Classes Of Singular Varieties Research Articles

GROTHENDIECK GROUPS AND A CATEGORIFICATION OF ADDITIVE INVARIANTS

A topologically-invariant and additive homology class is mostly not a natural transformation as it is. In this paper we discuss turning such a homology class into a natural transformation; i.e. a "categorification" of it. In a general categorical set-up we introduce a generalized relative Grothendieck group from a cospan of functors of categories and also consider a categorification of additive invariants on objects. As an example, we obtain a general theory of characteristic homology classes of singular varieties.

Chern class identities from tadpole matching in type IIB and F-theory

In light of Sen's weak coupling limit of F-theory as a type IIB orientifold, the compatibility of the tadpole conditions leads to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. We present the physical argument leading to the identity, and a mathematical derivation of a Chern class identity which confirms it, after taking into account singularities of the relevant loci. This identity of Chern classes holds in arbitrary dimension, and for varieties that are not necessarily Calabi-Yau. Singularities are essential in both the physics and the mathematics arguments: the tadpole relation may be interpreted as an identity involving stringy invariants of a singular hypersurface, and corrections for the presence of pinch-points. The mathematical discussion is streamlined by the use of Chern-Schwartz-MacPherson classes of singular varieties. We also show how the main identity may be obtained by applying `Verdier specialization' to suitable constructible functions.

MULTIPLIER IDEAL SHEAVES ON FIBRED PRODUCTS OF CERTAIN SINGULAR VARIETIES

In [3], we discussed the compatibility of multiplier ideal sheaves with fibred products. Multiplier ideal sheaves on singular varieties generalize in a natural way the ℚ-divisors on nonsingular varieties. However, there was one central result in [3] which we proved only for nonsingular varieties. In this paper we extend this to a certain class of singular varieties.

Limits of Chow Groups, and a New Construction of Chern-Schwartz-MacPherson Classes

We define an `enriched' notion of Chow groups for algebraic varieties, agreeing with the conventional notion for complete varieties, but enjoying a functorial push-forward for arbitrary maps. This tool allows us to glue intersection-theoretic information across elements of a stratification of a variety; we illustrate this operation by giving a direct construction of Chern-Schwartz-MacPherson classes of singular varieties, providing a new proof of an old (and long since settled) conjecture of Deligne and Grothendieck.

Differential forms with logarithmic poles and Chern-Schwartz-MacPherson classes of singular varieties

We express the Chern-Schwartz-MacPherson class of a possibly singular variety in terms of the total Chern class of a bundle of differential forms with logarithmic poles. As an application, we obtain a formula for the Chern-Schwartz-MacPherson class of a hypersurface of a nonsingular variety, in terms of the Chern-Mather class of a suitable sheaf.

The Hodge conjecture for a certain class of singular varieties

In [J] there is stated a version of the Hodge conjecture for complex singular varieties (reduced separated algebraic schemes over C), generalizing the popular version of the conjecture. A statement of the general Hodge conjecture for the singular case, extending the amended version by Grothendieck, is given in [L2], using the techniques in [J]. The statement is this. Let Z be a reduced separated algebraic scheme over C. The lowest weight part W−i Hi (Z ,Q) of the BorelMoore homology Hi (Z ,Q) defines a Hodge structure of weight −i . Recall the filtration by niveau, given by