Claudio Arezzo

Extremal metrics on blowups

In this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of Kahler manifolds which already carry an extremal metric. As a particular case, we construct extremal metrics on

Quantization of Kähler manifolds and the asymptotic expansion of Tian-Yau-Zelditch

\mathbb {P}^2

Symmetries, quotients and Kähler-Einstein metrics

blown-up k points in general position, with

Stable bundles and the first eigenvalue of the Laplacian

k \lt m+2

Singularities and K-semistability

.

On the Kummer construction for Kcsc metrics

In this paper we study the link between the asymptotic expansion of Tian–Yau–Zelditch [J. Diff. Geom. 32 (1990) 99] and the quantization of compact Kahler manifolds carried out in [J. Geophys. 7 (1990) 45; Trans. Am. Math. Soc. 337 (1993) 73].

On the K-stability of complete intersections in polarized manifolds

Abstract We consider Fano manifolds M that admit a collection of finite automorphism groups G 1, …, Gk , such that the quotients M/Gi are smooth Fano manifolds possessing a Kähler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kähler-Einstein metric too.

Geometric Constructions of Extremal Metrics on Complex Manifolds

AbstractIn this article we study the first eigenvalue of the Laplacian on a compact manifold using stable bundles and balanced bases. Our main result is the following: Let M be a compact Kähler manifold of complex dimension n and E a holomorphic vector bundle of rank r over M. If E is globally generated and its Gieseker point Te is stable, then for any Kähler metric g on M % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS% baaSqaaiaaigdaaeqaaOGaaiikaiaad2eacaGGSaGaam4zaiaacMca% cqGHKjYOdaWcaaqaaiaaisdacqaHapaCcaWGObWaaWbaaSqabeaaca% aIWaaaaOGaaiikaiaadweacaGGPaaabaGaamOCaiaacIcacaWGObWa% aWbaaSqabeaacaaIWaaaaOGaaiikaiaadweacaGGPaGaeyOeI0Iaam% OCaiaacMcaaaGaeyyXIC9aaSaaaeaadaaadaqaaiaadoeadaWgaaWc% baGaaGymaaqabaGccaGGOaGaamyraiaacMcacqWIQisvcaGGBbGaeq% yYdCNaaiyxamaaCaaaleqabaGaamOBaiabgkHiTiaaigdaaaGccaGG% SaGaai4waiaad2eacaGGDbaacaGLPmIaayPkJaaabaGaaiikaiaad6% gacqGHsislcaaIXaGaaiykaiaacgcacaWG2bGaam4BaiaadYgacaGG% OaGaamytaiaacYcacaGGBbGaeqyYdCNaaiyxaiaacMcaaaaaaa!6D89!

Generalized Connected Sum Constructions for Resolutions of Extremal and Kcsc Orbifolds

\lambda _1 (M,g) \leqslant \frac{{4\pi h^0 (E)}}{{r(h^0 (E) - r)}} \cdot \frac{{\left\langle {C_1 (E) \cup [\omega ]^{n - 1} ,[M]} \right\rangle }}{{(n - 1)!vol(M,[\omega ])}}