Monge-Ampère measures on subvarieties
Let \(\Omega\subset \mathbb C^n\) be a bounded hyperconvex domain, let \(\mathcal E(\Omega)\) be the maximal class of negative plurisubharmonic functions for which the complex Monge-Ampère operator \((dd^c\cdot)^n\) is well defined. For a complex manifold \(X\) let NEWLINE\[NEWLINE \mathcal E(X)=\big\{u\in \text{PSH}(X): \forall \, x\in X,\, \exists \text{ a neighborhood } W \text{ of }z\text{ such that } u\in \mathcal E(W)\big\}. NEWLINE\]NEWLINENEWLINENEWLINEIn this paper the authors study the Dirichlet problem for the complex Monge-Ampère operator \((dd^cu)^n=\mu\), for singular measures \(\mu\) (i.e., for measures supported by pluripolar sets).NEWLINENEWLINEThe main theorem is the following one. Let \(S\) be a subvariety in \(\Omega\) with dimension \(k<n\). If \(v\in \mathcal E(S)\) satisfies \((dd^cv)^k(\{v>-\infty\})=0\) then there exists \(u\in \mathcal E(\Omega)\) such that \((dd^cu)^n=(dd^cv)^k\).NEWLINENEWLINEThe toric case is also considered. Let \(\Delta\) be the unit ball in \(\mathbb C\). Let \(u\in \mathcal E(\Delta^n)\) be such that \newline \(u(z_1,\dots,z_k,z_{k+1},\dots,z_n)=u(z_1,\dots,z_k,|z_{k+1}|,\dots,|z_n|)\). Then there exists a Borel pluripolar set \(E\subset \Delta^k\) such that \((dd^cu)^n((\Delta^k\setminus E)\times \{0\}))=0\). In particular there is no function \(u\in \mathcal E(\Delta^n)\) such that NEWLINE\[NEWLINE u(z_1,\dots,z_k,z_{k+1},\dots,z_n)=u(z_1,\dots,z_k,|z_{k+1}|,\dots,|z_n|) \;\text{ and } \;(dd^cu)^n=\mu\times \delta_0^{n-k}, NEWLINE\]NEWLINE where \(\mu\) is any non-negative Radon measure on \(\Delta^k\) vanishing on all pluripolar sets.
scientific article; zbMATH DE number 6362987
scientific article