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On the homotopy types of Kähler manifolds and the birational Kodaira problem. (English) Zbl 1102.32008

A theorem of Kodaira says that every compact Kähler surface admits small deformations which are projective. In an earlier paper [Invent. Math. 157, No. 2, 329–343 (2004; Zbl 1065.32010)] the author has shown that the situation is different in higher dimensions, proving that there are \(n\)-dimensional compact Kähler manifolds for all \(n\geq 4\) which do not have the homotopy type of a projective manifold. In the paper under review she considers the more general question whether every compact Kähler manifold has a smooth bimeromorphic model which admits a deformation into a projective manifold. The answer is negative and even stronger: There are examples of \(2n\)-dimensional compact Kähler manifolds \(X\) for any \(n\geq 5\) whose smooth bimeromorphic models do not have the homotopy type of a projective complex manifold. The given example \(X\) is a specific \(\mathbb P_1\times\mathbb P_1\)-bundle over a simply connected even-dimensional compact Kähler manifold \(B\), where \(B\) is constructed as in the article cited above.
The heart of the paper is the result that the Hodge structure on the rational cohomology group \(H^2(X',\mathbb Q)\) of any smooth bimeromorphic Kähler model \(X'\) of \(X\) cannot be polarized. In particular there can be no isomorphism between the cohomology algebras \(H^*(X',\mathbb Q)\) and \(H^*(Y,\mathbb Q)\) for any projective manifold \(Y\). A key argument in the proof is the existence of an irreducible endomorphism \(H^2(X',\mathbb Q)\rightarrow H^2(X',\mathbb Q)\) of Hodge structures, based on a Lemma of Deligne about the construction of rational sub-Hodge structures of a rational Hodge structure. Since the rational homotopy type of a simply connected Kähler manifold is determined by its rational cohomology algebra, see P. Deligne, Ph. Griffiths, J. Morgan and D. Sullivan [Invent. Math. 29, 245–274 (1975; Zbl 0312.55011)], the above result implies that even the rational homotopy types of \(X'\) and \(Y\) are different.

MSC:

32J27 Compact Kähler manifolds: generalizations, classification
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q55 Topological aspects of complex manifolds
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