Geometry of logarithmic forms and deformations of complex structures

Abstract

We present a new method to solve certain∂-equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a ∂∂-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at E 1 -level, as well as certain injectivity theorem on compact Kähler manifolds.Furthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic (n, q)-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by differential geometric method.Corollary 0.3 (=Corollary 3.1). With the same notations as in Theorem 0.1, if α ∈ A 0,0 (X, Ω p X (log D)) with∂∂α = 0, then ∂α = 0. It is well-known that Deligne's degeneracy of logarithmic Hodge to de Rham spectral sequences at E 1 -level [8] is a fundamental result and has great impact in algebraic geometry, such as vanishing and injectivity theorems. P. Deligne and L. Illusie [9] also proved this degeneracy by using a purely algebraic positive characteristic method. For compact Kähler manifolds, as the second application of Theorem 0.1, we can give a geometric and simpler proof of Deligne's degeneracy theorem.Theorem 0.4 (=Theorem 3.2). With the same notations as in Theorem 0.

Injectivity theorem for pseudo-effective line bundles and its applications

“…However, as we mentioned above, there is no analytic proof for Theorem 1.1. Difficulties lie in that the usual L 2 -method does not work for log canonical singularities, and that no transcendental methods are corresponding to the theory of mixed Hodge structures (see [MaS8,No,LRW] for some approaches). The transcendental method often provides some powerful tools not only in complex geometry but also in algebraic geometry.…”

Deformations of log Calabi-Yau pairs can be obstructed

“…(i) D is smooth. This is due to Iacono [Iac15] and to Sano [San14,Remark 2.5] independently (see also [Kon,KKP08,LRW19,Wan,FP]). Note that in this case the deformations of (X, D) are locally trivial and coincide with the log smooth deformations of the log scheme given by X equipped with the divisorial log structure associated to D. (ii) X is weak Fano, i.e.…”

Logarithmic Akizuki--Nakano vanishing theorems on weakly pseudoconvex Kähler manifolds

“…In corollary 1.3 above, if we replace the positive line bundle by the positive vector bundle in the sense of Nakano, the claim still be true. Also as Professor Ohsawa pointed in [Ohsa21], it may be interested to generalize the results in [LRW19] and [LWY19] to the weakly pseudoconvex situation. Acknowledgement: The author would like to thank Professor Shigeharu Takayama for his guidance and warm encouragement, and Professor Sheng Rao for his constant support.…”