Cohomology, symmetry, and perfection

Publicacions Ma emá iques, Vol 36 (1992), 407-420 . A bs ac 2 . 3 . COHOMOLOGY, SYMMETRY, AND PERFECTION EMILI BIFET * We explain he philosophy behind he compu a ions in [BDP] and place hem in a wide concep ual se ing . We also ou line, o o ic a ie ies, he esul ing equi a ian app oach o some key esul s in ha heo y . 1 . Symme y In many si ua ions ha a ise in Algeb aic Geome y one is in e es ed in compu ing he mul iplica i e s uc u e o he cohomology ing H* (X), wi h a ional coe icien s say, o some algeb aic a ie y X . Examples o such si ua ions include o ic a ie ies, comple e quad ics, comple e symme ic a ie ies [DP1-2], . . . Some imes, as in he examples jus men ioned, he a ie y X is endowed wi h symme ies ha e lec he ac ion o some algeb aic g oup G on i . In hese cases he e is a ecipe, inspi ed by he wo k o M . F . A iyah and R . Bo [AB1], ha o en wo ks : Find a s ongly G-pe ec decomposi ion o X (c . Sec ion 3 below o a p ecise s a emen .) In he examples abo e, his is simply he decomposi ion in o o bi s . In gene al he e is a na u al candida e : he Kemp -Hesselink s a i ica ion o X [H], [K], [N], a na u al ou g ow h o D . Mum o d's Geome ic In a ian Theo y . Wi h he help o his decomposi ion compu e he equi a ian co- homology ing HG(X) . (A his poin i may also be na u al o apply he machine y o he localiza ion heo em [AB2], [Hs] ; in so doing one usually ob ains o he in e es ing desc ip ions o Hc(X) . ) Reco e H* (X) om HG (X) . *This pape is dedica ed o he memo y o my iend Pe e Menal . I chose he p esen opic o his occasion because i was he subjec o ou las con e sa ion . I miss him e y much . 40 8  E . BIFET This ecipe is jus one mo e ins ance o he old philosophy o using any symme ies ha may be p esen in he p oblem in o de o simpli y i . We shall wo k, o simplici y, wi h equi a ian cohomology de ined in e ms o homo opy quo ien s Le . he Bo el cons uc ion (see Sec ion 2 below .) Bu he knowledgeable eade could subs i u e h ougliou HG(X) by he smoo h ~-adic cohomology o he algeb aic s ack de e - mined by he G- a ie y X . He would hus gain he ad an age o ha ing a comple ely algeb aic heo y wi h, as a bonus, an in e es ing a i hme ic wis [B2] . In he examples men ioned ea lie , he ecipe wo ks and gi es e y explici esul s . We shall conside below in some de ail he case o o ic a ie ies, bu he eade is ad ised o look a [BDP] o a ho ough ea men o h ee examples om he p esen poin o iew . In ac one o he aims o his pape is o be e explain he philosophy behind hose compu a ions and o place hem in a wide concep ual se ing . Ano he aim o he pape is o ou line in he las sec ion an "equi a i- an " app oach o some key esul s in he heo y o o ic a ie ies . This app oach cla i ies, I belie e, he na u e o h ee esul s . The ex o he i s h ee sec ions ollows closely a alk deli e ed a he Uni e si y o Copenhagen in July 1 .989 on he occasion o he Zeu hen Symposium . I would like o hank S . Kleiman and A . Tho up o o ganizing ha con e ence and c ea ing a e y iendly a hmosphe e . 2 . Cohomology Suppose ha a opological g oup G ac s p ope ly on a space X . The (equi a ian ) cohomology o he G-space X should be he cohomology o he quo ien X/G . Un o una ely, unless G is ac ing eely on X, he usual opological quo ien does no p o ide a use ul heo y . In gene al, i is necessa y o ind he igh no ion o quo ien . The bes no ion is p obably ob ained by aking he quo ien in he 2-ca ego y o oposes [SGA4] . Howe e , o simplici y, we shall wo k he e wi h a homo opy quo ien XG . ( In a pu ely algeb aic con ex he ole o XG would be played by he algeb aic s ack de e mined by he G- a ie y X ; he equi a ian cohomology would jus be he smoo h ~-adic cohomology o his s ack . ) Be o e we can desc ibe his no ion o quo ien , howe e , i is necessa y o look closely a he case whe e X is a poin Le . he heo y o cha ac e is ic classes . Recall ha a classi ying space o p incipal G-bundles is by de ini ion a space BG oge he wi h a uni e sal p incipal G-bundle EG o e i . By uni e sal we mean ha isomo phism classes o p incipal G- ib e bundles COHOMOLOGY, SYMMETRY, AND PERFECTION  409 o e a nice space X co espond na u ally o homo opy classes o maps om X o BG . The co espondence is gi en by pulling-back he uni e sal bundle EG . The cohomology ing H* (BG) is by de ini ion he ing o cha ac e is ic classes o G . Example 1 . G= C' . This is simply he heo y o line bundles, and he classi ying space is he in ini e p ojec i e space PC . I s cohomology ing H*(BG) = Z[cl] is a polynomial ing in one a iable o deg ee wo . Example 2 . G=T = C lx . . . x C" (an algeb aic o us .) In his case we ha e : BT- BC x x . . . x BC x and i s cohomology is a polynomial algeb a in se e al a iables (as many as ac o s .) In ac , i X(T) deno es he g oup o algeb aic cha ac e s o T, hen H*(BT) - Sym*X(T) . In pa icula H 2 (BT) - X (T) . Example 3 . G = GL  (C) . The classi ying space is he in ini e di- mensional G assmannian and H* (BG) = Z[cl, . . . , c,,] whe e he a iables e2, 1 <_ i <_ n, ha e deg ee 2i and co espond o he Che n classes . In gene al, i T is a maximal o us in G, we ha e ( wi h a ional coe icien s ) H* (BG) =H*(BT ) W whe e W = NG(T)/T is he Weyl g oup o (G, T) and he igh hand side deno es he sub ing o in a ian s . We a e now eady o desc ibe he homo opy quo ien men ioned ea - lie . This is gi en by he Bo el cons uc ion XG ob ained a e exchang- ing he ib e G o he uni e sal bundle EG wi h X Le . XG=EGxGX=(EGxX)/G whe e G ac s by g - (e, x) = (eg -1 , gx) . De ini ion . The equi a ian cohomology HG(X) is by de ini ion he cohomology o he Bo el cons uc ion XG . No e ha he e is a ib a ion (2 .1)  X -+ XG --> BG . 410  E . BIFET The spec al sequence o his ib a ion is he key e he heed s ep in he ecipe . This is based on wo k o P . Deligne [De], V . A . Ginzbu g [G], F . Ki wan [K], . . . He e ollow some o he p ope ies o equi a ian cohomology wi h a- ional coe icien s [Hs], [AB2] : a) I G ac s eely en X, hen HG(X) = H*(XIG) . b) I T is a maximal o us in G and W = NG(T)/T is he Weyl g oup, hen HG(X) = HT(X)w whe e he igh hand side is he sub ing o W-in a ian s . c) I X has a single o bi , hen HG(X) - H* (B H) whe e H is he s abilize o any poin . d) I K is a maximal compac subg oup o G, hen Hc(X) = Hix(X) . One o he easons equi a ian cohomology is easie o compu e han o dina y cohomology is ha i has many mo e "poin s" . Le me y o explain his . Mos succes ul calcula ions o cohomology achie e hei objec i e by exp essing he cohomology o he space unde conside a ion (e .g . p ojec i e space P') in e ms o ha o spaces o which i is al eady known (e .g . cells .) Ul ima ely, howe e , hey educe he compu a ion e ha o he cohomology o a poin . I one hinks o o dina y cohomology as being he case G= 1 o he equi a ian one, hen i is clea ha he poin s coincide wi h he o bi s . Thus in he equi a ian heo y e e y o bi gi es ise o a "poin ", and he e a e as many poin s as he e a e conjugacy classes o subg oups in G . The equi a ian cohomology o such a poin H is p ecisely he ing o H-cha ac e is ic classes Le . he cohomology o he classi ying space o H . I ollows ha in he equi a ian heo y he e is much mo e eedom o mo emen . Ano he impo an ea u e o equi a ian cohomology is ha he e is a heo y o equi a ian Che n classes . A G-linea iza ion o a ec o bundle F o e X is an ac ion u : G x F -> F which is linea en he ib es and u ns he p ojec ion 7 : F ---> X in o a G-equi a ian map Le . 7 (g - x) = g - 7 (x) o e e y g E G and e e y xE F . No e ha he homo opy quo ien FG p o ides us wi h a ec o bundle o e XG . The equi a ian Che n classes o (F, u) a e by de ini ion he Che n classes o FG . This akes a mos simple o m o a line bundle o e an o bi . In his case he equi a ian Che n class c(L, u) is de e mined by he (3 .2) COHOMOLOGY, SYMMETRY,AND PERFECTION  411 iso opy ac ion (cha ac e ) o he s abilize on he ib e o L o e he poin . Ac ually hese no ions ind hei mos na u al o mula ion when exp essed in e ms o algeb aic s acks . Fo example a G-linea ized Gx- module is simply a module o he s uc u e shea o he algeb aic s ack de e mined by he G- a ie y X . 3 . Pe ec ion Le X be a smoo h complex algeb aic a ie y, and le he algeb aic g oup G ac on X . Suppose S C X is a closed G-in a ian smoo h sub a ie y and le U = X -S be he complemen a y open se . Unde hese condi ions, he e is a long exac sequence ( he equi a ian Thom- Gysin sequence, see [AB1] o example ) . . .  HG 2codimS(S) is HG(X) - HG(U)  . . . Mo eo e he composi e o he maps hen, o example, we ha e HG 2codimS(S) HG(X) 1 es ic ion Hc(S) is mul iplica ion by he Eule class e(NSIx) (= op Che n class in his con ex ) o he no mal bundle N s 1 x . I his long exac sequence spli s in o sho exac sequences 0 -> HG 2codimS(S) _, Hc(X) _ HG(U) -> 0 bG(X) = bG(U) + bG 2codimS(S) and one can deduce he equi a ian Be i numbe s o X om hose o S and U . In [AB1] A iyah and Bo made he ollowing undamen al obse a- ion : I e(NS I x ) is no a ze o-di iso in he ing H* (S), hen he mo phisms s a e injec i e and he long exac sequences (3 .1) spli in o sho exac sequences (3 .2) . This mo i a es : 412  E . BIFET De ini ion . We say ha a decomposi ion X= Si U . . .USN is s ongly G-pe ec i : 1) Each S i is bo h smoo h and G-in a ian . 2) Fo each k, Xk =S1 U . . . USk is an open subse o X . 3) Fo each pai (Sk, Xk), k > 1, he Eule class e(NS,_IX,) is a non-ze o di iso . In [AB1] a decomposi ion is de ined o be G-pe ec i he long ex- ac sequence de e mined by each pai (Sk,Xk) spli s in o sho exac sequences . In his case one has an iden i y o equi a ian Poinca é se ies I is clea ha s ongly G-pe ec implies G-pe ec . An immedia e consequence o he de ini ion is P oposi ion . I {S2}1<á<N is a s ongly G-pe ec decomposi ion o X, hen o e e y pa ial union Xk he mo phism induced by he es ic- ions o he s a a (3 .3)  HG(Xk) - H Hc(SZ) 1<¡<k is injec i e . PC(x) _  2 .codimSi . PG(S2) 1<i<N P oo .. Fo k = 1, i is ob ious . Suppose i holds o k - 1 ; i su lces o show ha he mo phsm a e injec i e . Conside he diag am Hi-2codimSk (sk) G HG(Xk) - HG(Xk-1) x H * (Sk) HG(Xk-1 U sk)  HG(Xk-1) HG(Sk) - P oo . Since HT(P) . Bu , i COHOMOLOGY, SYMMETRY, AND PERFECTION  41 3 Now, i 71(a) = 0, hen he e is a b E HG 2codiMSk (Sk) such ha a = ~(b) . Bu , om 0 = (a) = («b)) = b U e(NS, 1x k ) i ollows ha b = 0 and he e o e a = « b) = 0 . a Thus, in p inciple, i ose knows he cohomology ings o he s a a and one con ols he injec ion abo e, i is possible o desc ibe he cohomology ing o X . This is he eason we singled ou his no ion o special conside a ion . A iyah and Bo also gi e an in ini esimal c i e ios o e(NSIx) o be a non-ze o di iso (see [AB1 P oposi ion 13 .4 .]) I is p o ed in [K] using his c i e ios ha he Kemp -Hesselink s a i ica ion [H] o a G- a ie y is s ongly G-pe ec in he abo e sense . Le us enuncia e his las c i e ios in he case o an o bi : P oposi ion . Le C be a G-o bi in he smoo h algeb aic a ie y X . Choose a poin P E C and iden i y C wi h G/Gp, whe e Gp is he s abilize o P . I he iso opie ac ion o a maximal o us T in Gp on he no mal space NOIx(P) has no non-ze o ixed poin s, hen e(NoIx) is a non-ze o di iso . Hc(0) = Hcp (P) - HT(P) is an embedding, i su ices o see ha e = e(N(DIx) is non-ze o in NoIx(P) =  ®  Cxi X¡EX(T) is he weigh decomposi ion, hen e=l1xi :7É o . Since all xz a e non-ze o . The conside a ions abo e mo i a e : De ini ion . We say ha X is a pe ec embedding ( egula embedding in [BDP], bu his was a bad choice o which I plead guil y) p o ided a) Each o bi closu e C is smoo h and i is he ans e sal in e sec- ion o he codimension one o bi closu es ha con ain i . b) Fo e e y P E 0, he s abilize Gp has a dense o bi in he no mal space NOIX(P) . To any pe ec embedding X we associa e a simplicial complex Cx = (V, S) as ollows : 41 4  E . BIFET 1 . V = { 1 O is an o bi o codimension one} . 2 . I' C V is a simplex i , and only i , E (No e ha ? is a simplex .) O 7 ~ ? . I is easy o show ha he simplexes a e in one- o-one co espondence wi h he o bi s . I is clea ha he decomposi ion o X in o o bi s is in his case s ongly G-pe ec . The algeb aic a ie ies men ioned abo e, namely o ic a ie ies and comple e symme ic a ie ies (in pa icula comple e quad ics), p o ide examples o pe ec embeddings . In [BDP] an explici desc ip ion, based on hese ideas, is gi en o he equi a ian cohomol- ogy ing o any pe ec embedding . I should be possible o ex end hese esul s o he case o a well beha ed G- a ie y and he Kemp -Hesselink s a i ica ion . 4 . An example : o ic a ie ies We shall illus a e he gene ali ies o he p eceding sec ions wi h he conc e e case o o ic a ie ies . Le T be an algeb aic o us . A o ic a ie y is a no mal algeb aic T- a ie y X con aining T as a dense o bi ( his includes he equi emen ha he s abilize a he poin s o his o bi be i ial .) Re e en es [D], [F], [O] p o ide e y good exposi ions o he basic heo y o hese a ie ies . A simple example is he a ine plane wi h he ac ion o T= C" x C X gi en by ( 1, 2)(xl>x2) = ( 1X1, 2x2) This ac ion has ou o bi s, namely he o igin, he wo punc u ed axes and he o us T i sel . This can be gene alized o he ac ion - (x1, x2) = ( x1 x l, x2 x2) de e mined by any in eg al basis {X1, X2} o he cha ac e g oup X (T) - Z 2 . This ac ion induces an ac ion on he ing o egula unc ions on he a ine plane, he ing o polynomials in wo a iables, gi en by ( ) (x) = ( -1 x) . The weigh unc ions (Le . he unc ions such ha = X o some cha ac e X called he weigh o ) a e p ecisely he monomials a-Xi 1 X2 2 and hei weigh is n1( - X1)+ n2 ( - X2) . These weigh s span a COHOMOLOGY, SYMMETRY, AND PERFECTION  41 5 cope a in X(T) ®R . Con e sely we can eco e he a ie y, including he T-ac ion, by aking he Spec um o he monoid algeb a C[a nX(T)] o , wha is essen ially he same, he algeb a homomo phisms om C[a n X(T)] o C . In gene al, a ine o ic a ie ies can be cons uc ed as ollows . We deno e Y(T) = Hom(C", T) he g oup o one-pa ame e subg oups o he algeb aic o us T . (Recall ha he e is a pai ing X(T) x Y(T) -~ Z gi en by aking (X, w) o be he unique in ege such ha X(M( )) = (X,w) o all E C X .) Fi s we conside a cone a = {Al il + . . . +  1 A2 E R, Ai > 0 o e e y i} in Y(T)R= Y(T) ®Z R whe e {mi, . . . ,m  ,,} a e ini ely many one- pa ame e subg oups (ac ually we also ask ha he cone o . ha e a e ex Le . a n (-a) = 0 .) Nex we in oduce i s dual in X(T)R= X(T) OZR gi en by a = {X E X(T)R 1 (X, ) > 0 o e e y mEa} . Then we exp ess he monoid a n X(T) in e ms o a ini e numbe o gene a o s (4 .1) o, nX(T)=N-XI+ . . .+N-XN ( ha his can always be done is a consequence o Go dan's lemma [D], [F], [O] .) Finally we cons uc he a ine model X Q by aking, as in he case o he a ine plane, he Spec um o he monoid algeb a o equi alen ly he (scheme heo e ic) closu e o he image o he map T -+ CN sending o ( Xl , . . . , XN ) . A o ic a ie y is ob ained by glueing oge he he a ine models abo e along T-in a ian open subse s . Think o example o he p ojec i e plane ob ained by glueing he e copies o he a ine plane along he open o bi and iden i ying he punc u ed axes in pai s . O cou se, he e we a e glueing no only he spaces bu also he T-ac ions so he ac ions on ou he e a ine planes ha e o be compa ible . Fo una ely he cones in o- duced ea lie allow his compa ibili y o be exp essed in simple e ms . De ini ion . Le T be an algeb aic o us . A collec ion E o cones as abo e in Y(T)R is said o be a an whene e i sa is ies he ollowing p ope ies a) E e y ace o a cone a in E also belongs o E . b) The in e sec ion o any wo cones in E is a ace o bo h o hem .