Dynamical systems of \(p\)-Bergman kernels (Q6616793)

From the text: ``In complex geometry, there have been introduced several intrinsic (pseudo) volume forms on complex manifolds. The \(p\)-Bergman kernel is defined as the supremum of the norm square of the \(L^p\)-holomorphic functions whose \(L^p\)-norm is less than \(1.\) For \(p=2,\) this is the usual Bergman kernel. For \(p\neq 2,\) \(p\)-Bergman kernels are considered to be less interesting than usual Bergman kernels. However we see that to handle pluricanonical forms there are several advantages to use \(p\)-Bergman kernels. For instance, for a compact complex manifold \(X\) with ample canonical bundle, the space of \(L^2\)-canonical forms is nothing but \(H^0 (X, K_X)\) and it does not contain enough information, unless \(K_X\) is very ample. Hence to study a canonically polarized manifold \(X\) we need to consider the graded ring \(\oplus_{m=0}^\infty H^0 (X, mK_X)\) instead of \(H^0 (X, K_X).\) We note that \(H^0 (X, mK_X)\) has the natural \(L^{\frac 2m}\)-structure with respect to the pseudonorm: \N\[\N\left \Vert \sigma \right \Vert=\left \vert \int_X (\sigma \wedge \bar{\sigma})^{\frac 1m}\right \vert^{\frac m2}.\N\]\NHence it is natural to consider the \(L^{\frac 2m}\)-space of pluricanonical forms to handle compact Kähler manifolds with pseudoeffective canonical bundles as in [\textit{M. S. Narasimhan} and \textit{R. R. Simha}, Invent. Math. 5, 120--128 (1968; Zbl 0159.37902)].''\N\NThe purpose of the paper under review is to investigate the approximation of Kähler-Einstein forms and twisted Kähler-Einstein forms by a dynamical system of \(p\)-Bergman kernels. Its main results are for projective manifolds with intermediate Kodaira dimension.\N\NFor the entire collection see [Zbl 1537.32001].