Some Remarks on Isolated Singularity and Their Application to Algebraic Manifolds

Abstract

In 1954 F. Hirzebruch [8] obtained an interesting formula which makes it possible to determine the alternating sum X] 5 ( -z) 9 dim H q (V, J2 p (L fc )) for any complete intersection V of hypersurfaces in a complex projective space and for any k^Z where L is the analytic line bundle over V induced by hyperplanesection. He further determined dimH^y, J2 P ) by using some vanishing theorem. In the author's knowledge, however, the genera] dim H q (V, J2 P (Z/)) seem not to have been determined yet. In this note we shall give a formula which determines directly Part I is concerned with the general theory of isolated singularity and is of preparatory nature. The readers who have known the standard of the theory (e.g. Greuel [4]) may bypass it after they become familiar with the terminology and notations. Part II begins with the study of C*-actions over isolated singularities. The main theme there is to compute the characters of the representations of C* over various cohomology groups attached to the singularities, we apply it to the cones associated with algebraic manifolds and prove the required formula finally.The almost all results obtained in this paper have already been announced in [11], [12].Part I: General Theory I. PreliminariesWe shall often denote by (X, x) the pair of an analytic space X

The Jacobian algebra of a graded Gorenstein singularity

“…The first map is 0 except in weight 0; by [13] [27], Theorem B). If A is graded and normal with isolated singularity, then interior multiplication by the Euler derivation induces a graded exact sequence 0 -") Hm () -') Hm (-l) + Hm} (-2).…”

“…The last assertion is subtle and is a characteristic 0 fact; the proof we give (Proposition 1.4) is essentially due to Naruki [13], and uses a Gysin map. (This approach also implies there are 1 + emdim A generators for the module of derivations (1.12).…”

Almost split sequences and Zariski differentials

“…J. Wahl pointed out that the "if" part of Conjecture 2.4 holds in the complex-analytic category. This follows from a result of Naruki [Nar,2.1] (see [Wahl,(1.2)]) and Proposition 2.2.…”

The homological index and the De Rham complex on singular varieties

“…Let X be an (n + 1)-dimensional isolated complete intersection singularity and X = f −1 (0), where f : X → C, f (o) = 0, is a flat holomorphic map such that f | X −{o} is regular. In other terms, the singularity X is the hypersurface section of X defined by f (see [27]). Then the sequence of…”