Abstract
Let (X,theta ) be a compact complex manifold X equipped with a smooth (but not necessarily positive) closed (1, 1)-form theta . By a well-known envelope construction this data determines, in the case when the cohomology class [theta ] is pseudoeffective, a canonical theta -psh function u_{theta }. When the class [theta ] is Kähler we introduce a family u_{beta } of regularizations of u_{theta }, parametrized by a large positive number beta , where u_{beta } is defined as the unique smooth solution of a complex Monge–Ampère equation of Aubin–Yau type. It is shown that, as beta rightarrow infty , the functions u_{beta } converge to the envelope u_{theta } uniformly on X in the Hölder space C^{1,alpha }(X) for any alpha in ]0,1[ (which is optimal in terms of Hölder exponents). A generalization of this result to the case of a nef and big cohomology class is also obtained and a weaker version of the result is obtained for big cohomology classes. The proofs of the convergence results do not assume any a priori regularity of u_{theta }. Applications to the regularization of omega -psh functions and geodesic rays in the closure of the space of Kähler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where beta appears as the inverse temperature. This point of view also leads to an interpretation of u_{beta } as a “transcendental” Bergman metric.
Highlights
Let X be a compact complex manifold equipped with a smooth closed (1, 1)-form θ on X and denote by [θ ] the corresponding class in the Bott–Chern cohomology group H 1,1(X, R).There is a range of positivity notions for such cohomology classes, generalizing the classical positivity notions in algebraic geometry
The algebro-geometric situation concerns the special case when X is projective variety and the cohomology class in question has integral periods, which equivalently means that the class may be realized as the first Chern class c1(L) of a line bundle L over X [25,26,27]
General cohomology classes in H 1,1(X, R) are some times referred to as transcendental classes and the corresponding notions of positivity may be formulated in terms of the convex subspace of positive currents in the cohomology class—the strongest notion of positivity is that of a Kähler class, which means that the class contains a Kähler metric, i.e. a smooth positive form
Summary
Let X be a compact complex manifold equipped with a smooth closed (1, 1)-form θ on X and denote by [θ ] the corresponding class in the Bott–Chern cohomology group H 1,1(X, R).There is a range of positivity notions for such cohomology classes, generalizing the classical positivity notions in algebraic geometry. By the results in [18], the corresponding complex Monge– Ampère equations admit a unique θ -psh function uβ with minimal singularities; in particular its singularities can only appear along a certain complex subvariety of X, determined by the class [θ ], whose complement is called the Kähler locus of [θ ] (or the ample locus) introduced in [17] (which in the algebro-geometric setting corresponds to the complement of the augmented base locus of the corresponding line bundle). In the case when the class [θ ] is assumed to be nef the solution uβ is known to be smooth on , as follows from the results in [18] In this general setting our main result may be formulated as follows: Theorem 1.2 Let θ be a smooth (1, 1)-form on a compact complex manifold X such that [θ ] is a big class. For special big classes [θ ], namely those which admit an appropriate Zariski decomposition on some resolution of X, the regularity and convergence problem can be reduced to the nef case (in the line bundle case this situation appears if the corresponding section ring is finitely generated)
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