Th�or�me de Pfister pour les vari�t�s et anneaux de Witt r�duits

Unramified cohomology and quadratic forms

“…It gives an affirmative answer to a question raised by Knebush [Kn1]. Theorem 1.1 (Mahé) [M2,Th.4.1] [M1] If X = Spec A is an affine variety, the cokernel of the signature map Λ is a 2-primary torsion group. More precisely, if Spec r A = C ∪ D where C, D are two disjoint open sets, there exist N ∈ N and q ∈ W (X) such thatq = 2 N on C, andq = 0 on D. In addition, there exists a function w : N → N with the following property: for any affine variety X of Krull dimension d over a real closed field, the cokernel of Λ is killed by 2 w(d) .…”

The Witt group of real surfaces

“…Now suppose that V is a variety over R, and V (R) has ν > 0 connected components. The torsion subgroup of W (V ) is 2-primary (Pfister [40]), and Mahé [30] and Brumfiel [8] proved that the signature W (V ) → Z ν maps the torsionfree part of W (V ) isomorphically onto a subgroup of finite index in Z ν . By [23, 2.4], the signature factors through W R(V top ).…”

“…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.…”

“…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.…”

On the Pythagoras numbers of real analytic surfaces☆

“…Firstly, in Section 2 we discuss sums of squares of totally positive elements, much inspired on Mahe's results in [Mh2]. Section 3 is devoted to the proof of Theorem 1.3, based on Section 2 and a relative algebrization lemma in the style of classification of singularities, based on Tougeron's Implicit Function Theorem.…”