A universal metric for the canonical bundle of a holomorphic family of projective algebraic manifolds (Q2472606)

This paper in complex geometry makes a contribution to the literature connected with Siu's proof of the invariance of the plurigenera of complex projective algebraic manifolds under deformation in families. Its main goal is formulated in Corollary~2. Corollary~2. If \(\pi:X\to\Delta\) is a holomorphic family of projective algebraic manifolds over the open unit disc \(\Delta\subset{\mathbb C}\), and \(X_0=\pi^{-1}(0)\) is its central fiber, then the parameter disc \(\Delta\) may be shrunk to a smaller disc \(0\in\Delta'\subset\Delta\) such that the restricted family \(X'=\pi^{-1}(\Delta')\to\Delta'\) admits a so-called universal canonical metric \(e^{-\kappa}\) with non-negative curvature current, i.e., \(e^{-\kappa}\) is a singular Hermitian metric on the canonical bundle \(K_{X'}\) of \(X'\) that renders every holomorphic central section \(s\in H^0(X_0,mK_{X_0})\) of the pluricanonical bundles \(mK_{X_0}\), \(m\geq1\), square summable: \(\int_{X_0}| s| ^2e^{-(m-1)\kappa}<\infty\). In the case of fibers of general type Corollary~2 was proved earlier by \textit{Y.-T. Siu} [Invent. Math. 134, No. 3, 661--673 (1998; Zbl 0955.32017)]. The author deduces Corollary~2 from a more general Theorem~1 (too long to state here) that treats, under the same umbrella, complex projective algebraic manifolds individually or in families. The proof of Theorem~1 relies in the one-tower argument of \textit{M. Paun} [J. Differ. Geom. 76, No. 3, 485--493 (2007; Zbl 1122.32014)] together with the author's version [`A Takayama-type extension theorem', preprint (2006)] of the Ohsawa-Takegoshi extension theorem of square summable holomorphic sections over hypersurfaces to square summable holomorphic sections over the whole space with precise control of the norms; an idea that was put to spectacular use by \textit{Y.-T. Siu} in [loc. cit. and Complex Geometry, 223--277 (2002; Zbl 1007.32010)]. The paper reads well and seems to proceed along paths already trodden in the literature of its topics.