Moment graphs and representations

Peter Fiebig

Description

Niveau: Supérieur, Doctorat, Bac+8
Moment graphs and representations Jens Carsten Jantzen* In a 1979 paper Kazhdan and Lusztig introduced certain polynomials that nowadays are called Kazhdan-Lusztig polynomials. They conjectured that these polynomials de- termine the characters of infinite dimensional simple highest weight modules for complex semi-simple Lie algebras. Soon afterwards Lusztig made an analogous conjecture for the characters of irreducible representations of semi-simple algebraic groups in prime charac- teristics. The characteristic 0 conjecture was proved within a few years. Concerning prime characteristics the best result known says that the conjecture holds in all characteristics p greater than an unknown bound depending on the type of the group. In both cases the proofs rely on the fact (proved by Kazhdan and Lusztig) that the Kazhdan-Lusztig polynomials describe the intersection cohomology of Schubert varieties. It was then quite complicated to link the representation theory to the intersection coho- mology. In the characteristic 0 case this involved D–modules and the Riemann-Hilbert correspondence. The proof of the weaker result in prime characteristics went via quantum groups and Kac-Moody Lie algebras. In these notes I want to report on a more direct link between representations and cohomology. Most of this is due to Peter Fiebig. An essential tool is an alternative description of the intersection cohomology found by Tom Braden and Robert MacPherson. A crucial point is that on one hand one has to replace the usual intersection cohomology by equivariant intersection cohomology, while on the other hand one has to work with deformations of representations, i.

• intersection cohomology


• group


• b? ?


• mapping any


• prime characteristics


• inducing all


• open subset


• kazhdan-lusztig polynomials


• kac-moody algebras

Sujets

• Ergué-Armel
• Aarhus University
• Macpherson
• Intersection homology
• Open set
• Kazhdan?Lusztig polynomial

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 8
• Langue: English

Extrait

Moment graphs and representations

Jens Carsten Jantzen*

In a 1979 paper Kazhdan and Lusztig introduced certain polynomials that nowadays are called Kazhdan-Lusztig polynomials. They conjectured that these polynomials de- termine the characters of infinite dimensional simple highest weight modules for complex semi-simple Lie algebras. Soon afterwards Lusztig made an analogous conjecture for the characters of irreducible representations of semi-simple algebraic groups in prime charac- teristics. The characteristic 0 conjecture was proved within a few years. Concerning prime characteristics the best result known says that the conjecture holds in all characteristicsp greater than an unknown bound depending on the type of the group. In both cases the proofs rely on the fact (proved by Kazhdan and Lusztig) that the Kazhdan-Lusztig polynomials describe the intersection cohomology of Schubert varieties. It was then quite complicated to link the representation theory to the intersection coho- mology. In the characteristic 0 case this involvedD–modules and the Riemann-Hilbert correspondence. The proof of the weaker result in prime characteristics went via quantum groups and Kac-Moody Lie algebras. In these notes I want to report on a more direct link between representations and cohomology. Most of this is due to Peter Fiebig. An essential tool is an alternative description of the intersection cohomology found by Tom Braden and Robert MacPherson. A crucial point is that on one hand one has to replace the usual intersection cohomology byequivariantintersection cohomology, while on the other hand one has to work with deformationsof representations, i.e., with lifts of the modules to a suitable local ring that has our original ground field as its residue field. Braden and MacPherson looked at varieties with an action of an (algebraic) torus; under certain assumptions (satisfied by Schubert varieties) they showed that the equivari- ant intersection cohomology is given by a combinatorially defined sheaf on a graph, the moment graphof the variety with the torus action. Fiebig then constructed a functor from deformed representations to sheaves on a moment graph. This functor takes projective indecomposable modules to the sheaves defined by Braden and MacPherson. This is then the basis for a comparison between character formulae and intersection cohomology.

In Section 4 of these notes I describe Fiebig’s construction in the characteristic 0 case. While Fiebig actually works with general (symmetrisable) Kac-Moody algebras, I have restricted myself here to the less complicated case of finite dimensional semi-simple Lie algebras. The prime characteristic case is then discussed in Section 5, but with crucial proofs replaced by references to Fiebig’s papers.

* Mathematics Institute, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark

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The two middle sections 2 and 3 discuss moment graphs and sheaves on them. I de- scribe the Braden-MacPherson construction and follow Fiebig’s approach to a localisation functor and its properties. The first section looks at some cohomological background. A proof of the fact that the Braden-MacPherson sheaf describes the equivariant intersection cohomology was be- yond the reach of these notes. Instead I go through the central definitions in equivariant cohomology and try to make it plausible that moment graphs have something to do with equivariant cohomology. For advice on Section 1 I would like to thank Michel Brion and Jørgen Tornehave.

Moment graphs and representations

1 Cohomology

3

For general background in algebraic topology one may consult [Ha]. For more infor- mation on fibre bundles, see [Hm]. (I actually looked at the first edition published by McGraw-Hill.)

1.1.(A simple calculation) Consider the polynomial ringS=k[x 1 , x 2 , x 3 ] in three indeterminates over a fieldk. Setα=x 1 −x 2 andβ=x 2 −x 3  us determine the. Let followingS–subalgebra ofS 3 =S×S×S:

Z={(a, b, c)∈S 3 |a≡b(modSα), b≡c(modSβ), a≡c(modS(α+β))}

We have clearly (c, c, c)∈Zfor allc∈S; it follows thatZ=S(1,1,1)⊕Z ′ with

(1)

Z ′ ={(a, b,0)∈S 3 |a≡b(modSα), b∈Sβ, a∈S(α+β)} Any triple (b(α+β), bβ,0) withb∈Sbelongs toZ ′ . This yieldsZ ′ =S(α+β, β,0)⊕Z ′′ whereZ ′′ consists of all (a,0,0) witha∈S α∩S(α+β). Sinceαandα+βare non- associated prime elements in the unique factorisation domainS, the last condition is equivalent toa∈S α(α+β we get finally). So

Z=S(1,1,1)⊕S(α+β, β,0)⊕S(α(α+β),0,0)

(2)

SoZis a freeS–module of rank 3. ConsiderSas a graded ring with the usual grading doubled; so eachx i is homogeneous of degree 2. Then alsoS 3 andZ (2) says that we have an Noware naturally graded. isomorphism of gradedS–modules

Z≃S⊕Sh2i ⊕Sh4i

(3)

where quite generallyhniindicates a shift in the grading moving the homogeneous part of degreeminto degreen+m. The point about all this is that we have above calculated (in casek=C) the equi- variant cohomologyH T • (P 2 (C);C) whereTis the algebraic torusT=C × ×C × ×C × acting onP 2 (C) via (t 1 , t 2 , t 3 )[x:y:z] = [t 1 x:t 2 y:t 3 z] in homogeneous coordinates. Actually we have also calculated the ordinary cohomologyH • (P 2 (C);C) that we get (in this case) asZmZwheremis the maximal ideal ofSgenerated by thex i , 1≤i≤3. So we regain the well-known fact thatH 2r (P 2 (C);C)≃Cfor 0≤r≤2 while all remaining cohomology groups are 0.

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1.2.(Principal bundles) LetG Recall that abe a topological group.G–spaceis a topological spaceXwith a continuous actionG×X→XofGonX. IfXis aG–space, then we denote byXGthe space of all orbitsGxwithx∈Xendowed with the quotient topology: Ifπ:X→XGtakes anyx∈Xto its orbitGx, thenU⊂XGis open if and only ifπ −1 (U) is open inX. It then follows thatπis open sinceπ −1 (π(V)) = S g∈G gV for anyV⊂X. A (numerable)principalG–bundleis a triple (E, p, B) whereEis aG–space,Ba topological space andp:E→Bcontinuous map such that there exists a numerablea covering ofBby open subsetsUsuch that there exists a homeomorphism ϕ U :U×G→p −1 (U) withp◦ϕ U (u, g) =uandϕ U (u, gh) =g ϕ U (u, h) (1) for allu∈Uandg, h∈G numerability condition is automatically satisfied if. (TheBis a paracompact Hausdorff space. We assume in the following all bundles to be numerable.) Note that these conditions imply that the fibres ofpare exactly theG–orbits onE, that each fibrep −1 (b) withb∈Bis homeomorphic toG, and thatGacts freely onE: Ifg∈Gandx∈Ewithg x=x, theng also follows that It= 1.Gx7→p(x) is a homeomorphism fromEGontoBand thatpis open. For example the canonical mapp:C n+1 \ {0} →P n (C) is a principal bundle for the multiplicative groupC ×  we restrict. Ifpto the vectors of length 1, then we get a principal bundleS 2n+1 →P n (C) for the groupS 1 of complex numbers of length 1. IfGis a Lie group andHa closed Lie subgroup ofG, then the canonical map G→GHis a principal bundle forHacting onGby right multiplication. is a This fundamental result in Lie group theory. If (E, p, B) is a principal bundle for a Lie groupGand ifHis a closed Lie subgroup ofG, then (E, p, EH) is a principal bundle forHwherep:E→EHmaps anyv∈Eto itsH-orbitH v.

1.3.(Universal principal bundles () LetE, p, B) be a principal bundle for a topo- logical groupGand letf:B ′ →B onebe a continuous map of topological spaces. Then constructs aninducedprincipal bundlef ∗ (E, p, B) = (E ′ , p ′ , B ′  takes): OneE ′ as the fibre product E ′ =B ′ × B E={(v, x)∈B ′ ×E|f(v) =p(x)} and one definesp ′ as the projectionp ′ (v, x) =v. The action ofGonE ′ is given by g(v, x) = (v, gx); this makes sense asp(gx) =p(x) =f(v). Consider an open subsetU inBsuch that there exists a homeomorphismϕ U as in 1.2(1). ThenV:=f −1 (U) is open inB ′ , we have (p ′ ) −1 (V)⊂V×p −1 (U) and id V ×ϕ U induces a homeomorphism {(v, u, g)∈V×U×G|f(v) =u}→−(p ′ ) −1 (V),

hence using (v, g)7→(v, f(v), g) a homeomorphismψ V :V×G→(p ′ ) −1 (V) satisfying p ′ ◦ψ V (v, g) =vandg ψ V (v, h) =ψ V (v, gh) for allv∈Vandg, h∈G. One can show: Iff 1 :B ′ →Bandf 2 :B ′ →Bare homotopic continuous maps, then the induced principal bundlesf 1 ∗ (E, p, B) andf 2 ∗ (E, p, B) are isomorphic overB ′ . Here

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two principalG–bundles (E 1 , p 1 , B) and (E 2 , p 2 , B) are calledisomorphic overBif there exists a homeomorphismϕ:E 1 →E 2 withp 2 ◦ϕ=p 1 andϕ(g x) =g ϕ(x) for allx∈E 1 . A principal bundle (E G , p G , B G ) for a topological groupGis called auniversal prin- cipal bundleforGif for every principalG–bundle (E, p, B) there exists a continuous map f:B→B G such that (E, p, B) is isomorphic tof ∗ (E G , p G , B G ) overBand iffis uniquely determined up to homotopy by this property. Milnor has given a general construction that associates to any topological group a universal principal bundle. A theorem of Dold (inAnn. of Math.78(1963), 223–255) says that a principalG–bundle (E, p, B) is universal if and only ifEis contractible. In caseG=S 1  for any positive ConsiderMilnor’s construction leads to the following: integernthe principalG–bundlep n :E n G =S 2n+1 →B G n =P n (C have We) as in 1.2. natural embeddingsE G n →E G n+1 andB G n →B G n+1 induced by the embeddingC n →C n+1 mapping any (x 1 , x 2 ,    , x n ) to (x 1 , x 2 ,    , x n , embeddings are compatible with0). These the action ofGand with the mapsp n andp n+1  now the limits. Take E G = limE G n =S ∞ andB G = limB G n =P ∞ (C) −→−→ with the inductive topology. We get a mapp G :E G →B G inducing allp n , and (E G , p G , B G ) is then a universal principal bundle forG=S 1 . For the multiplicative groupG=C × get a universal principal bundle by aone can similar procedure: One sets nowE G n =C n+1 \ {0}andB G n =P n (C) with the canonical mapp n  Soand takes the limit as above. one gets nowE G =C ∞ \ {0}andB G =P ∞ (C). SinceS ∞ is a deformation retract ofC ∞ \ {0}and sinceS ∞ is contractible (e.g., by the preceding example and Dold’s theorem*), alsoC ∞ \ {0}is contractible. Therefore (E G , p G , B G ) is a universal principal bundle forG=C × . Remarks: 1) ( LetE G , p, B G ) be a universal principal bundle for a topological groupG. Dold’s theorem implies thatB G is pathwise connected. Furthermore one gets from the long exact homotopy sequence of this fibration: IfGis connected, thenB G is simply connected. 2) LetGbe a Lie group andHa closed Lie subgroup ofG. If (E G , p, B G ) is a universal principalG–bundle, then (E G , p, E G H) withp(x) =H xfor allx∈E G is a universal principalH–bundle: We noted at the end of 1.2 that we get here a principalH–bundle; it is universal by Dold’s theorem.

1.4.(Equivariant cohomology) LetGbe a topological group andXaG–space. We can associate to each principalG–bundle (E, p, B) a fibre bundle (X E , q, B): We letG act onX×Ediagonally, i.e., viag(x, y) = (gx, gy), and setX E equal to the orbit space (X×E)G. We defineqbyq(G(x, y)) =p(y). Letπ:X×E→X E denote the map sending each element to its orbit underG. Consider an open subsetUinBwith a homeomorphismϕ U :U×G→p −1 (U) as in 1.2(1). We haveq −1 (U) = (X×p −1 (U))Gandπ −1 (q −1 (U)) =X×p −1 (U). Now b ψ U :X×U×G−→π −1 (q −1 (U)),(x, u, g)7→(gx, ϕ U (u, g)) * Aten.wikipedia.org/wiki/Contractibility of unit sphere in Hilbert space you can find the standard proofs.

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is a homeomorphism (the composition of (x, u, g)7→(gx, u, g) with id X ×ϕ U ) that isG– equivariant if we letGact onX×U×Gviah(x, u, g) = (x, u, hg this action). ForX×U is homeomorphic to (X×U×G)G, mapping (x, u) toG(x, u,1). It follows that we get a homeomorphism

ψ U :X×U−→q −1 (U) =π −1 (q −1 (U))G,(x, u)7→G(x, ϕ U (u,1))

This shows that (X E , q, B) is a locally trivial fibration with all fibres homeomorphic toX. We get also that (X×E, π, X E ) is a principalG any open subset–bundle: ForUas aboveψ b U ◦(ψ U −1 ×id G ) is a homeomorphism fromq −1 (U)×Gontoπ −1 (q −1 (U)) satisfying the conditions in 1.2(1).

Apply this construction to a universal principalG–bundle (E G , p, B G ). In this case we use the notationX G = (X×E G )Gand get thus a fibre bundle (X G , q, B G  then). We define theequivariant cohomologyH • G (X;C) of theG–spaceXas the ordinary cohomology ofX G : H • G (X;C) =H • (X G ;C)(1) (We could of course also use other coefficients thanC.)

At first sight it looks as if this definition depends on the choice of the universal principalG–bundle (E G , p, B G  Suppose us show that this choice does not matter.). Let that (E ′ G , p ′ , B ′ G ) is another universal principalG–bundle. ConsiderX×E G ×E ′ G as a G–space withGacting on all three factors. We get natural maps q 1 : (X×E G ×E ′ G )G−→(X×E G )Gandq 2 : (X×E G ×E ′ G )G−→(X×E ′ G )G These are locally trivial fibrations with fibres homeomorphic toE ′ G (forq 1 ) or toE G (forq 2  example the construction above of a fibre bundle yields). Forq 1 if we start with the G–spaceE ′ G and the principalG–bundle given by the orbit mapX×E G →(X×E G )G. Since the fibreE ′ G of the locally trivial fibrationq 1 is contractible, the long exact homotopy sequence of the fibration shows thatq 1 induces isomorphisms of all homotopy groups. Now a result of Whitehead implies thatq 1 ∗ is an isomorphism of cohomology algebrasH • ((X×E G )G;C)− ∼ →H • ((X×E G ×E ′ G )G;C same argument applies). The toq 2 and get thus an isomorphism (q 2 ∗ ) −1 ◦q 1 ∗ :H • ((X×E G )G;C)− ∼ →H • ((X×E ′ G )G;C) Denote this isomorphism for the moment byα(E ′ G , E G ). If now (E ′ G ′ , p ′′ , B ′ G ′ ) is a third universal principalG–bundle, then one checks thatα(E ′ G ′ , E G ) =α(E ′ G ′ , E ′ G )◦α(E ′ G , E G ). We can now formally defineH • G (X;C) as the limit of the family of allH • ((X×E G )G;C) and of allα(E ′ G , E G ) over all universal principalG–bundles (E G , p, B G ).

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1.5.(Elementary properties) Let againGbe a topological group and (E G , p, B G ) a universal principalG–bundle. Iff:X→Yis a morphism ofG–spaces (i.e., a continuous G–equivariant map), thenf×id is a morphismX×E G →Y×E G ofG–spaces and induces a continuous mapf:X E →Y E ,G(x, z)7→G(f(x), z We) of the orbit spaces. get thus a homomorphism f ∗ :H • G (Y;C)→H • G (X;C) (1) in the equivariant cohomology. IfXis a point, then (X×E G )G= ({pt} ×E G )Gidentifies withE G G≃B G e.W get thus H • G (pt;C)≃H • (B G ;C),(2) an isomorphism of algebras. More generally, ifGacts trivially on a topological spaceX, then (X×E G )Gis homeomorphic toX×(E G G) underG(x, y)7→(x, Gydleiyalus)S.K¨unotheformneth an isomorphismH • G (X;C)≃H • (X;C)⊗H • (B G ;C). For an arbitraryG–spaceXthe map sending all ofXto a point is a morphism ofG we get a homomorphism–spaces. ThereforeH • (B G ;C)→H • G (X;C) that makes H • G (X;C) into anH • (B G ;C)–algebra. Anyf ∗ as in (1) is then a homomorphism of H • (B G ;C)–algebras. IfHis a closed subgroup ofG, then we can regardGHas aG–space. We get then a homeomorphism

(GH×E G )G−→E G H, G(gH, x)7→H g −1 x The inverse map takes anyH–orbitH xtoG(1H, x). IfGis a Lie group and ifHis a closed Lie subgroup ofG, thenE G →E G His a universal principalH–bundle, as noted in 1.3. So in this case we can takeB H =E G Hand get an isomorphism H • G (GH;C)− ∼ →H • (B H ;C)(3) The structure as anH • G (pt;C)–algebra onH • G (GH;C) is induced by the homomor- phismq ∗ withq:E G H→B G ,q(H x) =p(x). 1.6.(Tori) IfGandG ′ are topological groups, if (E, p, B) is a (universal) principalG– bundle and if (E ′ , p ′ , B ′ ) is a (universal) principalG ′ –bundle, then (E×E ′ , p×p ′ , B×B ′ ) is a (universal) principal (G×G ′ )–bundle. Consider an (algebraic) torusT=C × ×C ×  ×  × C × (d we getfactors). Then principalT–bundles (E T n , p n , B T n ) settingE n T = (C n+1 \ {0}) d andB n T =P n (C) d , cf. 1.3, and we get a universal principalT–bundle (E T , p T , B T ) withE T = (C ∞ \ {0}) d and B T =P ∞ (C) d  in 1.3 we can identify. AsE T andB T as inductive limits of allE T n and allB T n respectively. IfXis aT–space, thenX T = (X×E T )Tis the inductive limit of allX T n = (X× E T n )T. Since we are looking at cohomology with coefficients in a field, we get therefore H T i (X;C) = limH i ((X×E n T )T;C all) fori∈N, (1) ←−

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cf. [Ha], Thm. 3F.5. We get in particular that eachH i (B T ;C) is the inverse limit of theH i (P n (C) d ;C), n∈N. There is an isomorphism of graded rings C[x 1 , x 2 ,    , x d ](x 1 n+1 , x 2 n+1 ,    , x n d +1 )− ∼ →H • (P n (C) d ;C)(2) HereC[x 1 , x 2 ,    , x d ] is the polynomial ring overCindindeterminates, graded such that eachx i  So forhas degree 2.n≥1 the map in (2) sends P d i=1 Cx i bijectively to H 2 (P n (C) d ;C the inclusion). Furthermoreι n :P n (C) d →P n+1 (C) d induces forn≥1 an isomorphismι ∗ n :H 2 (P n+1 (C) d ;C)− ∼ →H 2 (P n (C) d ;C that one gets). (RecallP n+1 (C) fromP n (C) by adjoining a (2n and use the cell decomposition to compute the+ 2)-cell cohomology.) It follows that we get an isomorphism of graded rings C[x 1 , x 2 ,    , x d ]− ∼ →H • (B T ;C)(3) Consider the maximal compact subgroupK=S 1 ×S 1 ×    ×S 1 (dfactors) ofT. SettingE K n = (S 2n+1 ) d ⊂E T n andB n K =B T n we get principalK–bundles (E n K , p ′ n , B n K ) withp ′ n the restriction ofp n  we get a universal principal. SimilarlyK–bundle (E K , p K , B K ) withE K = (S ∞ ) d ⊂E T andB K =B T . As in the case ofTwe get for anyK–spaceX • thatH K (X;C) is the inverse limit of allH • ((X×E K n )K;C). We can regard anyT–spaceXas aK–space by restricting the action ofTtoK. The inclusion ofE n K = (S 2n+1 ) d intoE n T = (C n+1 \ {0}) d induces a continuous map of orbit spaces ψ: (X×E n K )K→(X×E n T )T We can coverB n T by open subsetsUsuch that there exists a homeomorphismϕ U :U×T→ p n −1 (U) as in 1.2(1) and such thatϕ U restricts to a similar homeomorphismϕ ′ U :U×K→ (p ′ n ) −1 (U the inverse images of). ThenUboth in (X×E n K )Kand in (X×E T n )Tidentify withX×U, cf. 1.4. Under this identificationψcorresponds to the identity map. Therefore ψand induces an isomorphism of cohomology groups.is a homeomorphism  inverse Taking limits (or working directly withE K andE T ) we get thus an isomorphism H T • (X;C)− ∼ →H • K (X;C)(4) 1.7.(Line bundles need a more canonical description of the isomorphism 1.6(3).) We This involves (complex) line bundles and their Chern classes. LetGbe a topological group and (E, p, B) a principalG associate to any–bundle. We continuous group homomorphismλ:G→C × a line bundleL(λ) =L(λ;E) onBas follows: Denote byC λ theG–space equal toCas a topological space such thatg a=λ(g)afor all g∈Ganda∈C. SetL(λ) = (C λ ×E)Gand defineq λ :L(λ)→Bbyq λ (G(a, x)) =p(x). Then (L(λ), q λ , B Each fibre It is in fact a line bundle:) is a fibre bundle as described in 1.4. − q λ 1 (p(x)) withx∈Egets a vector space structure such thatC→q λ −1 (p(x)),a7→G(a, x) is an isomorphism of vector spaces. This structure is independent of the choice ofx inp −1 (p(x)) =GxsinceGacts linearly onC λ . The homeomorphismsψ U as in 1.4 are compatible with this structure.

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If alsois a continuous group homomorphismG→C × , then one gets an isomorphism of line bundles L(λ)⊗ L()− ∼ → L(λ+) (1)

where we use an additive notation for the group of continuous group homomorphisms fromGtoC × ; so we have (λ+) (g) =λ(g)(g). Let alsoG ′ be a topological group with a principalG ′ –bundle (E ′ , p ′ , B ′ ). Suppose that we have continuous mapsϕ:B ′ →Bandψ:E ′ →Eand a continuous group homo- morphismα:G ′ →Gsuch thatp◦ψ=ϕ◦p ′ andψ(hy) =α(h)ψ(y) for allh∈G ′ and y∈E ′ . Then the pull-backϕ ∗ L(λ) of the line bundleL(λ) underϕis isomorphic to the line bundleL(λ◦α): L(λ◦α)− ∼ →ϕ ∗ L(λ)(2) Recall that the pull-backϕ ∗ L(λ) is the fibre product ofB ′ andL(λ) overB, hence consists of all pairs (b ′ , G(a, x)) withb ′ ∈B ′ ,a∈C, andx∈E. The isomorphism in (2) sends any orbitG ′ (a, y) witha∈Candy∈E ′ to (p(y), G(a, ψ(y))). This result implies in particular for any inner automorphism Int (g):h7→ghg −1 ofG that L(λ◦Int (g))− ∼ → L(λ)(3) Take aboveG ′ =Gand (E ′ , p ′ , B ′ ) = (E, p, B the assumptions are satisfied by). Then ϕ= id B andα= Int (g) if we setψ(x) =gxfor allx∈E. Consider for exampleG=C × and the principal bundle (C n+1 \ {0}, π,P n (C)) with the canonical mapπ. Chooseλ:C × →C × as the mapg7→g −1  the map. Then C λ ×(C n+1 \ {0})→C n+1 ×P n (C),(a, v)7→(av,Cv) is constant on the orbits ofGand induces an isomorphism of line bundles L(λ)− ∼ → {(w,Cv)∈C n+1 ×P n (C)|w∈Cv}(4) Here the right hand is usually known as thetautologicalline bundle onP n (C). In algebraic geometry this bundle is usually denoted byO(−1).

1.8.(Chern classes) Ifq:L →Bis a (complex) line bundle on a topological spaceB, then the Chern class ofLis an elementc 1 (L)∈H 2 (B;C it is an element). (Actually inH 2 (B;Z) that we here replace by its image inH 2 (B;C line bundles).) Isomorphic have the same Chern class. IfL ′ is another line bundle onB, then we havec 1 (L ⊗ L ′ ) = c 1 (L) +c 1 (L ′ ). Iff:B ′ →Bis a continuous map, then one getsf ∗ (c 1 (L)) =c 1 (f ∗ L) in H 2 (B ′ ;C). LetGbe a topological group and (E, p, B) a principalG a continuous–bundle. Given group homomorphismλ:G→C × we get a line bundleL(λ) onBas in 1.7, hence a Chern classc 1 (λ) =c 1 (L(λ)) inH 2 (B;C also). If:G→C × is a continuous group homomorphism, then we get

c 1 (λ+) =c 1 (λ) +c 1 ()

(1)