Local contributions to global deformations of surfaces

Abstract

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