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Quasi-isometry and deformations of Calabi-Yau manifolds. (English) Zbl 1318.32017

The authors provide several interesting results concerning the Hodge theory and the Kodaira-Spencer-Kuranishi deformation theory on compact Kähler manifolds.
The main results are the following. In Theorem 2.2, the authors prove a quasi-isometry result in \(L^2\)-norm for bundle-valued differential forms with respect to the operator \(\overline\partial^*\circ\mathbb{G}\) on a compact Kähler manifold. (Here, \(\mathbb{G}\) denotes the Green operator in the Hodge decomposition with respect to the operator \(\overline\partial\) and with respect to the Kähler metric.) Using the operator \(\overline\partial^*\circ\mathbb{G}\), an explicit \(\overline\partial\)-inverse formula for the solutions of some vector-bundle-valued \(\overline\partial\)-equations over compact Kähler manifolds is provided in Proposition 2.3, which is considered as a sort of bundle-valued version of the \(\partial\overline\partial\)-Lemma. The previous results are proven just using Hodge theory, therefore they can be extended to general Kähler manifolds as long as Hodge theory can be applied. In Theorem 3.4 and Corollary 3.5, explicit formulas for the deformed differential operators for bundle-valued differential forms on the deformation spaces of complex structures of Kähler manifolds are provided. As an application, recursive methods are used to construct Beltrami differentials in Kodaira-Spencer-Kuranishi deformation theory. An \(L^2\)-global canonical family of Beltrami differentials (namely, a globally convergent power series of Beltrami differentials in \(L^2\)-norm) on Calabi-Yau manifolds is constructed in Theorem 4.3. This is done by showing that the (sufficiently small) radius of convergence is in fact at least \(1\). (See Theorem 4.4 for the multi-parameter version.) This result allows to construct an \(L^2\)-global canonical family of holomorphic \((n,0)\)-forms on the deformation spaces of Calabi-Yau manifolds, see Theorem 5.2. The global expansion of the canonical family of \((n,0)\)-forms on the deformation spaces of Calabi-Yau manifolds in cohomology classes given in Theorem 5.3 has applications in studying the global Torelli problem. Analogue results for deformation spaces of compact Kähler manifolds are proven in Theorem 5.5 and Corollary 5.6, under the assumption of the existence of a \(L^2\)-global canonical family of Beltrami differentials.
More applications to the Torelli problem and the extension of twisted pluricanonical sections are expected in a sequel paper.

MSC:

32G05 Deformations of complex structures
32J27 Compact Kähler manifolds: generalizations, classification
58A14 Hodge theory in global analysis
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q25 Calabi-Yau theory (complex-analytic aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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