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The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics. (English) Zbl 0714.32007

The authors address some fundamental questions concerning the structure of moduli spaces of (polarised) compact Kähler manifolds, in particular the existence of natural Kähler metrics on such moduli spaces. The basic idea for constructing such a metric is to realise the polarisation by a distinguished Kähler metric with good functorial behaviour which allows one to define a metric on the base of any holomorphic family of such distinguished metrics. This idea has previously been used by N. Koiso [Invent. Math. 73, 71-106 (1983; Zbl 0515.53040)] with Kähler- Einstein metrics, while Calabi-Yau metrics were used by the second author [Invent. Math. 71, 295-307 (1983; Zbl 0521.32019)] to construct the moduli space of polarised Kähler manifolds with vanishing first Chern class. The distinguished metrics used here are the extremal metrics, defined for any polarised compact Kähler manifold (X,\(\gamma\)) of dimension n as follows: let U be the space of all \(C^{\infty}\) Kähler forms \(\omega\) on X which represent the polarisation \(\gamma\) ; then \(\omega\) is called extremal if it is a critical point for the functional \(\int R^ 2\omega^ n\), where \(R=R(\omega)\) is the scalar curvature of \(\omega\). (In particular, any Kähler-Einstein metric is extremal.) The authors begin by developing a deformation theory of extremal Kähler manifolds X (that is, compact Kähler manifolds equipped with extremal Kähler metrics) under the additional condition (A) that the automorphism group of X has compact components; in this case, a Kähler metric is extremal if and only if it has constant scalar curvature. The basic fact is the unique extension property for extremal metrics (Theorem 6.3), from which the existence of a coarse moduli space for the set \({\mathcal M}_ e\) of extremal compact Kähler manifolds satisfying (A) is deduced; in fact \({\mathcal M}_ e\) has a natural structure of Hausdorff reduced complex space (Theorem 6.6). The remainder of the paper is concerned with the construction of Kähler metrics on \({\mathcal M}_ e\) and on an associated moduli space \({\mathcal M}_{H,e}\) of extremal Hodge manifolds (Theorem 7.10); these are the generalized Weil-Petersson metrics of the title. For Hodge manifolds, the Weil-Petersson form can be realised as the Chern class of a Hermitian line bundle on \({\mathcal M}_{H,e}\) (Theorem 11.1).
Essential ingredients of the proofs are a fibre integral formula (Theorem 8.1), determinant bundles and Quillen metrics (§ 10 Theorem BGS [J. M. Bismut, H. Gillet and Ch. Soulé, Commun. Math. Phys. 115, No.1, 49-78; 79-126; No.2, 301-351 (1988; Zbl 0651.32017)]) and a generalization of Theorem BGS to singular base spaces (Theorem 10.1 and § 12).
Reviewer: P.E.Newstead

MSC:

32G13 Complex-analytic moduli problems
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32J27 Compact Kähler manifolds: generalizations, classification
32Q15 Kähler manifolds
14J15 Moduli, classification: analytic theory; relations with modular forms
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