Riemann-Roch and smoothings of singularities

Abstract

IF A4 is a (not necessarily compact) complex n-manifold with the property that for some compact K c M the subalgebra of H*(M -K) generated by the rational Chern classes is trivial in dim 2 n, then Chern numbers for M not involving the top class c,(M) may be defined almost as usual and-assuming that M has finite Betti numbers--c,(M] may be replaced by the euler characteristic x(M). We encounter this situation when dealing with isolated singularities of complex spaces. Specifically, let X be a contractible Stein space of pure dimension n with isolated singular point x E X and let 7~: x -+ X be a resolution with exceptional set E = IE- l(x)_ Then a deep result of0. Gabber implies that H*(z) --* H*(x - E) is the zero map in dim 2 n (if no coefficient groups are mentioned I mean rational cohomolog). Hence _? admits Chern numbers. From this it is not hard to see that the Milnor fibre X, of any smoothing of X (which after all can be piven the same boundary as 2) also admits Chern numbers. An interesting point is that, since H*(X,) is trivial in dim > n, any nonzero Chern number of X, is a universal linear combination of the euler characteristic and (if n = 2m is even) ci The purpose of this paper is to relate these Chern numbers with analytic invariants of the smoothing or of the singularity itself. Our most general result in this direction is a Riemann-Roth defect theorem (3.3), whose proof depends on a globalization property of smoothings which we establish in an appendix. This globalization property (which was known to hold for certain classes of singularities, e.g. complete intersections and curve singularities) has been previously used for similar purposes by Milnor, Deligne, Laufer, and Wahl[14]. The fact that any smoothing can be globalized is especially relevant to this last paper, as many of its assertions need this as an extra hypothesis (see the remark at the end of this paper). In practice it may be hard to use our results for singularities of dimension n > 2. Nevertheless we do have some applications to higher dimensional singularities. For instance, we prove that for any isolated singularity X pure odd dimension n, and smoothing component S of X, dim (S) + l/(n - l)! (euler characteristic of a general fibre over S) is independent of S. Furthermore, we find a new smoothability condition for odd dimensional X.