Roy Joshua

Riemann–Roch for Algebraic Stacks: I

In this paper we establish Riemann–Roch and Lefschtez–Riemann–Roch theorems for arbitrary proper maps of finite cohomological dimension between algebraic stacks in the sense of Artin. The Riemann–Roch theorem is established as a natural transformation between the G-theory of algebraic stacks and topological G-theory for stacks: we define the latter as the localization of G-theory by topological K-homology. The Lefschtez–Riemann–Roch is an extension of this including the action of a torus for Deligne–Mumford stacks. This generalizes the corresponding Riemann–Roch theorem (Lefschetz–Riemann–Roch theorem) for proper maps between schemes (that are also equivariant for the action of a torus, respectively) making use of some fundamental results due to Vistoli and Toen. A key result established here is that topological G-theory (as well as rational G-theory) has cohomological descent on the isovariant etale site of an algebraic stack. This extends cohomological descent for topological G-theory on schemes as proved by Thomason.In this paper we establish Riemann–Roch and Lefschtez–Riemann–Roch theorems for arbitrary proper maps of finite cohomological dimension between algebraic stacks in the sense of Artin. The Riemann–Roch theorem is established as a natural transformation between the G-theory of algebraic stacks and topological G-theory for stacks: we define the latter as the localization of G-theory by topological K-homology. The Lefschtez–Riemann–Roch is an extension of this including the action of a torus for Deligne–Mumford stacks. This generalizes the corresponding Riemann–Roch theorem (Lefschetz–Riemann–Roch theorem) for proper maps between schemes (that are also equivariant for the action of a torus, respectively) making use of some fundamental results due to Vistoli and Toen. A key result established here is that topological G-theory (as well as rational G-theory) has cohomological descent on the isovariant étale site of an algebraic stack. This extends cohomological descent for topological G-theory on schemes as proved by Thomason.

Higher Intersection Theory on Algebraic Stacks: II

This is the second part of our work on the intersection theory of algebraic stacks. Here we establish the existence of Chern classes and Chern character for all Artin stacks of finite type over a field with values in our Chow groups. We also extend these to higher Chern classes and a higher Chern character for perfect complexes on an algebraic stack, taking values in cohomology theories of algebraic stacks that are defined with respect to complexes of sheaves on the big smooth site. We also provide an integral intersection pairing for all smooth Artin stacks which we show reduces to the known intersection pairing on the Chow groups of smooth Deligne-Mumford stacks modulo torsion. This involves showing the existence of Adams operations on the rational étale K-theory of all smooth Deligne-Mumford stacks. As a by-product of our techniques we also provide an extension of higher intersection theory to all schemes of finite type over a field.

Intersection cohomology of reductive varieties

We extend the methods developed in our earlier work to algorithmically compute the intersection cohomology Betti numbers of reductive varieties. These form a class of highly symmetric varieties that includes equivariant compactifications of reductive groups. Thereby, we extend a well-known algorithm for toric varieties.

Vanishing of odd dimensional intersection cohomology II

Abstract. For a variety where a connected linear algebraic group acts with only finitely many orbits, each of which admits an attractive slice, we show that the stratification by orbits is perfect for equivariant intersection cohomology with respect to any equivariant local system. This applies to provide a relationship between the vanishing of the odd dimensional intersection cohomology sheaves and of the odd dimensional global intersection cohomology groups. For example, we show that odd dimensional intersection cohomology sheaves and global intersection cohomology groups vanish for all complex spherical varieties.

Derived functors for maps of simplicial spaces

Abstract In this paper, we discuss in detail a site for simplicial spaces which is particularly suitable for defining derived functors for maps between simplicial spaces. It is shown that the derived category of sheaves on this site is closely related to the derived category of sheaves on another well-known site. Applications to algebraic group actions in positive characteristics are also discussed.

Künneth decompositions for quotient varieties

Abstract In this paper we discuss Kunneth decompositions for finite quotients of several classes of smooth projective varieties. The main result is the existence of an explicit (and readily computable) Chow-Kunneth decomposition in the sense of Murre with several pleasant properties for finite quotients of abelian varieties. This applies in particular to symmetric products of abelian varieties and also to certain smooth quotients in positive characteristics which are known to be not abelian varieties, examples of which were considered by Enriques and Igusa. We also consider briefly a strong Kunneth decomposition for finite quotients of projective smooth linear varieties.

Equivariant intersection cohomology of semi-stable points

The main result of the paper is that the equivariant intersection cohomology of the semi-stable points on a complex projective variety, for the action of a complex reductive group, may be determined from the equivariant intersection cohomology of the semi-stable points for the action of a maximal torus. It extends the work of Brion who considered the smooth case using equivariant cohomology. Equivariant intersection cohomology is a theory due to Brylinski and the second author. As an application, a surprising relation between the intersection cohomology of Chow hypersurfaces is established in the last section of the paper.

Toric residue codes: I

In this paper, we begin exploring the construction of algebraic codes from toric varieties using toric residues. Though algebraic codes have been constructed from toric varieties, they have all been evaluation codes, where one evaluates the sections of a line bundle at a collection of rational points. In the present paper, instead of evaluating sections of a line bundle at rational points, we compute the residues of differential forms at these points. We show that this method produces codes that are close to the dual of those produced by the first technique. We conclude by studying several examples, and also discussing applications of this technique to the construction of quantum stabilizer codes and also to decryption of toric evaluation codes.

The Intersection Cohomology and Derived Category of Algebraic Stacks

The present paper provides an extension of the theory of perverse sheaves to algebraic stacks and therefore to moduli problems ℚ-varieties, algebraic spaces, etc. We also include a detailed study of the intersection cohomology of algebraic stacks and their associated moduli spaces. Smooth group scheme actions on singular varieties and the associated derived category turn up as special cases of the more general results on algebraic stacks.

Spanier-Whitehead duality in étale homotopy

We construct a (mod-I) Spanier-Whitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. The Thom space of the normal bundle to imbedding any compact complex manifold in a large sphere as a real submanifold provides a Spanier-Whitehead dual for the disjoint union of the manifold and a base point. Our construction generalises this to any characteristic. We also observe various consequences of the existence of a (mod-I) Spanier-Whitehead dual. Introduction. In this paper we establish the existence of a (mod-i) SpanierWhitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. This generalises the familiar construction of the Spanier-Whitehead dual for a compact complex manifold. In the forthcoming papers [J-2 and J-3] we make use of this to establish a Becker-Gottlieb type transfer for proper and smooth maps of smooth quasi-projective varieties. Recall that associated to every finite spectrum X there exists another spectrum, denoted DX, and called the Spanier-Whitehead dual of X, which is characterised by the following property. Let EO denote the sphere spectrum, while E denotes any arbitrary spectrum. Then there exists a map ,u: E* X A DX of spectra which induces isomorphisms [u]: h-q(X, E) -* hq(DX, E) and [T,p]: h-q(DX, E) -* hq(X, E) for all q. Here r is the map interchanging the two factors X and DX while h* ( , E) (h* ( , E)) is the generalised homology (generalised cohomology, respectively) with respect to the spectrum E. (See [Sw, pp. 321-335] for a general reference on the familiar notion of Spanier-Whitehead duality in topology.) If X happens to be the suspension spectrum associated to a compact closed real manifold M, there exists an explicit geometric construction of its SpanierWhitehead dual. If a is the normal bundle to imbedding M in a large sphere as a smooth closed submanifold, then a suitable desuspension of the Thom space of this bundle forms a Spanier-Whitehead dual for M+. We observe that this construction therefore provides a Spanier-Whitehead dual for any compact complex manifold, by merely forgetting its complex structure. This construction is generalised here for projective and geometrically unibranched varieties over any algebraically closed field. Received by the editors November 14, 1984 and, in revised form, June 23, 1985. 1980 Mathematics Subject Classification. Primary 14F35; Secondary 14F20, 55P25, 55P42, 55N20. (@)1986 American Mathematical Society 0002-9947/86