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Sheng Rao

Publications and source records attributed to Sheng Rao.

25 records · Page 2Linked to original sources
arXiv2018

Geometry of logarithmic forms and deformations of complex structures

We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-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 Kahler manifolds. Furthermore, for a family of logarithmic deformations of complex structures on Kahler 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.

math.AG↗
arXiv2018

On local stabilities of $p$-Kähler structures

By use of a natural extension map and a power series method, we obtain a local stability theorem for p-Kähler structures with the $(p,p+1)$-th mild $\partial\bar\partial$-lemma under small differentiable deformations.

math.CV↗
arXiv2018

Dolbeault cohomologies of blowing up complex manifolds

We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic invariance of $(\bullet,0)$- and $(0,\bullet)$-Hodge numbers on a compact complex manifold, and obtain the equality for the numbers of the blow-ups and blow-downs in the weak factorization of the bimeromorphic map between two compact complex manifolds with equal $(1,1)$-Hodge number or equivalently second Betti number. Many examples of the latter one are listed. Inspired by these, we obtain the bimeromorphic stability for degeneracy of the Frölicher spectral sequences at $E_1$ on compact complex threefolds and fourfolds.

math.AG↗
arXiv2017

Several special complex structures and their deformation properties

We introduce a natural map from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the infinitesimal deformations of this complex manifold. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the first author. As direct corollaries, we prove several deformation invariance theorems for Hodge numbers. Moreover, we also study the Gauduchon cone and its relation with the balanced cone in the Kähler case, and show that the limit of the Gauduchon cone in the sense of D. Popovici for a generic fiber in a Kählerian family is contained in the closure of the Gauduchon cone for this fiber.

math.CV↗
arXiv2014

Quasi-isometry and deformations of Calabi-Yau manifolds

We prove several formulas related to Hodge theory and the Kodaira-Spencer-Kuranishi deformation theory of Kähler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi-Yau manifolds and also a construction of global canonical family of holomorphic $(n,0)$-forms on the deformation spaces of Calabi-Yau manifolds. Similar constructions are also applied to the deformation spaces of compact Kähler manifolds.

math.DG↗
arXiv2013

New proofs of the Torelli theorems for Riemann surfaces

In this paper, by using the Kuranishi coordinates on the Teichmüller space and the explicit deformation formula of holomorphic one-forms on Riemann surface, we give an explicit expression of the period map and derive new differential geometric proofs of the Torelli theorems, both local and global, for Riemann surfaces.

math.DG↗