Every K 3 surface is Kähler. (English) Zbl 0557.32004

Siu, Y.-T.

Invent. Math. 73, 139-150 (1983).

Kodaira made the conjecture that every compact complex surface with even first Betti number is Kähler. The only open case was that of K-3 surfaces; so the result of this paper completes the confirmation of Kodaira’s conjecture. A previous proof by A. N. Todorov [Invent. Math. 61, 251-265 (1980; Zbl 0472.14006)] of the same result presents some serious gaps so that this is the first complete proof. Two important ingredients of the approach followed here are the surjectivity of the period map for K-3 surfaces due to Todorov (loc. cit.) and E. Looijenga [Geometry, Proc. Symp., Utrecht 1980, Lect. Notes Math. 894, 107-112 (1981; Zbl 0473.53041)] and the global Torelli theorem for Kähler K-3 surfaces due to D. Burns jun. and M. Rapoport [Ann. Sci. Éc. Norm. Super., IV. Sér. 8, 235-273 (1975; Zbl 0324.14008)]. One has to prove that a K-3 surface which has the same periods as a Kähler K-3 surface is necessarily analytically isomorphic to it.
The program has serious difficulties that the author overcomes with deep and subtle arguments. First he proves the existence of a closed 2-form \(\xi\) on any K-3 surface M with positive definite (1,1)-component. Then that the orthogonal projections \(\theta\) to \(H_ R^{1,1}(M)\) (with respect to the quadratic form defined by the cup product) of the cohomology class of \(\xi\) belongs to the positive cone of \(H_ R^{1,1}(M)\) and moreover if M is Kähler \(\theta\) lies in the same component of the positive cone of \(H_ R^{1,1}(M)\) as any Kähler class of M. The derived result is obtained from the previous facts and a clever use of the before recalled local and global Torelli theorems for K-3 surfaces.

Reviewer: F.Gherardelli

MSC:

32J15	Compact complex surfaces
14J25	Special surfaces
53C55	Global differential geometry of Hermitian and Kählerian manifolds
32J25	Transcendental methods of algebraic geometry (complex-analytic aspects)
14J15	Moduli, classification: analytic theory; relations with modular forms

Keywords:

every K3 surface is Kähler; Kodaira conjecture; compact complex surface with even first Betti number; surjectivity of the period map for K-3 surfaces; global Torelli theorem

Citations:

Zbl 0472.14006; Zbl 0473.53041; Zbl 0324.14008

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