Niveau: Supérieur, Doctorat, Bac+8
Appendix to I. Cheltsov and C. Shramov's article “Log canonical thresholds of smooth Fano threefolds” : On Tian's invariant and log canonical thresholds Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier The goal of this appendix is to relate log canonical thresholds with the ? invariant introduced by G. Tian [Tia87] for the study of the existence of Kahler-Einstein metrics. The approximation technique of closed positive (1, 1)-currents introduced in [Dem92] is used to show that the ? invariant actually coincides with the log canonical threshold. Algebraic geometers have been aware of this fact after [DK01] appeared, and several papers have used it consistently in the latter years (see e.g. [JK01], [BGK05]). However, it turns out that the required result is stated only in a local analytic form in [DK01], in a language which may not be easily recognizable by algebraically minded people. Therefore, we will repair here the lack of a proper reference by stating and proving the statements required for the applications to projective varieties, e.g. existence of Kahler-Einstein metrics on Fano varieties and Fano orbifolds. Usually, in these applications, only the case of the anticanonical line bundle L = ?KX is considered. Here we will consider more generally the case of an arbitrary line bundle L (or Q-line bundle L) on a complex manifold X , with some additional restrictions which will be stated later.
- standard andreotti-vesentini-hormander
- singular hermitian metric
- every compact
- kahler-einstein metrics
- compact complex
- evaluation linear
- numbers ?
- section ? ?
- any trivialization