Extended Deformation of Kodaira Surfaces

几乎复4维流形的柯西-亚罗维茨维度（Kodaira维度）

Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class

本文研究了具有 torsion 第一陈类的几乎复4维流形的柯西-亚罗维茨维度，证明了其可能的值为 0 或 -∞，并探讨了 K3 和 Enriques 表面的非可积微小变形。

On Kodaira dimension of almost complex 4-dimensional solvmanifolds without complex structures

The aim of this paper is to continue the study of Kodaira dimension for almost complex manifolds, focusing on the case of compact $4$-dimensional solvmanifolds without any integrable almost complex structure. According to the classification theory we consider: $\mathfrak{r}\mathfrak{r}_{3, -1}$, $\mathfrak{nil}^4$ and $\mathfrak{r}_{4, λ, -(1 + λ)}$ with $-1 < λ< -\frac{1}{2}$. For the first solvmanifold we introduce special families of almost complex structures, compute the corresponding Kodaira dimension and show that it is no longer a deformation invariant. Moreover we prove Ricci flatness …

Kodaira Families and Newton-Cartan Structures with Torsion

We describe the induced geometry on several classes of Kodaira moduli spaces of rational curves in twistor spaces. By constructing connections and frames on the moduli spaces we build and review twistor theories pertaining to relativistic and non-relativistic geometries. Focussing on the cases of three- and five-dimensional moduli spaces we establish novel twistor theories of Newton-Cartan spacetimes. We generalise a canonical class of connections on Kodaira moduli spaces to encompass torsion and prove that in three dimensions deformations of the twistor space's holomorphic structure induce Ne…

A Note on the Analytic Families of Compact Submanifolds of Complex Manifolds

In this paper, we prove a result related to the deformation of complex submanifolds, modifying a result of Kodaira (Ann. Math, 75(1), 146-162, 1962).

Deformations of calibrations, {Calabi-Yau, HyperKähler, $G_2$ and Spin(7) structures}

We shall develop a new deformation theory of geometric structures in terms of closed differential forms. This theory is a generalization of Kodaira -Spencer theory and further we obtain a criterion of unobstructed deformations. We apply this theory to certain geometric structures: Calabi-Yau, HyperKähler, $\G$ and $\Spin$ structures and show that these deformation spaces are smooth in a systematic way.

Theorem of existence and completeness for holomorphic Poisson structures

In this paper, we define a concept of a family of compact holomorphic Poisson manifolds on the basis of Kodaira-Spencer's deformation theory and deduce the integrability condition. We prove an analogue of their `Theorem of existence for complex analytic structures' under some analytic assumption, and establish an analogue of their `Theorem of completeness for complex analytic structures' in the context of holomorphic Poisson deformations.

Deformations and BBF form on non-Kahler holomorphically symplectic manifolds

In 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation theory of Bogomolov-Guan manifolds and show that it is similar to that of hyperkahler manifolds. We prove the local Torelli theorem, showing that holomorphically symplectic deformations of BG-manifolds are unobstructed, and the corresponding period map is locally a diffeomorphism. Using the local Torelli theorem, we prove the Fujiki formul…

Deformations of generalized complex and generalized Kahler structures

In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which admits deformations to generalized complex manifolds and obtain non-trivial generalized Kahler structures on Fano surfaces and toric Kahler manifolds. In particular, we show that holomorphic Poisson structures on a Kahler manifold induce deformations of generalized Kahler structur…

一类LVM三维流形上几何结构的形变

Deformations of geometric structures on a subclass of LVM threefolds

该研究针对三维复LVM流形的几何结构形变，结合Ehresmann-Thurston原理分析共振结构与复结构形变的关联，刻画其Kuranishi族并构造出完备的包含族。

Deformation of Dirac structures via $L_\infty$ algebras

The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra which depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty$ algebra instead. We develop a simplified method for describing this $L_\infty$ algebra and use it to prove that the $L_\infty$ algebras corresponding to different transversals are canonically $L_\infty$-isomorphic. In some cases, this isomorphism provides a formality map, as we show in several examples including (quasi)-Poisson geometry, Dirac structures on Lie grou…

复结构变形障碍与上同调收缩

Obstructions of deforming complex structures and cohomology contractions

本文从分析角度改进了Kodaira原理，通过幂级数方法和弗洛里切尔谱序列，揭示了复结构变形障碍与上同调收缩映射的关系，并给出了具有平凡canonical包的紧复流形的新无障碍准则。

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 inva…

一种杂化 Kähler 引力与距离猜想

A Heterotic Kähler Gravity and the Distance Conjecture

本文将杂化超势形变中的复结构场积分消去，得到描述复化 Hermitian 与规范自由度的新全纯场论，并借此探讨其与杂化弦 swampland 距离猜想的联系及新型辛上同调。

On the deformation of inversive distance circle packings, II

We show that the results in \cite{Ge-Jiang1} are still true in hyperbolic background geometry setting, that is, the solution to Chow-Luo's combinatorial Ricci flow can always be extended to a solution that exists for all time, furthermore, the extended solution converges exponentially fast if and only if there exists a metric with zero curvature. We also give some results about the range of discrete Gaussian curvatures, which generalize Andreev-Thurston's theorem to some extent.

Deformations of coisotropic submanifolds for fibrewise entire Poisson structures

We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the $L_\infty$-algebra introduced by Oh-Park (for symplectic manifolds) and Cattaneo-Felder. In the symplectic case, we recover results previously obtained by Oh-Park. Moreover we consider the extended deformation problem and prove its obstructedness.