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**Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type.**
*(English)*
Zbl 1007.32010

Bauer, Ingrid (ed.) et al., Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. Berlin: Springer. 223-277 (2002).

Summary: Let \(X\) be a holomorphic family of compact complex projective algebraic manifolds with fibers \(X_t\) over the open unit 1-disks \(\Delta\). Let \(K_{X_t}\) and \(K_X\) be respectively the canonical line bundles of \(X_t\) and \(X\).

We prove that, if \(L\) is a holomorphic line bundle over \(X\) with a (possibly singular) metric \(e^{-\varphi}\) of semipositive curvature current on \(X\) such that \(e^{-\varphi}|_{X_0}\) is locally integrable on \(X_0\), then for any positive integer \(m\), any \(s\in\Gamma(mK_{X_0}+ L)\) with \(|s|^2 e^{-\varphi}\) locally bounded on \(X_0\) can be extended to an element of \(\Gamma(X,mK_X+ L)\). In particular, \(\dim\Gamma(X_t, mK_{X_t}+ L)\) is independent of \(t\) for \(\varphi\) smooth. The case of trivial \(L\) gives the deformational invariance of the plurigenera.

The method of proof uses an appropriately formulated effective version, with estimates, of the argument in the author’s earlier paper on the invariance of plurigenera for general type. A delicate point of the estimates involves the use of metrics as singular as possible to \(pK_{X_0}+ a_pL\) on \(X_0\) to make the dimension of the space of \(L^2\) holomorphic sections over \(X_0\) bounded independently of \(p\), where \(a_p\) is the smallest integer \(\geq{p-1\over m}\). These metrics are constructed from \(s\). More conventional metrics, independent of \(s\), such as generalized Bergman kernels are not singular enough for the estimates.

For the entire collection see [Zbl 0989.00069].

We prove that, if \(L\) is a holomorphic line bundle over \(X\) with a (possibly singular) metric \(e^{-\varphi}\) of semipositive curvature current on \(X\) such that \(e^{-\varphi}|_{X_0}\) is locally integrable on \(X_0\), then for any positive integer \(m\), any \(s\in\Gamma(mK_{X_0}+ L)\) with \(|s|^2 e^{-\varphi}\) locally bounded on \(X_0\) can be extended to an element of \(\Gamma(X,mK_X+ L)\). In particular, \(\dim\Gamma(X_t, mK_{X_t}+ L)\) is independent of \(t\) for \(\varphi\) smooth. The case of trivial \(L\) gives the deformational invariance of the plurigenera.

The method of proof uses an appropriately formulated effective version, with estimates, of the argument in the author’s earlier paper on the invariance of plurigenera for general type. A delicate point of the estimates involves the use of metrics as singular as possible to \(pK_{X_0}+ a_pL\) on \(X_0\) to make the dimension of the space of \(L^2\) holomorphic sections over \(X_0\) bounded independently of \(p\), where \(a_p\) is the smallest integer \(\geq{p-1\over m}\). These metrics are constructed from \(s\). More conventional metrics, independent of \(s\), such as generalized Bergman kernels are not singular enough for the estimates.

For the entire collection see [Zbl 0989.00069].