Related papers: Kuranishi type Moduli Spaces for proper CR submers…
A theorem of Kuranishi tells us that the moduli space of complex structures on any smooth compact manifold is always locally a finite-dimensional space. Globally, however, this is simply not true; we display examples in which the moduli…
We prove a Kuranishi-type theorem for deformations of complex structures on ALE K\"ahler surfaces. This is used to prove that for any scalar-flat K\"ahler ALE surface, all small deformations of complex structure also admit scalar-flat…
Motivated by the definition of homotopy $L_\infty$ spaces, we develop a new theory of Kuranishi manifolds, closely related to Joyce's recent theory. We prove that Kuranishi manifolds form a $2$-category with invertible $2$-morphisms, and…
We construct the Kuranishi spaces, or in other words, the versal deformations, for the following classes of connections with fixed divisor of poles $D$: all such connections, as well as for its subclasses of integrable, integrable…
Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov-Schmidt reduction for…
Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in $\Bbb C^p$, for some $p>0$) or differentiable (parametrized by an open neighborhood of $0$ in $\Bbb…
A Kuranishi space is a topological space with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on moduli spaces of J-holomorphic curves in symplectic geometry. This paper is a brief introduction to the…
We introduce $L_{\infty}$-Kuranishi spaces by associating, to each chart, $L_{\infty}[1]$-algebras defined on open neighborhoods of the zero points of the Kuranishi section. We show that these objects collectively form a category, which…
A Kuranishi space is a topological space with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on moduli spaces of J-holomorphic curves in symplectic geometry. Let Y be an orbifold and R a commutative…
We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic…
This is a comment on the Kuranishi method of constructing analytic deformation spaces. It is based on a simple observation that the Kuranishi map can always be inverted in the category of $L_{\infty}$-algebras. The $L_{\infty}$-structure…
Moduli spaces of stably irreducible sheaves on Kodaira surfaces belong to the short list of examples of smooth and compact holomorphic symplectic manifolds, and it is not yet known how they fit into the classification of holomorphic…
In complete analogy with the Beltrami parametrization of complex structures on a (compact) Riemann surface, we use in this paper the Kodaira-Spencer deformation theory of complex structures on a (compact) complex manifold of higher…
We describe explicitly the possible degenerations of a class of double Kodaira fibrations in the moduli space of stable surfaces. Using deformation theory we also show that under some assumptions we get a connected component of the moduli…
Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such…
Cohesive modules give a dg-enhancement of the bounded derived category of coherent sheaves on a complex manifold via superconnections. In this paper we discuss the deformation theory of cohesive modules on compact complex manifolds. This…
This paper constructs a Kuranishi structure for the moduli stack of holomorphic curves in exploded manifolds. To avoid some technicalities of abstract Kuranishi structures, we embed our Kuranishi structure inside an ambient moduli stack of…
Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such…
We present the extended Kuranishi space for Kodaira surface as a non-trivial example to Kontsevich and Barannikov's extended deformation theory. We provide a non-trivial example of Hertling-Manin's weak Frobenius manifold. In addition, we…
This is the second in a series of five papers math.DG/0211294, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally…