Stable bundles and the first eigenvalue of the Laplacian

Abstract

In 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 Kahler 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 Kahler 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! $$\lambda _1 (M,g) \leqslant \frac{{4\pi h^0 (E)}}{{r(h^0 (E) - r)}} \cdot \frac{{\left\langle {C_1 (E) \cup [\omega ]^{n - 1} ,[M]} \right\rangle }}{{(n - 1)!vol(M,[\omega ])}}$$ where ω = ωg is the Kahler form associated to g. By this method we obtain, for example, a sharp upper bound for λ1 of Kahler metrics on complex Grassmannians.