Pluripotential theory on quaternionic manifolds (Q413696)

The author develops a theory of plurisubharmonic functions on a quaternionic manifold \(M^{4n}\) with \(n>1\), defined as sections of a real line bundle. NEWLINENEWLINENEWLINEIn the first section he reviews the aspects of the theory of quaternionic manifolds that he will need during the rest of the paper. Consider the subgroup NEWLINENEWLINE\[NEWLINE\mathcal{G}=\big\{(A,q)\in \mathrm{GL}_n(\mathbb{H})\times \mathrm{GL}_1(\mathbb{H})\,\big|\;\det A\cdot |q|=1\big\};NEWLINE\]NEWLINE it is assumed that the structure group of the tangent bundle \(TM\) lifts to \(\mathcal{G}\). Let \(G_0\) be the corresponding principal \(\mathcal{G}-\)bundle and define \(\mathcal{E}_0= G_0\times_{\mathcal{G}} \mathbb{H}^n\) and \(\mathcal{H}_0= G_0\times_{\mathcal{G}} \mathbb{H}\). These are a bundle of right \(\mathbb{H}\)-vector spaces of rank \(n\) and a line bundle of left \(\mathbb{H}\)-vector spaces, respectively, and they satisfy \(TM\cong\mathcal{E}_0\otimes_\mathbb{H} \mathcal{H}_0 \). NEWLINENEWLINENEWLINEThen he describes the construction of the holomorphic and of the smooth Bastion complexes of sheaves over a quaternionic manifold. One of the operators appearing in these complexes is a second order operator \(\Delta: \mathcal{C}^{\infty}(M, \det \mathcal{H}_0^*)\rightarrow \mathcal{C}^{\infty}(M, \wedge^2 \mathcal{E}_0^*\left[-2\right])\), which coincides with the quaternionic Hessian operator in the flat case. One of the new results of the paper is the proof of a new multiplicative property for such an operator.NEWLINENEWLINEAfter introducing a notion of positivity for the fibers of \(\wedge^2 \mathcal{E}_0^*\left[-2\right]_{\mathbb{R}}\), a plurisubharmonic function \(h\) is defined as a continuous section of \(( \det \mathcal{H}_0^*)_{\mathbb{R}}\) such that \(\Delta h\) is nonnegative. This definition and the properties of the Bastion operator are what is needed to prove an estimate of Chern-Levine-Nirenberg type.NEWLINENEWLINEA Monge-Ampère operator is defined on sections of \((\det\mathcal{H}_0^*)_\mathbb{R}\) via \(h\mapsto (\Delta h)^n\). The main theorem of the paper is the following: if \(\{ h_N\}\) is a sequence of \(\mathcal{C}^2\) plurisubharmonic sections converging to a \(\mathcal{C}^2\)-section \(h\) in the \(\mathcal{C}^0\)-topology, then \((\Delta h_N)^n\rightarrow (\Delta h)^n\) in the sense of measures.NEWLINENEWLINENEWLINEIn the last section the author proves that the theory in the flat case, which he developed in [Bull. Sci. Math. 127, No. 1, 1--35 (2003; Zbl 1033.15013)], is a special case of the theory of the present paper.