Solutions to degenerate complex Hessian equations
Given a compact Kähler manifold \((X,\omega)\) of complex dimension \(n\), let \(dd^c\) be the usual real differential operators. Fixed an integer \(m\), \(1\leq m\leq n\), one studies the degenerate complex Hessian equations of the form NEWLINE\[NEWLINE(\omega+ dd^c\varphi)^m\wedge \omega^{n-m}= F(x,\varphi)\omega^n,\tag{1}NEWLINE\]NEWLINE where the density \(F: X\times\mathbb{R}\to \mathbb{R}_+\) satisfies suitable conditions. Equation (1) is a generalization of both Laplace and Monge-Ampère equations.NEWLINENEWLINE Firstly the author introduces the class of \((\omega,m)\)-subharmonic functions and the concepts of \((\omega,m)\)-capacity for Borel subsets of \(X\) and of quasi-uniform convergence. This allows to define a suitable class of \((\omega,m)\)-subharmonic functions on which the complex Hessian operator is well-defined and continuous under quasi-uniform convergence. This class is denoted by \({\mathcal P}_m(X,\omega)\). Under suitable conditions for \(F\), the author proves existence and unicity, up to an additive constant, of a solution \(\varphi\) of (1) which belongs to \(\mathbb{P}_m(X,\omega)\cap{\mathcal C}^0(X)\). In particular, one obtains the Hölder continuity of the solution when \(X\) is homogeneous and \(\omega\) invariant under the Lie group action.
scientific article; zbMATH DE number 6236655
scientific article