# Kähler-Einstein metrics: Old and New

@article{Angella2017KhlerEinsteinMO, title={K{\"a}hler-Einstein metrics: Old and New}, author={Daniele Angella and Cristiano Spotti}, journal={Complex Manifolds}, year={2017}, volume={4}, pages={200 - 244} }

Abstract We 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.

## 2 Citations

### Constant $$\mu $$-Scalar Curvature Kähler Metric—Formulation and Foundational Results

- MathematicsThe Journal of Geometric Analysis
- 2022

We introduce mu-scalar curvature for a K"ahler metric with a moment map mu and start up a study on constant mu-scalar curvature K"ahler metric as a generalization of both cscK metric and…

### Equivariant stable sheaves and toric GIT

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2022

<jats:p>For <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$(X,\,L)$</jats:tex-math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png"…

## References

SHOWING 1-10 OF 170 REFERENCES

### Kähler-Einstein metrics with positive scalar curvature

- Mathematics
- 1997

Abstract. In this paper, we prove that the existence of Kähler-Einstein metrics implies the stability of the underlying Kähler manifold in a suitable sense. In particular, this disproves a…

### Kähler–Ricci flow, Kähler–Einstein metric, and K–stability

- MathematicsGeometry & Topology
- 2018

We prove the existence of Kahler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kahler-Ricci flows. The key ingredient is an algebro-geometric description of the…

### Kähler–Einstein metrics along the smooth continuity method

- Mathematics
- 2015

We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler–Einstein metric. This is a strengthening…

### Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2\pi\ and completion of the main proof

- Mathematics
- 2013

This is the third and final paper in a series which establish results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the…

### Kähler-Einstein Metrics on $$\mathbb{Q}$$-Smoothable Fano Varieties, Their Moduli and Some Applications

- Mathematics
- 2017

We survey recent results on the existence of Kahler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli…

### Degeneration of Riemannian metrics under Ricci curvature bounds

- Mathematics
- 2013

These notes are based on the Fermi Lectures delivered at the Scuola Normale Superiore, Pisa, in June 2001. The principal aim of the lectures was to present the structure theory developed by Toby…

### Scalar Curvature and Projective Embeddings, I

- Mathematics
- 2001

We prove that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introduced…

### ON GEODESICS IN THE SPACE OF KÄHLER METRICS

- Mathematics
- 2011

We show that geodesics in the space of Kähler metrics are of class C1,1, provided that the manifold has nonnegative bisectional curvature. X.X.Chen has proved that these geodesics have bounded mixed…

### Optimal bounds for the volumes of Kähler-Einstein Fano manifolds

- Mathematics
- 2015

abstract:We show that any $n$-dimensional Ding semistable Fano manifold $X$ satisfies that the anti-canonical volume is less than or equal to the value $(n+1)^n$. Moreover, the equality holds if and…