Virtual Betti numbers of real algebraic varieties

Abstract

The weak factorization theorem for birational maps is used to prove that for all i ≥ 0 the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti number βi defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then βi(X) = βi(X \ Y ) + βi(Y ).We define an invariant of the Grothendieck ring K 0 (V R ) of real algebraic varieties, the virtual Poincaré polynomial β(X, t), which is a ring homomorphism K 0 (V R ) → Z [t]. For X nonsingular and compact, β(X, t) is the classical Poincaré polynomial for cohomology with Z 2

Equivariant virtual Betti numbers

“…As in the non-equivariant case ( [17]), this is a consequence of Poincaré duality and the existence of the long exact sequence of a pair for equivariant homology. We recall briefly the proof for the convenience of the reader.…”

“…For G the trivial group, the equivariant virtual Poincaré series β G is equal to the virtual Poincaré polynomial β (cf. [17,7]). …”

Virtual Poincaré polynomial of the link of a real algebraic variety

“…By evaluation of the virtual Poincaré polynomial at u = −1 one recovers the Euler characteristic with compact supports [12]. Proposition 1.2.…”

“…The virtual Poincaré polynomial, denoted by β, has been introduced in [12] as an algebraic invariant for Zariski constructible real algebraic sets. Then the invariance of the virtual Poincaré polynomial has been established under Nash diffeomorphisms [6], and finally in [13] under regular homeomorphisms.…”

The motivic real Milnor fibres

“…If we restrict ourself to quasi-projective varieties, taking the fixed point of the complex points X(C) of an algebraic variety X over R under the complex conjugation gives a real algebraic variety in the sense of [2], and we obtain that way a ring morphism from the Grothendieck rings K 0 (Var R ) of algebraic varieties over R to the Grothendieck rings K 0 (Var R ) of real algebraic varieties considered already in [13]. In particular we may consider a realization of K 0 (Var R ) into the real polynomial ring in one variable via the virtual Poincaré polynomial [13].…”