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

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
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References:

[1] Bishop, E.: Conditions for the analyticity of certain sets. Michigan Math. J.11, 289-304 (1964) · Zbl 0143.30302 · doi:10.1307/mmj/1028999180
[2] Burns, D., Rapoport, M.: On the Torelli problem for Kähler K3 surfaces. Ann. Scient. Éc. Norm. Sup.8, 235-274 (1975) · Zbl 0324.14008
[3] Dunford, N., Schwartz, J.T.: Linear Operators, Vol. I. New York: Interscience 1958 · Zbl 0084.10402
[4] Harvey, F.R., Lawson, B.: An intrinsic characterization of Kähler manifolds. · Zbl 0553.32008
[5] Hitchin, N.: Compact four-dimensional Einstein manifolds. J. Diff. Geom.9, 435-441 (1974) · Zbl 0281.53039
[6] Kodaira, K.: On the structure of compact complex analytic surfaces, I. Amer. J. Math.86, 751-798 (1964) · Zbl 0137.17501 · doi:10.2307/2373157
[7] Kodaira, K., Morrow, J.: Complex Manifolds. New York: Holt, Rinehart & Winston 1971 · Zbl 0325.32001
[8] Kodaira, K., Spencer, D.C.: On deformations of complex analytic structures, III. Stability theorems for complex structures. Ann. of Math.71, 43-76 (1960) · Zbl 0128.16902 · doi:10.2307/1969879
[9] Kulikov, V.: Epimorphicity of the period mapping for surfaces of type K3. Uspehi Mat. Nauk32, (No. 4) 257-258 (1977) · Zbl 0449.14008
[10] Looijenga, E.: A Torelli theorem for Kähler-Einstein K3 surfaces. Lecture Notes in Math Vol. 892, pp. 107-112. Berlin-Heidelberg-New York: Springer 1980
[11] Looijenga, E., Peters, C.: Torelli theorems for Kähler K3 surfaces. compositio Math.42, 145-186 (1981) · Zbl 0477.14006
[12] Miyaoka, Y.: Kähler metrics on elliptic surfaces. Proc. Japan Acad.50, 533-536 (1974) · Zbl 0354.32011 · doi:10.3792/pja/1195518827
[13] Morrey, C.B., Jr.: Multiple integrals in the calculus of variations. New York: Springer 1966 · Zbl 0142.38701
[14] Piatetskii-Shapiro, I.I., Shafarevitch, I.R.: A Torelli theorem for algebraic surfaces of type K3. Izv. Akad. Nauk SSSR35, 547-572 (1971); (English Translation) Math. USSR Izvestija5, 547-588 (1971)
[15] Siu, Y.-T.: A simple proof of the surjectivity of the period map of K3 surfaces. Manuscripta Math.35, 311-321 (1981) · Zbl 0497.32019 · doi:10.1007/BF01263265
[16] Sommese, A.J.: Quaternionic manifolds. Math. Ann.212, 191-214 (1975) · Zbl 0299.53023 · doi:10.1007/BF01357140
[17] Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math.36, 225-255 (1976) · Zbl 0335.57015 · doi:10.1007/BF01390011
[18] Todorov, A.N.: Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of K3 surfaces. Invent. Math.61, 251-265 (1980) · Zbl 0472.14006 · doi:10.1007/BF01390067
[19] Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold, and the complex Monge-Ampère equation, I. Comm. Pure Applied Math.31, 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
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