Monge-Ampere equations in big cohomology classes

• 成果类型: Article
• 署名作者: Boucksom, Sebastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed
• 署名单位: Universite Paris Cite; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Aix-Marseille Universite
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-010-0054-7
• 发表日期: 2010
• 页码: 199-262
• 关键词: energy; definition; VARIETIES; EXISTENCE; capacity; volume

摘要

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kahler manifold X. Given a big (1, 1)-cohomology class alpha on X (i.e. a class that can be represented by a strictly positive current) and a positive measure mu on X of total mass equal to the volume of alpha and putting no mass on pluripolar sets, we show that mu can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in alpha. We then extend Kolodziedj's approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if mu has L (1+epsilon) -density with respect to Lebesgue measure. If mu is smooth and positive everywhere, we prove that T is smooth on the ample locus of alpha provided alpha is nef. Using a fixed point theorem, we finally explain how to construct singular Kahler-Einstein volume forms with minimal singularities on varieties of general type.