On Classification of Compact Complex Surfaces of Class VII

Georges Dloussky ()
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Georges Dloussky: Georges Dloussky Aix-Marseille Université, Institut de Mathématiques de Marseille (I2M), UMR 7373

A chapter in Real and Complex Geometry, 2025, pp 159-189 from Springer

Abstract: Abstract Let S be a minimal compact complex surface with Betti numbers b 1 ( S ) = 1 $$b_1(S)=1$$ and b 2 ( S ) ≥ 1 $$b_2(S)\ge 1$$ i.e. a compact surface in class VII 0 + $$_0^+$$ . We show that if there exists a twisted logarithmic 1-form θ ∈ H 0 ( S , Ω 1 ( log D ) ⊗ ℒ λ ) $$\theta \in H^0(S,\Omega ^1(\text{log} D)\otimes \mathcal L_\lambda )$$ , where D is a non zero divisor and ℒ ∈ H 1 ( S , ℂ ⋆ ) $$\mathcal L\in H^1(S,\mathbb C^\star )$$ , then S is a Kato surface. It is known that λ $$\lambda $$ is in fact real and we show that λ ≥ 1 $$\lambda \ge 1$$ and unique if S is not a Inoue-Hirzebruch surface. Moreover λ = 1 $$\lambda =1$$ if and only if S is a Enoki surface. When λ > 1 $$\lambda >1$$ these conditions are equivalent to the existence of a negative PSH function τ ̂ $$\hat \tau $$ on the cyclic covering p : Ŝ → S $$p:\hat S\to S$$ of S which is PH outside D ̂ : = p −1 ( D ) $$\hat D:=p^{-1}(D)$$ with automorphic constant being the same automorphy constant λ $$\lambda $$ for a suitable automorphism of Ŝ $$\hat S$$ . With previous results obtained with V.Apostolov it suggests a strategy to prove the GSS conjecture.

Keywords: Non Kähler; Classification; Compact complex surfaces; Complex foliation (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-92297-8_7

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DOI: 10.1007/978-3-031-92297-8_7

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