} -manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

Sign in / Sign up

Export Citation Format

Share Document