Deformations Of Complex Structure Research Articles

Local contributions to global deformations of surfaces

In [23], Schiffer and Spencer prove that all small deformations of complex structure on a compact Riemann surface may be realized by altering the complex structure only within an arbitrarily small neighborhood of a point on the surface. It seems interesting in general to consider whether it is possible to construct deformations of an algebraic variety or complex manifold from deformations of neighborhoods of certain subvarieties. Further motivation for trying to understand the role of subvarieties in deformations is suggested by the instability under deformation of the Neron-Severi group, i.e., the group of divisors modulo numerical equivalence. As an example, one may consider the family of affine surfaces Vt: x 2 + y 2 + z 2 = t z (t is a parameter; V o has a nodal singularity at the origin). This family admits a resolution {Xt} --, {V~}, with {X,} a smooth family of non-singular surfaces, and each X~ is a minimal resolution of V t ([4] or [5]). The exceptional curve E in X o is a IP 1 with self-intersection 2 which does not appear in any Xt, for t#:0. One may ask whether every smooth surface X with such a curve in it admits a one-parameter family of deformations arising from this local model. Furthermore, if X contains several disjoint such curves, does each one independently contribute one dimension to the moduli of X? The Hartogs ' theorem of [14] says that one cannot simply plumb in the local deformation, leaving the structure of X unchanged outside a small neighborhood of E. Moreover, old examples of Segre [26] show that the nodes on certain hypersurfaces V in IP 3 are not "independent", i.e., there aren't enough deformations of the resolution X of V to allow for a one-dimensional contribution from each node. Theorem (3.7) of this paper says that the regularity of the Kuranishi variety of X is sufficient for the deformations of X to realize independently

Deformations of Complex Structures on Certain Non-Compact Complex Manifolds

In this paper we continue to explore the ideas initiated in [7] and [8]. We show that it is possible to give a nice description of complex structures on a wide class of non-compact complex manifolds by extending Kuranishi's methods used in the compact case [101 and [11]. Our results can be summarized as follows. Let X be a complete Kahler complex manifold such that (a) the curvature is bounded, (b) every point of X admits a normal coordinate neighborhood (with respect to the fixed metric on X) whose geodesic radius is bounded from below.

ON THE THEORY OF VARIATION OF STRUCTURES DEFINED BY TRANSITIVE, CONTINUOUS PSEUDOGROUPS

The theory of deformations of structure was begun some years ago by Kodaira and Spencer [6], who laid the foundations for the theory of variation of complex structure. Later, together with Nirenberg, they established the fundamental existence theorem. On the other hand, it was soon realized that the deformations of complex structure was (at least conceptually) a special case of the theory of variation of the structure defined by a transitive, continuous pseudo-group Γ acting on a manifold X (briefly, we say a Γ-structure on X). In the complex analytic case, Γ = ΓGjL(M>c) is the pseudo-group of all local bi-holomorphic transformations in C. The theory of deformations of other special Γ-structures has been discussed in [4] and [6]. Following this, Spencer succeeded in [7] in establishing the basic mechanism for deformations of what might be called flat Γ-structures i.e. Γ-structures for which a certain jet bundle has coordinate cross-sections of a suitable type. Furthermore, he proved the existence theorems in case ΓdΓGZ,(W)O. On the other hand, in [I] (see also [9]), the deformation theory of certain non-flat Γ-structures has found important applications, and, in [3], the general theory of a (perhaps non-transitive) Γ-structure has been developed. The existence theorems for certain special cases have also been given in [3]. The purpose of the present paper is to discuss the deformations of a general Γ-structure. That this is possible is not surprising, and it may even be possible to extend [7] to the general case. However, we have chosen a somewhat different approach based on the following heurestic consideration : Let X have a Γ-structure and suppose that this structure is given by a certain Maurer-Cartain Φ defined on a suitable principal bundle Q-*X. Thus, Φ is a Lie algebra valued one form on Q which satisfies the Maurer-Cartan equation.

An Example of a Non-Kahlerian Complex-Analytic Deformation of Kahlerian Complex Structures

An Example of a Non-Kahlerian Complex-Analytic Deformation of Kahlerian Complex Structures