The weight filtration for real algebraic varieties II: classical homology

Abstract

We associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology with Z2 coefficients an analog of the weight filtration for complex algebraic varieties. This complements our previous definition of the weight filtration of Borel-Moore homology.We define the weight filtration of the homology of a real algebraic variety by first addressing the case of smooth noncompact varieties. As in Deligne's definition [5] of the weight filtration for complex varieties, given a smooth variety X we consider a good compactification, a smooth compactification X of X such that D = X \ X is a divisor with normal crossings. Whereas Deligne's construction can be interpreted in terms of the action of a torus (S 1 ) N , we use the action of a discrete torus (S 0 ) N to define a filtration of the chains of a semialgebraic compactification of X associated to the divisor D. The resulting filtered chain complex is functorial for pairs (X, X) as above, and it behaves nicely for a blowup with a smooth center that has normal crossings with D.We apply a result of Guillén and Navarro Aznar [6, Theorem (2.3.6)] to show that our filtered complex is independent of the good compactification of X (up to quasi-isomorphism) and to extend our definition to a functorial filtered complex, the weight complex, that is defined for all varieties and enjoys a generalized blowup property (Theorem 7.1). For compact varieties the weight complex agrees with our previous definition [9] for Borel-Moore homology.We work with homology rather than cohomology to take advantage of the topology of semialgebraic chains [9, Appendix]. We denote by H k (X) the kth classical homology group of X, with compact supports and coefficients in Z 2 , the integers modulo 2. The vector space H k (X) is dual to H k (X), the classical kth cohomology group with closed supports. On the other hand, let H BM k (X) denote the kth Borel-Moore homology group of X (i.e. homology with closed supports) with coefficients in Z 2 . Then H BM k (X) is dual to H k c (X), the kth cohomology group with compact supports. Our work owes much to the foundational paper [6] of Guillén and Navarro Aznar. In particular we have been influenced by the viewpoint of section 5 of that paper, on the theory of motives. Using Guillén and Navarro Aznar's extension theorems, Totaro [13] observed that there is a functorial weight filtration for the cohomology with compact supports of a real analytic variety with a given compactification. In [9] we developed this theory in detail for real algebraic varieties, working with Borel-Moore homology. Our task was simplified by the strong additivity property of Borel-Moore homology (or compactly supported cohomology) [9, Theorem 1.

Real intersection homology

“…The complex of semialgebraic chains of X has the following geometric definition (see [15], Appendix). For k ≥ 0, let S k (X) be the Z/2 vector space generated by the closed semialgebraic subsets of X of dimension ≤ k. The chain group C k (X) is the quotient of S k (X) by the following relations:…”

“…(We overlooked this result in [15].) We have ∂(C c k (X)) ⊂ C c k−1 (X), and the homology of the chain complex (C c * (X), ∂) is canonically isomorphic to the simplicial homology H * (X; Z/2) with respect to a (locally finite) semialgebraic triangulation.…”

“…Remark 7.3. The adapted resolution construction induces a filtration on the complex of semialgebraic chains; see [15]. Question 7.4.…”

Complexe de poids des variétés algébriques réelles avec action

“…[A] ∈ N −k+1 C k (X) (on utilise l'une deséquivalences de la preuve de [14] Corollary 3.12 : X est compacte non-singulière de dimension k).…”

“…On montre que l'on peut trouver une fonction génériquement Nash-constructible sur X/G, divisible par 2 k+α , qui représente π * c. Pour cela, on applique le critère de l'éventail [14] Theorem 4.10à chacune des composantes irréductibles de chaque carte affine de l'adhérence de Zariski Y de X/G età la fonction On peut de plus supposer f 0 génériquement constructible en dimension k, en remplaçant, dans le raisonnement précédent, X par l'adhérence de Zariski du support de c. On a donc π * c ∈ N α C k (X/G).…”