Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/pdc/2006/reports/Bondarko_3year.pdf Дата изменения: Tue Dec 29 19:17:42 2009 Дата индексирования: Sat Jan 2 04:01:33 2010 Кодировка: Windows-1251 Поисковые слова: р р р с с р р р с с с р р

The report of Mikhail Bondarko for 20072009 1 My plans (from the research proposal)
sn my reserh proposl s listed the following gols of the pro jetX 4he min gol of the pro jet is the study of di'erent tringulted tegories of motivesD their tEstrutures nd reliztionsF sn prtiulrD new method of tthing weights to ohomology funtors will e studiedY this inludes the study of soE lled 9weight strutures9D relted tEstruturesD nd qerstenEtype resoultionsF hese results would e pplied to the study of motivesD ohomologil funtors on shemes nd to uEtheoryF sn prtiulrD s pln to study @existing nd newA theories of reltive motivesF s lso pln to de(ne ertin nonEredued theories of motivesF he in(nitesmll prt of suh theory ould e relted with my previous results on forml groups nd group shemesF4

2

My successes and failures

s ws not le to de(ne ny 9resonle9 lol motives @those tht would e omptile with 9redued9 onesAF s hve developed the theory of weight strutures quite suessfullyF xow this is theory tht hs severl pplition to motives @inluding weights for ritrry ohomology of motivesAD nd lso to @topologilA spetr nd other tringulted tegoriesF he @very interesting3A onnetion of weight strutures with tEstrutures ws lso studied in detilF sn prtiulrD weight strutures were suessfully used to study qersten resE olutions nd oniveu spetrl sequenes @though to this end s hd to introdue new motivi tegory not mentioned in the originl pro jetAF s hve lso invented progrm to desrie weights for reltive motivesF st is desried in VFP of PY the detils hve to e written down stillF

3

Publications

sn PHHU!PHHW three of my ppers were pulishedX TD SD nd IF he (rst two re not relted to motives @though S hs some reltion with the 9lol9 prt of the pro jet tht ws not relly suessfullAF I is my (rst 9motivi9 pperF sn it the following results were estlished @tullyD for the proof of prt s of heorem QFI in its urrent form weight strutures re essentilY so in loFitF weker sttement ws provedD nd the present version ws proved in PAF
Theorem 3.1.

ef s e ful l desription of oevodsky9s DMgmf in terms of 9twisted9 uslin uil omplexes @in the sense of uprnov nd fondlA ws givenF sn prtiulrD for ny motivi omplex M @for instneD the uslin omplex of n ritrry vrietyA there exists qusiEisomorphi omplex M 9onstruted from9

I


the uslin omplexes of smooth projetive vrietiesY M is unique up to homotopy equivleneF ss oeovodsky9s DMgm Q is ntiEequivlent to the rnmur9s motivi tegoryF ef sss here exist onservtive ext weight omplex funtor t : DMgmf DMgm K b (C howef f ) K b (C how)F ef s t indues isomorphisms K0 (DMgmf ) K0 (C how) nd K0 (DMgm ) K0 (C how)Y they re isomorphisms of ringsF por ny ohomologil funtor H : DMgm A @here A is n elin teE goryA nd X Obj DMgm there exists weight spetrl sequene T : H i (P -j ) = H i+j (X ) where (P i ) is representtive of t(X )F T is nonil nd motivil ly funtoril strting from E2 F his yields heligne9s weight spetrl sequenes nd weight (ltrtions for mixed rodge nd ? etle ohomology of vrieties @nd lso ertin 9weight spetrl sequene9 for motivi ohomologyAF s e motif @n ojet of oevodsky9s DMgm A is mixed te one whenever its weight omplex isF
sn PHHW P ws epted y the tournl of uEtheoryY Q ws sumitted to houment wthF

4

Basic properties of weight structures

eight strutures @de(ned in P nd studied further in QY see R for surveyA were entrl in my reserh pro jetF s showed tht prts sss!s of heorem QFI follow from very generl relevnt formlism for tringulted tegoriesY this setting ws not previously desried in litertureF yne onsiders set of xioms tht re @in ertin senseA 9dul9 to the xioms of tEstruturesF everl properties of weight strutures re similr to those of tEstruturesY yet other ones re quite distintF felow C nd D will e tringulted tegoriesY A will e elinF e tegory C with weight struture w hs n dditive hert H w with the property tht there re no morphisms of positive degrees etween o jets of the hert in C F eny ounded weight struture yields onservtive weight omplex funtor to the wek homotopy tegory of omplexes over the hertF woreoverD w gives ostnikov tower of ny o jet of C whose 9ftors9 elong to H wY suh weight ostnikov tower is nonil nd funtoril 9up to ohomology zero mps9F epplying ny @oAhomologil funtor C A @A is n elin tegoryA to this tower one otins 9weight spetrl sequene9 whose E1 Eterms re @oAhomology of the orresponding o jets of the hertF his spetrl sequene is nonil nd funtoril strting from E2 F xow s desrie this theory in more detilF

he(nition RFI @eight strutures nd their hertsAF s e pir of sulsses C w0 , C w0 Obj C for tringulted tegory C will e sid to de(ne weight struture w if C w0 , C w0 stisfy the following onditionsX @iA C w0 , C w0 ontin ll diret summnds of their o jetsF
P


@iiA C w0 [1] C w0 D C w0 C w0 [1]F @iiiA por ny X C w0 D Y C w0 [1] we hve C (X, Y ) = 0F @ivA por ny X Obj C there exists distinguished tringle

B [-1] X A B

f

@IA

suh tht A C 0 , B C w0 F ss e tegory H w whose o jets re C w=0 = C w0 C w0 D H w(X, Y ) = C (X, Y ) for X, Y C w=0 D will e lled the hert of the weight struture wF

H w is dditiveF sn ontrst to the sitution with tEstruturesD there nnot e ny nonEtrivil 9C Eextensions9 of o jets of H wF he si exmple of weight struture is given y the stupid (ltrtion on the homotopy tegory of omplexes over n ritrry dditive tegory B F sts hert is the @esily desriedA idempotent ompletion of B in K (B )F eny weight struture yields weight omplex funtor t : C Kw (H w)Y here Kw (H w) is ertin ftor of K (H w) @we 9kill9 morphisms of the form df + g d for f , g eing olletions of rrows tht shift degrees y -1Y this does not hnge isomorphism lsses of o jets in K (H w)AF his funtor hs severl nie propertiesY in prtiulrD it is 9usully9 onservtive @t lestD this is the se when w is ounded iFeF C w0 [i] = C w0 [i] = {0}AF woreoverD one n 9often9 reple Kw (H w) y K (H w) @then t will e extAY this is the se for motives nd spetrF sf H : C A is ohomologil funtorD then for ny X Obj C one hs pq spetrl sequene T (H, X ) with E1 = H (X -p [-q ])Y here X i re the terms of @ny hoie of A the weight omplex of X F st @weklyA onverges to H (X [-p - q ])Y it is C Efuntoril in X strting from E2 F sn prtiulrD one otins funtoril @9weight9A (ltrtion on H (X )F xow we desrie the reltion of weight strutures with tEstruturesF vet : C op Ч D A e nie dulity of tringulted tegories @see he(nition PFSFI of PAF uppose lso tht C is endowed with weight struture wD D is endowed with tEstruture tD nd w is orthogonl to t @with respet to AF he esiest @ut not the only one existingA exmple of dulity isX D = C D A = AbD (X, Y ) = C (X, Y ) for X, Y Obj C F sn this se t is orthogonl to w whenever C w0 = C t0 Y we will sy tht t is djent to wF por some Y Obj D we onsider the funtor H = (-, Y )F hen one hs funtoril desription of T (H, -) @strting from E2 A in terms of tEtruntions of Y Y see heorem PFTFI of QF his is powerful tool for ompring spetrl sequenes @in this situtionAY it does not require onstruting ny omplexes @nd (ltrtions for themA in ontrst to the method of rnjpe @prolyD originting from heligneAF elsoD H w is 9dul9 to the hert of t in very interesting senseF e funtor right djoint to tEext funtor F : C D @with respet to some t for C nd t for DA is weightEext @in the nturl senseA with respet to the weight strutures djent to t nd t @if those existAY the onverse is lso trueF

Q


sf w is ounded nd H w is idempotent ompleteD then C is idempotent omplete lso nd K0 (C ) K0 (H w)F = eight strutures lso desend to loliztionsD nd n e glued @under ertin onditionsA in wys tht re similr to the orresponding ones for tE struturesF

5

Applications to motives

e hve two min 9motivi9 weight strutures @tht tully elong to sinE gle series of thoseAF hey orrespond to @ghowAEweight nd oniveu spetrl sequenesD respetivelyF xote tht oth of these spetrl sequenes were 9lssiE lly9 de(ned only for ohomology of vrietiesY still our pproh llows to de(ne them for ritrry oevodsky9s motivesD nd lso yields their motivi funtoriE lity @whih is very fr from eing ovious for oth of them if one uses their 9lssil9 de(nitions3AF e use some nottion from UF ef he (rst @9motivi9A weight struture is wC how Y it is de(ned on DMgmf ef f DMgm D its hert is C how C howF oD oevodsky9s motives ould e 9slied into piees9 tht re ghow motives 9nonilly up to homotopy equivlene9F ef xote hereX the orresponding weight omplex funtor t : DMgmf K b (C howef f ) ef f is onservtiveD wheres DMgm is very fr from eing isomorphi to K b (C howef f )F he weight spetrl sequene with respet to wC how is isomorphi to the heligne9s ones for H eing ? etle or singulr ohomology of vrietiesF et note tht TwC how (H, -) is de(ned for ny H @inluding motivi ohomology nd sinE ef gulr ohomology with integrl oe0ients3A nd is DMgmf Efuntoril strting from E2 3 ef here exists ghow tEstruture tC how for DM- f whose hert is AddFun(C howef f , Ab)F ef tC how is djent to the ghow weight struture for DM- f Y it is relted with unrmi(ed ohomologyF wC how is lso losely relted with the 9usul expettions from weights for oevodsky9s motives9Y see VFT of PF he seond 9motivi9 weight struture is the qersten weight struture w ef de(ned on the tegory Ds DMgmf @for ountle k AF rere Ds is full tringulted sutegory of ertin tegory D of omotivesF he ide is tht w should e orthogonl to the homotopy tEstruture on ef DM- f @rell tht the ltter is the restrition of the nonil tEstruture of the derived tegory of xisnevih sheves with trnsfersAF oD H w is 9generted9 y omotives of funtion (elds over k @note tht these re xisnevih pointsAY in ef ef prtiulrD it nnot e de(ned on DMgmf @or DM- f AF ef f ef he prolem with DM- DMgmf is tht there re no 9nie9 homotopy limits in themF sn order to hve these limits one needs 9nie9 @smllA produtsY ef one lso needs the o jets of DMgmf to e oompt @in this 9tegory of hoE ef f motopy limits9AF DM- de(nitely does not stisfy these onditionsF snsted in S of Q the tegory D of omotives ws onstrutedY there exists nie dulE R


ef ity Dop Ч DM- f AbF sn Ds D the @oAmotif of ny smooth vriety n e 9deomposed9 @in the sense of ostnikov towersA into te twists of omoE tives of its points @using tringulted nlogue of the usul gousin omplex onstrutionAF he generl theory of weight spetrl sequenes yields Tw (G, X ) for ny ohomologil funtors G : Ds AY the prolem here is tht Ds is 9lrge9 nd ef rther 9mysterious9 tegoryF etD ny H : DMgmf A hs 9nie9 extension to s s D @nd lso to D D A if A stis(es efS @see roposition RFQFI of QAF oD we n onsider weight spetrl sequenes T = Tw (H, X ) for ny suh H nd ny ef X Obj DMgmf or X Obj Ds F st turns out tht for X eing the motif of smooth vrietyD T is isomorphi to the oniveu spetrl sequene @orresponding to H A strting from E2 F oD we ll T oniveu spetrl sequene for ny X F hus @very similrly to the se of ghowEweight spetrl sequenesA s vstly generlized oniveu spetrl sequenesD nd proved tht they re motivilly funtorilF es well s for 9lssil9 oniveu spetrl sequenesD if H is represented ef y n o jet of DM- f D Tw (H, X ) ould e desried in terms of ohomology of X with oe0ients in the homotopy tEtruntions of H Y this ft extends the relted results of flohEygus nd rnjpeF e relted result isX torsion motivi ohomology of motives n e expressed in terms of ? etle ohomology @in ertin wyY here the reently proved feilinsonEvihtenum onjeture is usedAF elsoD s proved olletion of diret summnd resultsF sn prtiulrD the oE motif of smooth semiElol sheme @or ny primitive smooth shemeA is diret summnd of the omotif of its generi (reY omotives of (elds ontin s diret summnds twisted omotives of their residue (elds @for ny geometri vluE tionsAF rene similr results hold for ny ohomology of @semiElolA shemes mentionedF ef fesidesD w ould e restrited to the tegory DAT DMgmf of soElled ertinEte motives @this is the tegory generted y te twists of ertin moE tivesAF his yields 9eonomi9 desriptions of oniveu spetrl sequenes for suh motives @strting from E2 AF s lso onstruted @nonEexpliitlyA whole series of weight strutures for the tegory of omotivesF his series is indexed y single integrl prmeterY ll the strutures indue the sme weight struture on the tegory of irtionl omotives @iFeF the loliztion of D y D(1)AD nd for the iEth weight struture tensoring y Z(1)[i] is weightEext @iFeF C wi 0 (1)[i] C wi 0 nd C wi 0 (1)[i] C wi 0 AF sn prtiulrD for i = 2 one otins the ghow weight struture @for DAY for i = 1 one otins the qersten weight strutureF st is quite mzing tht spetrl sequenes tht re so distint from the geometri point of view di'er just y [1] @in this desriptionA3 ossilyD other memers of this series ould e lso interesting @espeilly the one orresponding to i = 0AF

S


6

Pedagogical activity

sn PHHU!PHHW s led student9s prtie in higher lger nd numer theory @in tF etersurg tte niversityAF fesides s tively prtiipted in the omposition of two ooks of prolemsX one in xumer theory nd one in pield theoryF he (rst one is is pulished nowF

7

Conferences

huring PHHU!PHHW s mde tlks @on motives nd weight struturesA t the folE lowing interntionl onferenesX IF erithmeti qeometryD intEetersurgD IQ!IWFHTFPHHUF PF snterntionl elgeri gonferene dedited to the IHHth nniversry of hF uF pddeevD intEetersurgD PR!PWFHWFPHHUF QF oung wthemtis in ussiD wosowD IP!IQFHIFPHHWF RF orkshop 4piniteness for wotives nd wotivi gohomology egensurgD W!IQFHPFPHHWF SF orkshop on wotivi romotopy heoryD wunsterD PU!QIFHUFPHHWF ? TF elgeri gonferene dedited to the THth enniversry of eF sF qenerlovD tF etersurgD P!QFHWFPHHWF s lso prtiipted in seminrs inX tF etersurg tte niversityD niversity ris IQD wx lnk snstitut fur wthemtikD niversity of lmnD nd ? snstitut de wth? emtiques de tussieuF

Список литературы
I fondrko wFFD hi'erentil grded motivesX weight omplexD weight (lE trtions nd spetrl sequenes for reliztionsY oevodsky vsF rnmuE rGG tF of the snstF of wthF of tussieuD vFV @PHHWAD noF ID QW!WUD see lso http://arxiv.org/abs/math.AG/0601713F P fondrko wFD eight strutures vsF tEstruturesY weight (ltrtionsD spetrl sequenesD nd omplexes @for motives nd in generlAD to pper in tF of uEtheoryD http://arxiv.org/abs/0704.4003F Q fondrko wFD wotivilly funtoril oniveu spetrl sequenesY diret summnds of ohomology of funtion (eldsD preprintD http://arxiv.org/abs/0812.2672F R fondrko wFFD eight strutures nd motivesY omotivesD oniveu nd ghowEweight spetrl sequenesX surveyD preprintD http://arxiv.org/abs/0903.0091F S fondrko wFFD hievsky eFFD xonEelin ssoited orders of wildly rmE i(ed lol (eld extensionsGG piski xuhnyh eminrov ywsD volF QSTD S!RSD PHHVD see http://www.p dmi.ras.ru/znsl/2008/v356.html T


T fondrko wFFD gnonil representtives in strit isomorphism lsses of forml groupsGG wthemtil xotesD vF VPD nF I!PD PHHUD ppF ISW!ITRF U oevodsky F ringulted tegory of motivesD inX oevodsky FD uslin eFD nd priedlnder iF gylesD trnsfers nd motivi homology theoriesD ennls of wthemtil studiesD volF IRQD rineton niversity ressD PHHHD IVV!PQVF

U