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Sheng Rao

Publications and source records attributed to Sheng Rao.

At least 19 recordsLinked to original sources
arXiv2026

On Pseudo-Effectivity and Volumes of Adjoint Classes in K\"ahler Families with Projective Central Fiber

This paper is devoted to studying the deformation behavior of pseudo-effective canonical divisors and volumes of adjoint classes in K\"ahler families. Based on recent developments in the K\"ahler minimal model program, for flat families with fiberwise canonical singularities, we establish the global stability of the pseudo-effectivity of canonical divisors and uniruledness, assuming in addition that one fiber is projective, while the same conclusion for K\"ahler threefolds is also true without the projectivity assumption of the central fiber. For a smooth K\"ahler family whose central fiber is projective with a big adjoint class, we show that its volume remains locally constant. Finally, using the (relative) minimal model program for K\"ahler threefolds, we verify the deformation invariance of volumes of adjoint classes and plurigenera for smooth families of K\"ahler threefolds, thereby confirming Siu's invariance of plurigenera conjecture in dimension three.

math.AG↗
arXiv2025

Deformation of nef adjoint canonical line bundles

Much inspired by J. A. Wi\'sniewski's nef-value function method, we prove that in a smooth projective family over the unit disk, if the adjoint bundle of the canonical line bundle with a relatively semiample line bundle is nef on one fiber, then it remains nef on all fibers. We further extend this result to the semiampleness of the adjoint canonical line bundles. Using these, we prove the deformation invariance of any generalized plurigenera by assuming that only one fiber admits the semiample canonical line bundle and improve the first author--Xiao-Lei Liu's recent deformation rigidity of projective manifolds with semiample canonical line bundles. In particular, also by E. Viehweg--K. Zuo's result on the minimal number of singular fibers in a family and the first author--X. Liu's isotriviality result, if a projective family over $\mathbb{P}^1$ or an elliptic curve has one fiber with the big and nef (or more generally semiample) canonical line bundle, then all fibers are isomorphic to this fiber. Next, much inspired by M. Andreatta--T. Peternell's deformation theoretical approach, we prove that, in a smooth K\"ahler family of threefolds, if the canonical line bundle of one fiber is not nef, then none of its small deformations admits a nef canonical line bundle either. This partially confirms a problem posed by F. Campana--T. Peternell and the global stability of semiampleness of canonical line bundles of threefolds under a K\"ahler smooth deformation.

math.AG↗
arXiv2025

Rigidity for compact hyperbolic complex manifolds

We study the deformation behavior of compact hyperbolic complex manifolds. Let $\pi:\mathcal{X}\rightarrow \Delta$ be a smooth family of compact complex manifolds over the unit disk in $\mathbb{C}$, and $H$ a compact hyperbolic complex manifold. Then the $H$-locus $\{t\in\Delta: X_t\cong H\}$ is either at most a discrete subset of $\Delta$ or the whole $\Delta$. For a smooth family over a compact Riemann surface $Y$, its $H$-locus is either at most finite or the whole $Y$. Furthermore, if $Y$ is isomorphic to $\mathbb{P}^1$ or an elliptic curve, then we conjecture that the $H$-locus is empty or the whole $Y$.

math.CV↗
arXiv2025

Characterization of fiberwise bimeromorphism and specialization of bimeromorphic types I: locally Moishezon case

Inspired by the recent works of M. Kontsevich--Y. Tschinkel and J. Nicaise--J. C. Ottem on specialization of birational types for smooth families (in the scheme category) and J. Koll{\'a}r's work on fiberwise bimeromorphism, we focus on characterizing the fiberwise bimeromorphism and utilizing the characterization to investigate the specialization of bimeromorphic types for non-smooth families in the complex analytic setting. We provide several criteria for a bimeromorphic map between two families over the same base to be fiberwise bimeromorphic. By combining these criteria with the relative Barlet cycle space theoretic argument motivated by D. Mumford--U. Persson, K. Timmerscheidt and T. de Fernex--D. Fusi, we establish the specialization of bimeromorphic types for locally Moishezon families with fibers having only canonical singularities and being of non-negative Kodaira dimension. These specialization results can easily lead to criteria for locally strongly bimeromorphic isotriviality. Throughout this paper, we unveil the connections among the four classical topics in bimeromorphic geometry: the deformation behavior of plurigenera (or even $1$-genus), fiberwise bimeromorphism, specialization of bimeromorphic types, and the bimeromorphic version of the deformation rigidity.

math.AG↗
arXiv2022

Invariance of plurigenera and Chow-type lemma

This paper answers a question of Demailly whether a smooth family of nonsingular projective varieties admits the deformation invariance of plurigenera affirmatively, and proves this more generally for a flat family of varieties with only canonical singularities and uncountable ones therein being of general type and also two Chow-type lemmata on the structure of a family of projective complex analytic spaces.

math.AG↗
arXiv2022

On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversely) $p$-K\"ahler structures

Inspired by a recent work of D. Wei--S. Zhu on the extension of closed complex differential forms and Voisin's usage of the $\partial\bar{\partial}$-lemma, we obtain several new theorems of deformation invariance of Hodge numbers and reprove the local stabilities of $p$-K\"ahler structures with the $\partial\bar{\partial}$-property. Our approach is more concerned with the $d$-closed extension by means of the exponential operator $e^{\iota_\varphi}$. Furthermore, we prove the local stabilities of transversely $p$-K\"ahler structures with mild $\partial\bar{\partial}$-property by adapting the power series method to the foliated case, which strengthens the works of A. El Kacimi Alaoui--B. Gmira and P. Ra\'zny on that of the transversely K\"ahler foliations with homologically orientability. We observe that a transversely K\"ahler foliation, even without homologically orientability, also satisfies the $\partial\bar{\partial}$-property. So even when $p=1$ (transversely K\"ahler), our results are new as we can drop the assumption in question on the initial foliation. Several theorems on the deformation invariance of basic Hodge/Bott--Chern numbers with mild $\partial\bar{\partial}$-properties are also presented.

math.CV↗
arXiv2022

Hodge cohomology on blow-ups along subvarieties

We establish a blow-up formula for Hodge cohomology of locally free sheaves on smooth proper varieties over an algebraically closed field of positive characteristic. For this, we introduce a notion of relative Hodge sheaves and study their behavior under blow-ups along smooth centers. In particular, as an application, we study the blow-up invariance of the $E_2$-degeneracy of the Hochschild--Kostant--Rosenberg spectral sequence for smooth proper varieties.

math.AG↗
arXiv2021

Power series proofs for local stabilities of Kähler and balanced structures with mild $\partial\bar\partial$-lemma

By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira-Spencer's local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an $n$-dimensional balanced manifold with the $(n-1,n)$-th mild $\partial\bar\partial$-lemma by power series method and the other one on $p$-Kähler structures with the deformation invariance of $(p,p)$-Bott-Chern numbers.

math.CV↗
arXiv2021

$L^2$ extension of $\bar\partial$-closed forms on weakly pseudoconvex Kähler manifolds

Combining V. Koziarz's observation about the regularity of some modified section related to the initial extension with J. McNeal--D. Varolin's regularity argument, we generalize two theorems of McNeal--Varolin for the $L^2$ extension of $\bar\partial$-closed high-degree forms on a Stein manifold to the weakly pseudoconvex Kähler case under mixed positivity conditions.

math.CV↗
arXiv2020

Deformation limit and bimeromorphic embedding of Moishezon manifolds

Let $π: \mathcal{X}\rightarrow Δ$ be a holomorphic family of compact complex manifolds over an open disk in $\mathbb{C}$. If the fiber $π^{-1}(t)$ for each nonzero $t$ in an uncountable subset $B$ of $Δ$ is Moishezon and the reference fiber $X_0$ satisfies the local deformation invariance for Hodge number of type $(0,1)$ or admits a strongly Gauduchon metric introduced by D. Popovici, then $X_0$ is still Moishezon. We also obtain a bimeromorphic embedding $\mathcal{X}\dashrightarrow\mathbb{P}^N\timesΔ$. Our proof can be regarded as a new, algebraic proof of several results in this direction proposed and proved by Popovici in 2009, 2010 and 2013. However, our assumption with $0$ not necessarily being a limit point of $B$ and the bimeromorphic embedding are new. Our strategy of proof lies in constructing a global holomorphic line bundle over the total space of the holomorphic family and studying the bimeromorphic geometry of $π:\mathcal{X}\rightarrow Δ$. S.-T. Yau's solutions to certain degenerate Monge--Ampère equations are used.

math.AG↗
arXiv2019

Extension formulas and deformation invariance of Hodge numbers

We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the second author. As a direct corollary of the extension formulas, we prove several deformation invariance theorems for Hodge numbers on some certain classes of complex manifolds, without use of Frölicher inequality or the topological invariance of Betti numbers.

math.CV↗
arXiv2018

Dolbeault cohomologies of blowing up complex manifolds II: bundle-valued case

We use a sheaf-theoretic approach to obtain a blow-up formula for Dolbeault cohomology groups with values in the holomorphic vector bundle over a compact complex manifold. As applications, we present several positive (or negative) examples associated to the vanishing theorems of Girbau, Kawamata-Viehweg and Green-Lazarsfeld in a uniform manner and study the blow-up invariance of some classical holomorphic invariants.

math.AG↗
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↗