Mark E. Walker

Semi-topological K -theory using function complexes

Abstract The semi-topological K -theory K ∗ semi (X) of a quasi-projective complex algebraic variety X is based on the notion of algebraic vector bundles modulo algebraic equivalence. This theory is given as the homotopy groups of an infinite loop space K semi (X) which is equipped with maps K alg (X)→ K semi (X) , K semi (X)→ K top (X an ) whose composition is the natural map from the algebraic K -theory of X to the topological K -theory of the underlying analytic space X an of X . The theory K semi (X) defined and studied here is equivalent (when X is projective and weakly normal) to the so-called “holomorphic K -theory”, K hol (X) , of projective varieties, which is studied by Cohen and Lima-Filho. We give an explicit description of K 0 semi ( X ) in terms of K 0 ( X ), a description of K q semi (−) in terms of K 0 semi (−) for projective varieties, a Poincare duality theorem for projective varieties, and a computation of K semi (X) whenever X is a product of projective spaces or a smooth complete curve. For X a smooth quasi-projective variety, there are natural Chern class maps from K ∗ semi (X) to morphic cohomology compatible with similarly defined Chern class maps from algebraic K -theory to motivic cohomology and compatible with the classical Chern class maps from topological K -theory to the singular cohomology of X an .

The

Recent advances in computational techniques for K-theory allow us to describe the K-theory of toric varieties in terms of the K-theory of fields and simple cohomological data.

K

The morphic Abel-Jacobi map is the analogue of the classical Abel-Jacobi map one obtains by using Lawson and morphic (co)homology in place of the usual singular (co)homology. It thus gives a map from the group of r-cycles on a complex variety that are algebraically equivalent to zero to a certain “Jacobian” built from the Lawson homology groups viewed as inductive limits of mixed Hodge structures. In this paper, we define the morphic Abel-Jacobi map, establish its foundational properties, and then apply these results to the study of algebraic cycles. In particular, we show the classical Abel-Jacobi map (when restricted to cycles algebraically equivalent to zero) factors through the morphic version, and show that the morphic version detects cycles that cannot be detected by its classical counterpart — that is, we give examples of cycles in the kernel of the classical Abel-Jacobi map that are not in the kernel of the morphic one. We also investigate the behavior of the morphic Abel-Jacobi map on the torsion subgroup of the Chow group of cycles algebraically equivalent to zero modulo rational equivalence.

-theory of toric varieties

We establish the existence of Adams operations on the members of a filtration of K-theory which is defined using products of projective lines. We also show that this filtration induces the gamma filtration on the rational K-groups of a smooth variety over a field of characteristic zero.

The morphic Abel–Jacobi map

We observe that there is an equivalence between the singularity category of an affine complete intersection and the homotopy category of matrix factorizations over a related scheme. This relies in part on a theorem of Orlov. Using this equivalence, we give a geometric construction of the ring of cohomology operators, and a generalization of the theory of support varieties, which we call stable support sets. We settle a question of Avramov about which stable support sets can arise for a given complete intersection ring. We also use the equivalence to construct a projective resolution of a module over a complete intersection ring from a matrix factorization, generalizing the well-known result in the hypersurface case.

Adams Operations for Bivariant K-Theory and a Filtration Using Projective Lines

We show that if X is a toric scheme over a regular ring containing a field then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was known in characteristic 0. The affine case of our result was conjectured by Gubeladze.

Matrix factorizations in higher codimension

In this paper, we introduce the “semi-topological K-homology” of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasiprojective complex variety Y coincides with the connective topological Khomology of the associated analytic space Y an. From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the “Bott inverted” semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of Xan. In combination with a result of Friedlander and the author [12, 3.8], this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason’s celebrated theorem that “Bott inverted” algebraic K-theory with Z/n coefficients coincides with topological K-theory with Z/n coefficients.

The K-theory of toric varieties in positive characteristic

We present a novel proof of Thomasons theorem relating Bott inverted algebraic K-theory with finite coefficients and etale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic K-homology, and proving it satisfies etale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varieties. Then, we prove algebraic K-homology and algebraic K-theory (each taken with finite coefficients) coincide on smooth varieties upon inverting the Bott element.

Semi-Topological K-Homology and Thomason's Theorem

Abstract Using elementary observations concerning the simplicial group obtained by applying the functor GL to a connected simplicial ring, we obtain computations of the weight zero pieces of two proposed filtrations of the K -theory space of a regular ring.

Thomason's theorem for varieties over algebraically closed fields

Hochsters theta invariant is defined for a pair of finitely generated modules on a hypersurface ring having only an isolated singularity. Up to a sign, it agrees with the Euler invariant of a pair of matrix factorizations. Working over the complex numbers, Buchweitz and van Straten have established an interesting connection between Hochsters theta invariant and the classical linking form on the link of the singularity. In particular, they establish the vanishing of the theta invariant if the hypersurface is even-dimensional by exploiting the fact that the (reduced) cohomology of the Milnor fiber is concentrated in odd degrees in this situation. In this paper, we give purely algebraic versions of some of these results. In particular, we establish the vanishing of the theta invariant for isolated hypersurface singularities of odd dimension in characteristic p > 0, under some mild extra assumptions. This confirms, in a large number of cases, a conjecture of Hailong Dao.