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On the homotopy types of compact Kähler and complex projective manifolds. (English) Zbl 1065.32010

It is a classical problem whether every compact Kähler manifold has the same homotopy type as a projective complex manifold. In 1960 Kodaira proved that every compact Kähler surface is even deformation equivalent to a complex projective surface. The examples in the present paper show that the situation is quite different in higher dimensions. For every \(m\geq 2\) there exists a \(2m\)- dimensional compact Kähler manifold \(X\), birational equivalent to a compact complex torus, with the following property: For any elliptic curve \(F\) neither of the graded integral cohomology rings \(H^*(X,\mathbb Z)\) and \(H^*(X\times F,\mathbb Z)\) is isomorphic to the graded integral cohomology ring of a complex projective manifold. The problem is still open in dimension three. The manifold \(X\) is constructed by a sequence of blow ups of \(T\times T\), where \(T\) is a specific compact complex torus. It is shown that the Albanese torus of a compact Kähler manifold \(X'\) with \(H^*(X',\mathbb Z)\) isomorphic to \(H^*(X,\mathbb Z)\) (resp. \(H^*(X\times F,\mathbb Z))\) is not an abelian variety. The author applies a Lemma of P. Deligne to show that the latter remains true if the integral cohomology rings in the assumption are replaced by the rational cohomology rings. Moreover the manifold \(X\) can be modified by a sequence of blow ups to a compact Kähler manifold \(X_1\) such that the graded cohomology algebra \(H^*(X_1,\mathbb C)\) is not isomorphic to the cohomology algebra of a complex projective manifold.
Finally the author constructs examples of \(2m\)-dimensional simply connected compact Kähler manifolds \(X\) (for \(m\geq 3\)) whose rational cohomology ring is not isomorphic to that of a complex projective manifold. Starting with a torus \(T\) as above let \(K\) be the desingularization of the quotient of \(T\) by the \(-1\)-involution. \(K\) is a simply connected Kähler manifold and \(X\) is constructed by a sequence of blowing ups of \(K\times K\). The Hodge index theorem is a main tool in the proof.

MSC:

32J27 Compact Kähler manifolds: generalizations, classification
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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References:

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