Semi-topological K-theory of Real Varieties

The semi-topological K-theory of real varieties, KR(−), is an oriented multiplicative (generalized) cohomology theory which extends the authors’ earlier theory, K(−), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles (sharpened to real semi-topological equivalence), consideration of Z/2-equivariant mapping spaces of morphisms of algebraic varieties to Grassmannian varieties, and consideration of the algebraic K-theory of real varieties. The authors verify that the semi-topological K-theory of a real variety X interpolates between the algebraic K-theory of X and Atiyah’s Real K-theory of the associated Real space of complex points, XR(C). The resulting natural maps of spectra K (X) → KR(X) → KRtop(XR(C)) satisfy numerous good properties: the first map is a mod-n equivalence for any projective real variety and any n > 0; the second map is an equivalence for smooth projective curves and flag varieties; the triple fits in a commutative diagram of spectra mapping via total Segre classes to a triple of cohomology theories. The authors also establish results for the semi-topological K-theory of real varieties, such as Nisnevich excision and a type of localization result, which were previously unknown even for complex varieties. Introduction In the papers [FW2], [FW3], we introduced and studied semi-topological Ktheory, which is a spectrum valued theory K(−) defined on the category of quasi-projective complex varieties. The definition of K was originally suggested in [F3], and more recently an equivalent theory defined for smooth, projective complex varieties, called holomorphic K-theory, has been studied by R. Cohen and P. Lima-Filho in [CL2]. The semi-topological K-theory of a complex variety X fits in between the algebraic K-theory of X and the topological K-theory of the associated topological space of complex points X(C): K(X) → K(X) → Ktop(X(C)). The theory K(X) is a good interpolation between K(X) and K(X): the map K(X) → K(X) induces a isomorphism on homotopy groups with finite coefficients, while the map K(X) → Ktop(X(C)) apparently induces an isomorphism on homotopy groups once the action of the so-called Bott element in K 2 (SpecC) is inverted. Moreover, we view K (X) as having intrinsic interest, Both authors were partially supported by the N.S.F. and the N.S.A. Typeset by AMS-TEX 1 2 ERIC M. FRIEDLANDER AND MARK E. WALKER ∗ for it can be viewed as the stabilization of function complexes of algebraic morphisms of X into Grassmannians, a topic of study in papers such as [Ki], [CLS]. Thus, computations of K(X) arising from either topological or algebraic Ktheory can provide information about such morphisms, whereas computations of invariants of certain moduli spaces (such as those in [Ki]) can provide information about algebraic K-theory. In this paper, we extend the definition of K to real varieties, establish numerous foundational properties of our theory, and compute various examples. We also show how this new theory, KR, is related to various other constructions in algebraic geometry and topology. As was the case for K(X), with a complex variety X , the initial motivation for the definition of KR(X), with X a real variety, is our intention of constructing a theory based on algebraic vector bundles and algebraic equivalence. A subtlety arises in that we must use the real analytic topology in our definition of the equivalence relation giving KR 0 (X) as a quotient of K 0 (X), for a real variety X . This equivalence relation, real semi-topological equivalence, provides an invariant finer than ordinary algebraic equivalence (as suggested by [Fu; 10.3]) and thus KR 0 (X) is an invariant more closely approximating K 0 (X) (cf. Proposition 1.6). As for most constructions of higher K-groups, to define KR q (X) for q > 0 we require some machinery to provide a suitable homotopy-theoretic group completion; we find the “machine” using E∞-operads convenient for most our purposes (see [M1]), although the equivalent “machine” stemming from Segal’s notion of a Γ-space (see [Se]) is used instead in Section 5. Thanks to a stabilization theorem proved in Section 7, we find that this homotopy-theoretic group completion can be viewed as a colimit of more familiar spaces of algebraic morphisms. In analogy with K, the theory KR fits in between the algebraic K-theory, K, of real varieties and the so-called Atiyah’s Real K-theory, KRtop, of Real spaces – i.e., spaces equipped with continuous involutions (cf. [At]). Namely, if X is a quasi-projective real variety, then writing XR(C) for the analytic space of complex points equipped with the involution given by complex conjugation, we have natural maps of spectra K(X) → KR(X) → KRtop(XR(C)). We mention three theorems which should give the reader some impression of how KR(X) relates to these otherK-theories and also to cohomology theories arising from cycles. To prove these theorems, we follow [FW3] in introducing a variation of KR(X), written K(∆top ×R X), which is weakly equivalent to KR (X) whenever X is projective (and weakly normal), but which is better behaved for arbitrary quasi-projective real varieties. Theorem 0.1. (cf. Corollary 3.10) For a projective real variety X, the natural map K q (X ;Z/n) → KR semi q (X ;Z/n) is an isomorphism for all q ≥ 0, n > 0. Theorem 0.2. (cf. Propositions 6.1 and 6.2) Suppose X is one of the following projective real varieties: (1) a smooth, projective real curve, or (2) G/P , where G SEMI-TOPOLOGICAL K-THEORY OF REAL VARIETIES 3 is one of the linear algebraic groups GLn,R, SLn,R, Spinn,R, or Sp2n,R and P is a parabolic subgroup containing a split Borel subgroup. Then the natural map KR q (X) → KR −q top(X) is an isomorphism for all q ≥ 0. Theorem 0.3. (cf. Theorem 8.8) Let X be a smooth, projective real variety. Then there is a natural commutative diagram K i (X) −−−−→ K alg i (∆ • top ×R X) −−−−→ KR −i top(XR(C))