Algebraic geometry and algebraic topology

Transkrypt

1 Algebraic geometry and algebraic topology voevodsky connecting two worlds of math bringing intuitions from each area to the other coding and frobenius quantum information theory and quantum mechanics joint with Aravind Asok and Jean Fasel and Mike Hill

2 1890s-1970s: Many problems in mathematics were understood to be problems in algebraic topology/homotopy theory. This was due in large measure to the homotopy invariance of bundle theory. long entwined relation between fields allowing radically different intuitions to be brought to bear eg Frobenius vs complex analysis coding and eigenvalues of Frobenius quantum physics and quantum computing Story of Milnor in Dec. Theme of conference. Joyal s talk. Milnor of conj -> cat thy -> equations summarized by diagrams a la Joyal you can take that as an affirmation that we are all connected in this great universe, that the things that seem to divide us are illusions and whether by virtue of low entropy or a spirit in the sky all beings are but one being. and sometimes I have the reaction, of

3 1890s-1970s: Many problems in mathematics were understood to be problems in algebraic topology/homotopy theory. 1970s: Abstract homotopy theory Today: Thanks in large measure to Voevodsky, more and more mathematical questions are being understood to have a homotopy theoretic component, often rooted in abstract homotopy theory.

4 Question: Which complex vector bundles on complex algebraic varieties have algebraic structures?

5 1970s: Griffiths investigated this question on smooth affine varieties in connection with the Hodge Conjecture. Griffiths approach was via complex analysis. By a theorem of Grauert, every complex vector bundle on a Stein manifold has a unique holomorphic connection. The obstruction to the existence of an algebraic structure can be expressed in terms of the growth rate of the connection form. Griffiths used value distribution theory (Nevanlinna theory) to express this obstruction. 1990s - present: The work of Morel and Voevodsky let s us approach this question using the methods of homotopy theory.

6 Methods in homotopy theory f Homotopy groups Sn-1 Cell attachment X

7 Methods in homotopy theory Cell decompositions

8 Methods in homotopy theory Eilenberg-MacLane spaces Spectra Postnikov towers Brown-Gitler spectra Dyer-Lashof algebra Steenrod algebra

9 First examples

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sha1_base64="dwdlvlkknx3pp42u2oddk44ftzg=">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</latexit> <latexit sha1_base64="hysgfqeugyyuia4nfv89gip+e9o=">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</latexit> <latexit sha1_base64="aighdljjz4keqnxdf/in5emwfti=">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</latexit> <latexit sha1_base64="afbbfurkkvevd16zghhuq/g7b04=">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</latexit> Jouanoulou s Device Spec is the space of rank 1 projection operators on <latexit sha1_base64="kopu9eign8mm90vext7c+yoervs=">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</latexit> The map <latexit sha1_base64="nx5gct1+blxtjknq//zh9xzjolo=">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</latexit> sending a projection operator to its image is an (algebraic) homotopy equivalence.

14 <latexit sha1_base64="b69ezvvggxswl97bgvy4yacjfr0=">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</latexit> <latexit sha1_base64="iyzfj2i+9q9gcf+3+1zwzxtkmvy=">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</latexit> Algebra vs topology The pullbacks of and to J 2 are isomorphic, so the classification of algebraic vector bundles is not a problem in homotopy theory. But maybe it has to do with splitting exact sequences of projective modules, and it is homotopy theoretic when restricted to affine varieties.

15 <latexit sha1_base64="mr2nyoksowjcj9aex6cvtcky12u=">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</latexit> <latexit sha1_base64="buvfpsbxdbtqlkrjl1rjoemkez4=">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</latexit> <latexit sha1_base64="miyqx/s1onwsw6pztscnlhly9fm=">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</latexit> <latexit sha1_base64="wydzt2vqmvbrebtfckfp1sou0hi=">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</latexit> Algebra vs topology (dimension 2) Let I be the ideal sheaf of Since there is a unique non-trivial extension defining a rank 2 vector bundle V on P 2.

16 <latexit sha1_base64="xzchq9hskvvjisq+o+vlfnvxvl8=">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</latexit> Algebra vs topology V is non-trivial: V is topologically trivial: (c 1 (V) = c 2 (V) = 0) Question: Is V trivial on J 2?

17 <latexit sha1_base64="i5grr7ulumt3f1mfhofvnnsaj94=">aaaeehicfvndb9mwfpw2ain8bfaidxzttu0qxviygw9ikzakpya2bpvqmqpynnvemmnhjrmmsvkh+dw8jp/ce07xjrzdwlj0fc699/ja9wypz5l23eufxaxgvfsplh86jx4/efpszfx5aszzregesi7veuay4ezaiwaaw3mqgcqbh7pgcr/mz65azuyk77pmozeqslabo0rbql9ysn8t+m6r6hstxpq7lvzau1mfyr7pvl2tdeznigrvc+dcp6hu00ij33ib/zu1r+2ofr413pa7wwqnjddxf3xplr9kmicgnouky7qem+qeiuozyqfyfa2fhrjqxx/ftne2mxd8piou0essqdeagisq9czoasrcteiib1lzltqeodmrhirzviab9uyijrn5rgb/xxuj5z1tlnwsvb7s9gwtaa5b0bv1qc6xlrh+yhwybvtz0hqekmzlwjqmilbtp8jxmk38bcg0jeuaamylma1wkfn1btisteq1yxnihbos2xrmbju6ywp05kogcg7qkqmc5jra5ijrgci8zhwhfals4gx0qpiox83sx4rfbpkosbotvqlmng5yxhs2+r+pz7adxjxdcapbiao/mahpotiqcioz6yuw2969msynmhdtaudyoi6a+nw6tyz55qiibsbupe4gbbveh/dextvyusjreow/i2gthodtsgvflhmugilasr7tymqmgtn8sqz1uybgsgwsebeap5bjhd+05h3+ylk1hcuch1l2mrsyc0galyrhzpp7d4zmjukxp52257a9r+7a3t54spbrs/qabsap7aa9diio0qmi6af6ia7rr6xfddxyb2zeuc4ujgneojnv6pwbvtd2qa==</latexit> <latexit sha1_base64="4hkhfzaejlabaopsw++y7ssm2ti=">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</latexit> <latexit sha1_base64="7yyty62u5gcz6ojiay+haikg8xa=">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</latexit> Algebra vs topology Observation: In the ring the ideal can be generated by two elements a and b of degree -1.

18 <latexit sha1_base64="ild6wcikazltolv4xtdx6/kstao=">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</latexit> <latexit 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sha1_base64="fxihtyfrcxzytpui1cyetb3mqjm=">aaad8nicfvllbtnafj02pip5tbbkm6kkkkkk7nlssqhuqsxyilui+pdqei3hn86o4xlr5rqjzxnlr7bdblufbx/c3zb2kqpjesnzorr3npvyitiplpr+n4xf1q3bd+4u3fpup3j46phyypnjq3pd4yhrqc1pxcxioeaibuo4zqywnjjwep3v1/mtczbwapuziwy6kuuu6avo0iv6y9se63tu8+krahsuoiqn2viyqxed3vkq3/gbr2+cyajwyeqd9lywf4ex5nkkcrlk1p4ffobdkhkuxellhqgjhioyb7uvo9tbqnlhbifj/jwlcoagyinybtksv9g2i8s0r437fnimel1rsttaio0cm2u4spo5oviv3bmtsluoimq2pfz3uqvqwy6g+lh7p5cuna3pr2nhgkmshgdcclcs5qnmged3zm9rt+mnoua+qaujcjts0aefzl8yoikwkqktribfgbdttjczbtnjbnxmpbjyceyirbz29egov4lrgobdekdowoukw8cucvvfrdwfmhkeawbiapjwnerm2jurclqv/8p44kyi5ggx9j3hmn9rrjkhqjrjvjv+z7oxw7rf2rmdquhmnaviqyct1zrzrv6f1pltjqzaxbw6kdryg7dvgnp3cj4v6hxkckahk2hawzdrnguqlsodqgxrhmeshlsec3sw7+2b4hije/id4opm6t7expdl5bl5ttziqlbjhnlpdskr4eqrusq/ya8wtr61vrd+jkmlcxpnuzlzwpd/aceeqrm=</latexit> <latexit sha1_base64="l61kahy76sqkpbeoli8s8n5uw34=">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</latexit> <latexit sha1_base64="cgpanmkf+fueyjmqixrzw+tdhms=">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</latexit> <latexit sha1_base64="gs4xnlxqp5dgdc6wcqbzkkbk1ji=">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</latexit> Algebra vs topology Choose satisfying set and One can check that a and b generate (x 0, x 1 ) and have degree -1.

19 <latexit sha1_base64="zhb+/mr9dezvegvmrtz8dnzagpg=">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</latexit> Question: Is every topologically trivial vector bundle on J n trivial? Question: Is the map a bijection?

20 Homotopy invariance of bundle theory in algebraic geometry

21 Serre (1955) Faisceaux algébriques cohérents Vector bundles on X projective modules over O X Note that when X=K r (so that O X =K[x 1,,x r ]) it is unknown if there exist finitely generated projective modules which are not free. Serre s problem

22 Serre (1955) Faisceaux algébriques cohérents Serre s problem was solved independently by Quillen and Suslin Theorem (Quillen, Suslin 1977): If k is a field then every finitely generated projective module over is free. k[x 1,,x n ]

23 Bass-Quillen conjecture Bass-Quillen Conjecture: If A is a commutative regular ring of finite Krull dimension, then every finitely generated projective module over A[x 1,,x n ] is extended from A, ie where

24 Bass-Quillen conjecture Theorem (Lindel, 1981): The Bass-Quillen conjecture is true when A is a finitely generated (commutative regular) algebra over a field. (the geometric case) Lam, T. Y. Serre's problem on projective modules

25 Algebraic and motivic homotopy Theory

26 Classical homotopy theory C W the category of topological spaces the class of (weak) homotopy equivalences C top Classical homotopy theory

27 Algebraic homotopy theory Ken Brown ( 73), Abstract homotopy theory and generalized sheaf cohomology Sm C the category of (simplicial) presheaves on Sm

28 Algebraic homotopy theory Ken Brown ( 73), Abstract homotopy theory and generalized sheaf cohomology Sm C W the category of (simplicial) presheaves on Sm the class of Nisnevich (hyper-)coverings. C alg Algebraic homotopy theory

29 Motivic Homotopy Theory (Morel-Voevodsky 2001) Sm C W the category of (simplicial) presheaves on Sm the class of Nisnevich (hyper-)coverings together with the maps C mot Motivic homotopy theory

30 Abstract homotopy theory realization functors

31 Algebraic and Motivic Vector Bundles

32 Classifying spaces Classifying space for principal G-bundles: BG Bundle theory Adams was referring to Topology

33 Classifying spaces For X a smooth variety, there is an isomorphism (local description of vector bundles in terms of charts and transition functions)

34 Algebraic and motivic vector bundles motivic vector bundles Realization maps

35 Vector bundles on affine varieties Fabien Morel (2012) M. Schlichting (2015) Aravand Asok, Marc Hoyois, Matthias Wendt (2015) Theorem: For smooth affine X, the map is an isomorphism. affine homotopy invariance of algebraic bundle theory

36 Obstruction Theory

37 Cell decompositions

38 Obstruction theory Algebraic Homotopy Theory:

39 Obstruction theory Motivic Homotopy Theory: go to town here mention our example (rational chern classes) relation with griffiths, Interesting obstruction theory Morel and Asok-Fasel have investigated splitting free summands off of projective modules over regular rings and classification of projective modules over smooth algebras of Krull dimension less than or equal to 3.

40 The Rees Bundles

41 The Rees bundles Obstruction theory one needs to understand odd homotopy groups of S 3

42 The Rees bundles The first non-zero odd homotopy group of S 3 (localized at p) is Rees considers the bundle ξ p corresponding to Theorem (Rees): The bundle ξ p is non-trivial for all p. Note: The bundle ξ p has no Chern classes.

43 The Rees bundles Question: Do the Rees bundles exist over J 2p-1?

44 The Rees bundles Answer (Asok, Fasel, H.): Yes.

45 <latexit sha1_base64="kcwy+qdntjv/jkytpikv8phmrp0=">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</latexit> <latexit sha1_base64="lqg7czpm3p14udj1p12p67pd9nw=">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</latexit> <latexit sha1_base64="8emd/kornqdfhzjrba81ddep3n8=">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</latexit> Motivic Rees bundles The motivic Rees bundle corresponds to a rank 2 projective module over the degree zero part, R 0, of the ring Homotopy theory implies that it arises as the kernel of a map for some unimodular row [a,b,c] in R 0.

46 Construction of the motivic Rees bundles

47 Do you have 15 min left? yes go for it! no pretend the construction is waaay too hard to explain and skip it requires years of training in meditation and the martial arts at secret remote himalayan dojos

48 Construction of ζ p Key step leads to a map which may be composed with its suspension to give

49 Motivic spheres dimension S p,q (p q) Hodge weight S 1,1 = A 1 -{0} S 1,0 = A 1 /{0,1} S p,q = Other examples

50 Motivic homotopy groups π a,b (X) = [S a,b,x] mot topological realization

51 Movitic α p and ζ p We need a diagram

52 Suslin s n! theorem (unimodular rows) Theorem (Suslin): Let R be a commutative ring, and elements of R, satisfying and If is a sequence of positive integers whose product is divisible by (n-1)!, then there exists a unimodular matrix whose first row is

53 Movitic α p and ζ p (requires (-1) to be a sum of squares) Suslin s theorem gives a diagram Leads, as in topology, to a map (the case n=p+1) which when composed with its suspension gives (having order p)

54 Rank 2 vector bundles on P n the motivic Rees bundle should arise as from a map But we produced The Hodge weight is wrong.

55 Motivic ρ There is a map in motivic homotopy theory which changes weight, and realizes to the identity

56 Motivic Rees bundles The motivic map ζ p is killed by p, so it can be multiplied by ρ, and we can form and lift the Rees bundles to motivic vector bundles.

57 <latexit sha1_base64="kcwy+qdntjv/jkytpikv8phmrp0=">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</latexit> <latexit sha1_base64="lqg7czpm3p14udj1p12p67pd9nw=">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</latexit> <latexit sha1_base64="8emd/kornqdfhzjrba81ddep3n8=">aaaeq3icfvplbtqwfhujjxiebwejc4tqrctvo4rswhzilvqklkatjz7uzhg5zp3eqmnhttozymqxsivp4sp4bnailrlodkz0pghlky7ouq/fm+o44eybipg+c827fupmrdnb/p279+7pzs88oncyvbt2qershcvea2cc9g0zhi4kbssporzgp1unfnggsjmpppqqge5ousf6jbljqo783nl7brd8arvhrmihu/olytsyhnwbgvbzswyrjc5ed8f7hcwsljkiqznr+iqmctoxrblgodr+zgbg+iwx2avv9voae35uaigipsupndgosa66y4ez1ljlmat3phkfmhjixs6wjne6ymmxmrot6wmtif+lnrdoo3zq1zpttw+jy5kosgocnnfvlry7zttbwgltqa2vhcbumtcsphlrhbq3u99vtfchpjm0axig4cbfricrwt5vgfpjrnoiriy2gdotlj9x2+edc6atv7jqclbnee23lrodujaqe5imurkyrwpxwipjcrpn1uxwrvhplilk6phgojxk0jzlmdhl/4l445whpgis6dl/df+fjxkjtvvpxnuxl1ac9sy5gbtkkqcczmv/ldwouqhegdezykovglhodtvplqhfvgib2cfzvgpgsnfeq1dwg5wrfbx14m4bihinrs6rtr0a07520iupencnms+jsgyuyxys7zh4rd91atooeo53nxpzox5pcyypa9+9p+dvm5og42eonrxdob2+cxy3n0cvaxy9qk/qegrrotpeo2gp7soksvqzfuffvw/ed++n9+s89nrmkochmjje7z/nlgaa</latexit> Motivic Rees bundles The motivic Rees bundle corresponds to a rank 2 projective module over the degree zero part, R 0, of the ring Homotopy theory implies that it arises as the kernel of a map for some unimodular row [a,b,c] in R 0.

58 <latexit sha1_base64="zhb+/mr9dezvegvmrtz8dnzagpg=">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</latexit> Question: Is the map a bijection?

59 Unstable Adams-Novikov resolution <latexit sha1_base64="t7dun9tnyfncq7chfc788vm5nns=">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</latexit>

60 The Wilson Space Hypothesis

61 Homologically even spaces Many naturally occurring spaces have cell decompositions with only even dimensional cells: homologically even spaces For such a space X i) H 2n-1 X = 0 ii) H 2n (X) is a free abelian group.

62 <latexit sha1_base64="a+tcebdptnnz2sqjirmu2z+ovpm=">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</latexit> <latexit sha1_base64="jt53nyvmvqwgokoav9kolxmylss=">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</latexit> Homotopically even spaces Definition: A space X is homotopically even if i) ii)

63 Wilson spaces (even spaces) Definition: An even space is a space which is both homologically and homotopically even. Examples: CP BU BSU

64 <latexit sha1_base64="1m7nznjz2r2sy4fcfm11v7qbj7s=">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</latexit> <latexit sha1_base64="zqdjz2ay5bzvnpmiwkqxoqeajx0=">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</latexit> Wilson s Theorem Theorem (Steve Wilson): The classifying spaces for complex cobordism are even spaces. Consequence. If X is homologically even then is even.

65 <latexit sha1_base64="t7dun9tnyfncq7chfc788vm5nns=">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</latexit> Unstable Adams-Novikov resolution Start with X homologically even even Leads to a resolution of X by K(Z,2n)

66 <latexit sha1_base64="gbjycvaxqndckbbotc/yciug0bu=">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</latexit> <latexit sha1_base64="bcmjolbmrwwqsfadypvtoiqw9sw=">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</latexit> <latexit sha1_base64="rjkikovfum5nys61q68hpswdgmu=">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</latexit> Motivic spaces Notation: S 2n,n = A n /A n -{0} S 2n-1,n = A n -{0} Definition: A motivic space X is homologically even if Definition: A motivic space X is homotopically even if

67 <latexit sha1_base64="l6kqibeavmqvjoshwbhde27fwua=">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</latexit> Motivic spaces Definition: A motivic space X is even if it is both homologically and homtopically even. Wilson space hypothesis: For every n, the motivic space is even.

68 <latexit sha1_base64="1e7pn0kgwihzgcgjwrcejv4dzfc=">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</latexit> Equivariant Wilson spaces Theorem (Hill, H.): For every integer n, the real space is an even real space. complex cobordism with complex conjugation A friend sent me this yesterday. I know it s not possible but I take some comfort in imagining it to be a sighting.

69