On compact Kähler surfaces.

*(English)*Zbl 0926.32025The conjecture of Kodaira that every compact complex surface \(X\) with first Betti number \(b_1(X) \equiv 0\pmod 2\) admits a Kähler metric follows partially from the Enriques-Kodaira classification. The remaining cases of elliptic and \(K3\) surfaces were affirmatively answered by Y. Miyaoka [Proc. Jap. Acad. 50, 533-536 (1974; Zbl 0354.32011)] and Y.-T. Siu [Invent. Math. 73, 139-150 (1983; Zbl 0557.32004)] respectively.

The paper under review contains a general proof of the conjecture which does not make any use of the Enriques-Kodaira classification. The author also gets criteria analogous to those of Nakai-Moishezon for elements in \(H_\mathbb{R}^{1,1} (X)\) to be representable by positive closed (1,1)-forms. The proofs are based on Hodge theory and \(L_2\)-cohomology. Main ingredients are results of P. Gauduchon on the existence of \(\partial \overline \partial\)-closed positive (1,1)-forms on compact complex surfaces, the theorem of Siu on the analyticity of the sets associated with the Lelong numbers of closed positive currents and results of Demailly about the smoothing of positive closed (1,1)-currents.

The paper under review contains a general proof of the conjecture which does not make any use of the Enriques-Kodaira classification. The author also gets criteria analogous to those of Nakai-Moishezon for elements in \(H_\mathbb{R}^{1,1} (X)\) to be representable by positive closed (1,1)-forms. The proofs are based on Hodge theory and \(L_2\)-cohomology. Main ingredients are results of P. Gauduchon on the existence of \(\partial \overline \partial\)-closed positive (1,1)-forms on compact complex surfaces, the theorem of Siu on the analyticity of the sets associated with the Lelong numbers of closed positive currents and results of Demailly about the smoothing of positive closed (1,1)-currents.

Reviewer: E.Oeljeklaus (Bremen)

##### MSC:

32J15 | Compact complex surfaces |

14J15 | Moduli, classification: analytic theory; relations with modular forms |

32J27 | Compact Kähler manifolds: generalizations, classification |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

32C30 | Integration on analytic sets and spaces, currents |

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\textit{N. Buchdahl}, Ann. Inst. Fourier 49, No. 1, 287--302 (1999; Zbl 0926.32025)

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