The Witt group of real surfaces

Abstract

Let V be an algebraic variety defined over R, and V top the space of its complex points. We compare the algebraic Witt group W (V ) of symmetric bilinear forms on vector bundles over V , with the topological Witt group W R(V top ) of symmetric forms on Real vector bundles over V top in the sense of Atiyah, especially when V is 2-dimensional. To do so, we develop topological tools to calculate W R(V top ), and to measure the difference between W (V ) and W R(V top ).If V is a smooth algebraic surface defined over the real numbers R, the Witt group W (V ) is a finitely generated abelian group, whose rank equals the number ν of components of the 2-manifold V (R) [30]. Sujatha has given formulas for the torsion subgroup in [48]; although the formulas she gives are not always easy to compute, they involvé etale cohomology and are thus mostly topological in nature. Indeed, Cox' theorem [14] states that H nHere V top is the 4-manifold underlying the complex variety V × R C, G is the cyclic group G = Gal(C/R) of order 2, acting on V top by complex conjugation, and H * G is Borel cohomology. Together with Schlichting, we introduced the topological Witt group W R(X) of a G-space X in [23] (see Definition 2.1 below), and showed that the natural map W (V ) → W R(V top ) is always an isomorphism modulo bounded 2-primary torsion; for curves it is an isomorphism; for surfaces, the kernel and cokernel of the map W (V ) → W R(V top ) have exponent 2 by [23, Thm. 8.7].One of the main goals of this paper is to develop topological tools to compute W R(X) when X has dimension ≤ 4, since if V is an algebraic surface then W R(V top ) is very close to W (V ). The second goal is to more precisely measure the difference between these two invariants in various examples.Weibel was supported by NSA and NSF grants.Proof. It is classical [15, Thm. E] that I d+1 (F ) is a torsionfree abelian group. Hence the subgroup I d+1 (V ) is also torsionfree. Since the element '2' of W (V ) lies in I(V ) and I n (V ) ⊆ I n (V ), 2 d+1 ∈ 2I d (V ) ⊆ I d+1 (V ). Finally, (iii) is immediate from (ii) and the fact that the torsionfree part has rank ν.When dim(V ) = 2, I 3 (V ) is torsionfree and I(V ) tors has exponent 4.Following Sujatha [48], let H 0 tors (V, H n ) denote the kernel of the stable map H 0 (V, H n ) → H 0 (V, H d+1 ) ∼ = (Z/2) ν . Sujatha described the torsion subgroup I(V ) tors as follows. Proposition 1.8 (Sujatha). When V is a smooth surface over R, there is a short exact sequence 0 → H 0 tors (V, H 2 ) → I(V ) tors → H 0 tors (V, H 1 ) → 0. Proof. Sujatha proves in [48, 2.1, 2.2] that the maps I(V ) tors /I 2 (V ) tors → H 0 tors (V, H 1 ), I 2 (V ) tors /I 3 (V ) tors → H 0 tors (V, H 2 ) are isomorphisms, and I 3 (V ) tors = 0 by Proposition 1.7.

Real K-theories

“…One considers a space X with involution and complex vector bundles on X, with an involutive antilinear action compatible with the involution on X, Atiyah denoted this theory by KR(X) and, among other things, he showed the interest of this theory in real operator theory and in real Algebraic Geometry. This last point of view was illustrated in recent publications [10] [11].…”