THE WEIGHT FILTRATION FOR REAL ALGEBRAIC VARIETIES II: CLASSICAL HOMOLOGY
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THE WEIGHT FILTRATION FOR REAL ALGEBRAIC VARIETIES II: CLASSICAL HOMOLOGY CLINT MCCRORY AND ADAM PARUSINSKI 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 on Borel-Moore homology. We define the weight filtration of the homology of a real algebraic variety by first addressing the case of smooth, not necessarily compact, 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 an action of a torus (S1)N on a neighborhood of the divisor at infinity, we use an action of a discrete torus (S0)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 Guillen 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
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- weight complex
- deligne's definition
- double cover
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- varieties
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THE WEIGHT FILTRATION FOR REAL ALGEBRAIC VARIETIES CLASSICAL HOMOLOGY
´ CLINT MCCRORY AND ADAM PARUSINSKI
Abstract.algebraic variety a filtered chain complex, theWe associate to each real weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology withZ2coefficients an analog of the weight filtration for complex algebraic varieties. This complements our previous definition of the weight filtration on Borel-Moore homology.
II:
We define the weight filtration of the homology of a real algebraic variety by first addressing the case of smooth, not necessarily compact, varieties. As in Deligne’s definition [5] of the weight filtration for complex varieties, given a smooth varietyXwe consider a good compactification, a smooth compactificationXofXsuch thatD=X\Xis a divisor with normal crossings. Whereas Deligne’s construction can be interpreted in terms of an action of a torus (S1)Ninfinity, we use an action of aon a neighborhood of the divisor at discrete torus (S0)Nto define a filtration of the chains of a semialgebraic compactification ofXassociated to the divisorD. 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 withD. WeapplyaresultofGuille´nandNavarroAznar([6]Theorem(2.3.6))toshowthat our filtered complex is independent of the good compactification ofX(up to quasi-isomorphism) and to extend our definition to a functorial filtered complex, theweight complex, which is defined for all varieties, and which enjoys a generalized blowup prop-erty (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 (cf.[9], Appendix). We denote byHk(X) thekth classical homology group ofX, with compact supports and coefficients inZ2 The, the integers modulo 2. vector spaceHk(X) is dual toHk(X), the classicalkth cohomology group with closed supports. On the other hand, letHkBM(X) denote thekth Borel-Moore homology group ofX, with closed supports and coefficients inZ2. ThenHkBM(X) is dual toHck(X), the kth cohomology group with compact supports. Ourworkowesmuchtothefoundationalpaper[6]ofGuill´enandNavarroAznar.In particular we have been influenced by the viewpoint of section 5 of that paper, on the theoryofmotives.UsingGuill´enandNavarroAznar’sextensiontheorems,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);cf. classical homology or cohomology one[9], Theorem 1.1. For
Date: February 14, 2012. 2000Mathematics Subject Classification.Primary: 14P25. Secondary: 14P10. 1
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´ C. MCCRORY AND A. PARUSINSKI
does not have such an additivity property, and so the present construction of the weight filtration is more involved.
In section 1 below we define the weight filtration of a smooth, possibly noncompact, varietyX, using a good compactificationXwith divisorD we define aat infinity. First semialgebraic compactificationX0, thecorner compactificationofX, and then we use a canonical action of a discrete torus{1−1}NonX0to define a filtration of the semialgebraic chain group ofX0. The filteredweight complexis obtained by an algebraic construction, theDeligne shift section . In2 we analyze the relation of the weight complex to the homologicalGysin complexof the divisorD. Section 3 contains the proof of the crucial fact that the weight complex is functorial for pairs (X X). Sections 4, 5, and 6 treat the blowup properties of the weight complex of a smooth variety. In section 7 we use the theoremsofGuill´enandNavarroAznartoextendthedefinitionoftheweightcomplexto singular varieties, and we describe some elementary examples. The appendix (section 8) is devoted to a canonical filtration of theZ2group algebra of a discrete torus group. This is in effect a local version of the weight filtration.
0.1.Real algebraic varieties.By areal algebraic varietywe mean an affine real alge-braic variety in the sense of Bochnak-Coste-Roy [3]: a topological space with a sheaf of real-valued functions isomorphic to a real algebraic setX⊂RNwith the Zariski topology and the structure sheaf of regular functions. Aregular function onXis the restriction a rational function onRNthat is everywhere defined onX. By aregular mappingwe mean a regular mapping in the sense of Bochnak-Coste-Roy [3]. For instance, the set of real points of a reduced projective scheme overR, with the sheaf of regular functions, is an affine real algebraic variety in this sense. This follows from the fact that real projective space is isomorphic, as a real algebraic variety, to a subvariety of an affine space ([3] Theorem 3.4.4). We also adopt from [3] the notion of an algebraic vector bundle. We recall that such a bundle is, by definition, a subbundle of a trivial vector bundle, and hence it is the pullback of the universal vector bundle on the Grassmannian, and its fibers are generated by global regular sections;cf.[3] Chapter 12. By asmooth real algebraic varietywe mean a nonsingular affine real algebraic variety.
1.The weight filtration of a smooth variety
In this section we define the weight filtration of the classical homology of a smooth varietyX. We use a smooth compactificationXwith a normal crossing divisor at infinity to define a semialgebraic compactificationX0ofXand a surjective mapπ:X0→Xwith finite fibers. This map is used to define the weight filtration of the semialgebraic chain complex ofX0withZ2coefficients. Thus we obtain the weight filtration of the homology ofX0, which is canonically isomorphic to the homology ofX will prove in Section 7. We that this filtration ofH∗(X) does not depend on the choice of compactificationX. 1.1.The corner compactification.LetMbe a compact smooth real algebraic variety and letD⊂Mbe a smooth divisor. Associated toDthere is an algebraic line bundleL overMthat has a sectionssuch thatDis the variety of zeroes ofs. LetS(L) be the space of oriented directions in the fibers ofL. It can be given the structure of a real algebraic variety as follows. By [3] Remark 12.2.5,Lis isomorphic to an algebraic subbundle of the trivial bundleM×RN. Denote by Ψ :L→RNthe regular map defined by this isomorphism. The scalar product onRNdefines a regular metric onL identify. WeS(L) with the unit zero-sphere bundle ofL; that is, with the real algebraic variety Ψ−1(SN−1).
WEIGHT FILTRATION II 3 This structure is uniquely defined. Indeed, the standard projectionL\M→Ψ−1(SN−1) is a regular map, and therefore two such unit sphere bundles are biregularly isomorphic. Finally,Lis the pullback of the universal line bundle onPN−1under the regular map M→PN−1induced by Ψ. ThusS(L) is a smooth real algebraic variety, and the projectionπL:S(L)→Mis an algebraic double covering. Now the subvarietyπL−1DofS(L) is the zero set of the regular functionϕ:S(L)→Rdefined byϕ(x `)∙`=s(x), wherex∈Mand`is a unit vector in the fiberLx=πL−1(x). Note that the generatorτof the group of covering transformations ofS(L) changes the sign ofϕ, forϕ(τ(x `)) =ϕ(x−`) =−ϕ(x `). LetXbe a smoothn-dimensional variety, and letXbe agood compactificationofX (cf.[12], p. 89):Xis a compact smooth variety containingX, andD=X\Xis a divisor with simple normal crossings. ThusDis a finite union of smooth codimension one subvarietiesDiofX, (1.1)D=[Di i∈I and the divisorsDi Notemeet transversely. that we do not assume the divisorsDiare irreducible. Fori∈I, letLibe the line bundle onXassociated toDiand letsibe a section of e Lithat defines the divisorDi. Letπe:X→Xbe the covering of degree 2|I|defined e as the fiber product of the double coversπLi:S(Li)→X, and letϕei:X→Rbe the pullback of the functionϕi:S(Li)→Rcorresponding to the sectionsi, so that the variety eπ−1Diis the zero space ofϕei. Thecorner compactificationofXassociated to the good e compactification (X D) is the semialgebraic setX0⊂Xdefined by e X0= Closure{xe∈X|ϕei(ex)>0 i∈I}. In the terminology of [11] (§3.2),X0is the varietyXcut along the divisorD. Letπ: e X0→Xbe the restriction of the covering mapπe:X→X. e LetTbe the group of covering transformations of the covering spaceeπ:X→X, with τei∈Tthe pullback of the nontrivial covering transformationτiof the double coverπLi: S(Li)→X. There is a canonical isomorphismθ:T→G, whereGis the multiplicative group of functionsg:I→ {1−1}, given byθ(τei) =gi, withgi(i) =−1 andgi(j) = 1 for i6=j . Toemphasize the role of the groupGwe prefer to consider e ξ= (eπ:X→X) as aprincipalG-bundlefor the groupG={1−1}Iand then ei(gi∙ex) =−ϕei(ex) (1.2)eϕj(gi∙ex) =ϕej(xe) i6=j. ϕ IfU⊂Xis contractible thenξ|Uis trivial, i.e. (1.3)πe−1(U)'U×G
as aGuniquely defined by a choice of isomorphism is Thisprincipal bundle.x∈Uand a pointex∈πe−1(x), which we identify via (1.3) with (x1)∈U×G. Proposition 1.1.The semialgebraic mapπ:X0→Xis surjective. Ifx∈XletJ(x) = {i∈I|x∈Di}andG(x) ={g∈G|g(i) = 1 i /∈J(x)}. The fiberπ−1(x) ={ex∈ πe−1(x);ϕei(ex)>0fori∈J(x)}. Thusπ−1(x)is a regular orbit of the action ofG(x)on Xe; i.e., aG(x)-torsor. Hence the number of points inπ−1(x)is2|J(x)|.
