A singular Demailly-Păun theorem (Q282722)

The Demailly-Păun theorem [\textit{J.-P. Demailly} and \textit{M. Paun}, Ann. Math. (2) 159, No. 3, 1247--1274 (2004; Zbl 1064.32019)] gives a precise numerical description of the Kähler cone \(\mathcal{K}(X)\) of a compact Kähler manifold \(X\). The cone \(\mathcal{K}(X)\) is a connected component of the set of all real \((1,1)\)-cohomology classes \(\{\alpha\}\) on \(X\) with the property \(\int_Y\alpha^{\dim Y}>0\) for every positive-dimensional irreducible analytic subvariety \(Y\) in \(X\). Moreover \(\{\alpha\}\) lies in the closure of \(\mathcal{K}(X)\) if and only if there exists a Kähler metric \(\omega\) on \(X\) such that \(\int_Y\alpha^k\wedge\omega^{\dim Y-k}\geq 0\) for all those \(Y\) and \(1\leq k\leq \dim Y\).NEWLINENEWLINEThe authors of the article under review present an extension of this theorem to compact analytic subvarieties \(E\) of (not necessarily compact) Kähler manifolds \((M,\omega)\). Let \(\alpha\) be a closed smooth real \((1,1)\)-form on \(M\) with the property \(\int_V\alpha^k\wedge\omega^{\dim V-k}>0\) for all positive-dimensional irreducible analytic subvarieties \(V\subset E\) and for all \(1\leq k\leq\dim V\). Then there exists an open neighborhood \(U\) of \(E\) in \(M\) and a smooth function \(\varphi:U\rightarrow\mathbb R\) such that \(\alpha + i\partial\overline{\partial}\varphi\) is a Kähler metric on \(U\). If, in addition, \(M\) is an open subset of some projective variety, then the condition \(\int_V\alpha^{\dim V}>0\) is sufficient for the conclusion.NEWLINENEWLINEThe proof uses induction on \(\dim E\) and the resolution of the singularities of \(E\) by a suitable modification \(\hat{M}\rightarrow M\). Main tools are the Demailly-Păun theorem and results and techniques from the authors' paper [Invent. Math. 202, No. 3, 1167--1198 (2015; Zbl 1341.32016)]. The final construction of \(\varphi\) uses a refinement of a gluing procedure which goes back to \textit{R. Richberg} [Math. Ann. 175, 257--286 (1968; Zbl 0153.15401)].