Report
K\'ahler-Einstein metrics on families of Fano varieties
| العنوان: | K\'ahler-Einstein metrics on families of Fano varieties |
|---|---|
| المؤلفون: | Pan, Chung-Ming, Trusiani, Antonio |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Complex Variables, Mathematics - Algebraic Geometry, Mathematics - Differential Geometry |
| الوصف: | Given a one-parameter family of $\mathbb{Q}$-Fano varieties such that the central fibre admits a unique K\"ahler-Einstein metric, we provide an analytic method to show that the neighboring fibre admits a unique K\"ahler-Einstein metric. Our results go beyond by establishing uniform a priori estimates on the K\"ahler-Einstein potentials along fully degenerate families of $\mathbb{Q}$-Fano varieties. In addition, we show the continuous variation of these K\"ahler-Einstein currents, and establish uniform Moser-Trudinger inequalities and uniform coercivity of the Ding functionals. Central to our article is introducing and studying a notion of convergence for quasi-plurisubharmonic functions within families of normal K\"ahler varieties. We show that the Monge-Amp\`ere energy is upper semi-continuous with respect to this topology, and we establish a Demailly-Koll\'ar result for functions with full Monge-Amp\`ere mass. Comment: 40 pages; v2: exposition improved following the referee's suggestions; to appear in J. Reine Angew. Math. (Crelle's Journal) |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2304.08155 |
| رقم الانضمام: | edsarx.2304.08155 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |