Codimension 2
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| − | = | + | = Motivation = |
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| + | In the 1970's a beautiful story began, when mathematicians, with Spencer Bloch leading the way, began to discover a | ||
| + | beautiful connection between algebraic K-Theory, algebraic cycles and polylogarithms. The first work was restricted to the | ||
| + | case of codimension 2 cycles, K_2, and the dilogarithm (this next polylogarithm after the usual logarithm) and '''quite explicit'''. | ||
| + | This story, for the next 35 years until now, triggered an enormous amount of research and was one of the motivations for many modern (and quite abstract) developments and conjectures, as for example | ||
| + | the theory of higher algebraic K-Theory (Quillen), mixed motives (Voevodsky - the ideas going back to Grothendieck), motivic cohomology, Bloch-Kato conjectures, Beilinson conjectures ... | ||
| + | |||
| + | The intention of this seminar is to study the beginnings of this subject as they evolved historically. The hope is that people might appreciate and | ||
| + | understand better many of the recent developments in the field when they have some "easy" explicit examples in mind that go just a little bit beyond the | ||
| + | relations between line bundles, codimension 1 cycles (divisors), invertible functions and the logarithm that everybody knows well. (Nevertheless the latter will be rediscussed in the first three talks). | ||
| − | + | While the first half of the seminar focuses on the explicit relation between (algebraic) K_2, the second (algebraic) Chern class of vector bundles and codimension 2 cycles, the second half is devoted to the '''dilogarithm function''' which enters these relations in various ways. The dilogarithm became quite famous in many other areas of mathematics recently. Some of those application we want to discuss as well. For instance, the relation with zeta functions, the applications in physics, and in hyperbolic geometry (all of course not completely unrelated with each other). | |
| − | + | Let me quote Don Zagier <ref name="Zag07"/>: | |
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| − | + | ''Almost all of its ''[of the dilogarithm]'' appearances in mathematics, and almost all the formulas relating to it, have something of the fantastical in them, as if this function alone among all others possessed a sense of humor.'' | |
= Schedule = | = Schedule = | ||
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|- | |- | ||
| 27.4. | | 27.4. | ||
| − | |[[#Cech cohomology|Cech cohomology]] | + | |[[#Cech cohomology and the Hodge conjecture in codim 1|Cech cohomology and the Hodge conjecture in codim 1]] |
| − | | | + | |Vetere |
|- | |- | ||
| 4.5. | | 4.5. | ||
| − | |[[# | + | |[[#Chow groups and Chern classes|Chow groups and Chern classes]] |
| − | | | + | |Bertini |
|- | |- | ||
| 11.5. | | 11.5. | ||
| − | |[[# | + | |[[#Elementary algebraic K-Theory: K0 and K1|Elementary algebraic K-Theory: K0 and K1]] |
| − | | | + | |Schwald |
|- | |- | ||
| 18.5. | | 18.5. | ||
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| 25.5. | | 25.5. | ||
| − | |[[#Elementary algebraic K-Theory: | + | |[[#Elementary algebraic K-Theory: K2|Elementary algebraic K-Theory: K2]] |
| − | | | + | |Zaccanelli |
|- | |- | ||
| 1.6. | | 1.6. | ||
| − | |[[# | + | |[[#Cycles in codim 2|Cycles in codim 2, part I]] |
| − | | | + | |Voelkel |
|- | |- | ||
| 8.6. | | 8.6. | ||
| − | |[[# | + | |[[#Cycles in codim 2|Cycles in codim 2, part II]] |
| − | | | + | |Kelly |
|- | |- | ||
| 15.6. | | 15.6. | ||
| − | |[[# | + | |[[#The dilogarithm|The dilogarithm]] |
| − | | | + | |Bergner, Sartori |
|- | |- | ||
| 22.6. | | 22.6. | ||
| − | |[[# | + | |[[#A generalization of the exponential sequence|A generalization of the exponential sequence]] |
| − | | | + | |Soergel |
|- | |- | ||
| 29.6. | | 29.6. | ||
| − | |[[#The | + | |[[#The dilog, scissors congruences and volumes of hyperbolic spaces|The dilog, scissors congruences and volumes of hyperbolic spaces]] |
| − | | | + | |Lye |
|- | |- | ||
| 6.7. | | 6.7. | ||
| − | |[[#The | + | |[[#The dilog and zeta functions|The dilog and zeta functions]] |
| − | | | + | |Eberhardt |
|- | |- | ||
| 13.7. | | 13.7. | ||
| − | |[[#The | + | |[[#The dilog in physics|The dilog in physics]] |
| − | | | + | |Scheidegger |
|- | |- | ||
| 20.7. | | 20.7. | ||
| − | |[[#Mixed | + | |[[#Outlook: Mixed motives and periods|Outlook: Mixed motives and periods]] |
| − | | | + | |Hörmann |
|} | |} | ||
| − | = Cech cohomology = | + | |
| − | = The Hodge conjecture = | + | == General comments == |
| + | |||
| + | All talks have 60 min + 30 min time for discussions. | ||
| + | |||
| + | The following outlines of the talks are merely '''suggestions'''. People should feel free to change the content of their talk as they like and concentrate rather on things they found interesting during the preparation. This should not be a problem as the talks are relatively independent. There are some restrictions, however: | ||
| + | |||
| + | * The topics mentioned in the description of talks 2-5 (though largely independent from each other) are needed for talks 6, 7 and 9. In case of doubt ask me or the speakers of these talks whether they need something. | ||
| + | * Talk 8 (The dilogarithm) is needed for the remaining talks on the dilog. | ||
| + | |||
| + | = Introduction = | ||
| + | |||
| + | * Motivation for alg. K-Theory (with review of topological K-Theory) | ||
| + | * Cycle classes of subvarieties in (singular) cohomology | ||
| + | * Formulation of the Hodge conjecture | ||
| + | * Overview on the topics of the seminar | ||
| + | |||
| + | = Cech cohomology and the Hodge conjecture in codim 1 = | ||
| + | |||
| + | * Review of Cech cohomology and the long exact sequence, explicit construction for the <math>\delta</math> homomorphism. Treat the analytic and Zariski topology parallel. | ||
| + | * Classification of vector bundles by Cech-cocycles, Theorem I.5.10<ref name="Wei13"/>. | ||
| + | * Mention also that for a ''projective'' complex algebraic variety <math>X</math>, complex analytic (rank n) vector bundles, classified by <math>H^1(X^{an}, \mathrm{GL}_n(\mathcal{O}_X^{an}))</math> (analytic Cech cohomology), are the same as algebraic (rank n) vector bundles on <math>X</math>, classified by <math>H^1(X, \mathrm{GL}_n(\mathcal{O}_X))</math> (Zariski Cech cohomology). | ||
| + | * The exponential sequence | ||
| + | <math> 0 \rightarrow 2\pi i \mathbb{Z}_X \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^* \rightarrow 1</math> | ||
| + | * Proof of the Hodge conjecture for divisors using the long exact sequence associated with the exponential sequence | ||
| + | * Explain also Lemma 2.1<ref name="Blo74"/> on a non-Abelian version of the long exact sequence. (This will be needed later) | ||
| + | |||
= Chow groups and Chern classes = | = Chow groups and Chern classes = | ||
The classical construction of Chow groups and the Chern classes using the splitting principle. | The classical construction of Chow groups and the Chern classes using the splitting principle. | ||
| − | = | + | * Definition of the Chow groups of a smooth variety X |
| + | * A survey over its properties, including the intersection pairing | ||
| + | * Cycle class map to singular cohomology for varieties over <math>\mathbb{C}</math>. | ||
| + | * Splitting principle: Theorem 5.19<ref name="Wei13"/> | ||
| + | * Definition of algebraic Chern classes using the splitting prinicple <ref name="Gro58"/>, axiomatic characterization | ||
| + | * Mention also the compatibility (via the cycle class map to singular cohomology) with the complex analytic construction of Chern classes using metrics and connections | ||
| + | * The isomorphism: | ||
| − | + | <math>Pic(X) \cong CH^1(X)</math> | |
| − | = Elementary algebraic K-Theory: | + | (Maybe also isomorphism of CH^1(X) with quotient sheaf of the sheaf of meromorphic functions! I.5.12<ref name="Wei13"/>) |
| − | = | + | |
| − | = Bloch | + | = Elementary algebraic K-Theory: <math>K_0</math> and <math>K_1</math> = |
| − | = The | + | |
| − | = Bloch | + | * The definition of <math>K_0(R)</math> for a ring <math>R</math> and <math>K_0(X)</math> for a variety <math>X</math>. |
| − | = Bloch | + | * Chern classes are maps from <math>K_0(X)</math> |
| − | = | + | * Also mention that there is a map |
| − | = | + | |
| − | = The quantum dilog = | + | <math>\{\text{codim } n \text{ cycles}\} \rightarrow K_0(X) </math> |
| + | |||
| + | which is right inverse to the <math>n</math>-th Chern class up to a rational factor. | ||
| + | * The Whitehead Lemma, 3.1<ref name="Mil71"/> and the definition of <math>K_1(R)</math> | ||
| + | * The localization exact sequence, if <math>R</math> is a Dedekind ring, <math>K</math> its quotient field: | ||
| + | |||
| + | <math> \bigoplus_{\mathfrak{p}}K_1(R/\mathfrak{p}) \rightarrow K_1(R) \rightarrow K_1(K) \rightarrow \bigoplus_{\mathfrak{p}} K_0(R/\mathfrak{p}) \rightarrow K_0(R) \rightarrow K_0(K) \rightarrow 0 </math> | ||
| + | |||
| + | References: §1-§5<ref name="Mil71"/>, | ||
| + | II, §8<ref name="Wei13"/>, Chapter I<ref name="Sri91"/>, 1.1-1.2 and 2.1-2.2<ref name="Ros94"/> | ||
| + | |||
| + | = Elementary algebraic K-Theory: <math>K_2</math> = | ||
| + | |||
| + | (only for ''commuative'' rings) | ||
| + | |||
| + | * Motivation of higher K-groups by non-exactness of the localization sequence on the left | ||
| + | |||
| + | * For a ring <math>R</math> define the Steinberg groups <math>St_n(R)</math> and <math>St(R)</math>, Definition p.40<ref name="Mil71"/>. | ||
| + | * Definition of <math>K_2(R)</math> as the Kernel of the map <math>St(R) \rightarrow \mathrm{GL}(R)</math>, p.40<ref name="Mil71"/>. | ||
| + | * Sketch of proof Theorem 5.1<ref name="Mil71"/>. | ||
| + | * Mention the description as universal central extension. | ||
| + | * Mention that and how <math>K_2(R)</math> is a module over <math>K_0(R)</math>. | ||
| + | * Introduce the symbol <math>\{a, b\} \in K_2(R)</math> for a pair of elements <math>a, b \in R</math> of a commutative ring <math>R</math>. | ||
| + | Theorem 8.8<ref name="Mil71"/> interprets the symbol as bimultiplicative skew-symmetric map | ||
| + | |||
| + | <math>K_1(R) \times K_1(R) \rightarrow K_2(R)</math> | ||
| + | |||
| + | * Introduction of Milnor K-groups <math>K_n^M</math> and give the idea of the proof of the statement | ||
| + | |||
| + | <math>K_2^M(R) \cong K_2(R)</math> | ||
| + | |||
| + | for fields (cf. §11, §12<ref name="Mil71"/>). | ||
| + | |||
| + | * extension of the localization squence to <math>K_2</math> (the tame symbol). | ||
| + | |||
| + | References: §5-§12<ref name="Mil71"/>, Chapter III<ref name="Wei13"/>, Chapter I<ref name="Sri91"/>, Chapter 4<ref name="Ros94"/>. | ||
| + | A nice collection of facts about <math>K_2</math> can be found here: <ref name="Dal06"/>. | ||
| + | |||
| + | More advances references: <ref name="Sus84"/>, <ref name="Sus87"/> | ||
| + | . | ||
| + | |||
| + | = Cycles in codim 2 = | ||
| + | |||
| + | '''These are two talks.''' | ||
| + | |||
| + | * Explain how <math>K_2</math>, and <math>St</math> (Steinberg group) can be sheafified | ||
| + | * Prove Proposition 1.13<ref name="Blo74"/> | ||
| + | * Remind of Lemma 2.1<ref name="Blo74"/> which has been discussed in the talk on Cech cohomology | ||
| + | * Deduce the long exact sequence associated with the sequence of sheaves | ||
| + | |||
| + | <math> | ||
| + | 1 \rightarrow K_2 \rightarrow St \rightarrow \mathrm{SL} \rightarrow 1 | ||
| + | </math> | ||
| + | |||
| + | and obtain a map | ||
| + | |||
| + | <math> | ||
| + | CK_2: H^1(X, \mathrm{SL}(\mathcal{O}_X)) \rightarrow H^2(X, K_2(\mathcal{O}_X)) | ||
| + | </math> | ||
| + | |||
| + | of Cech cohomology groups. The purpose of the rest of the two talks is to see that | ||
| + | |||
| + | <math> H^2(X, K_2(\mathcal{O}_X)) \cong CH^2(X)</math> | ||
| + | |||
| + | in such a way that <math>CK_2</math> becomes identified with the second Chern class as explained in the third talk. | ||
| + | |||
| + | * Prove Proposition 2.4, Corollary 2.7, and Proposition 2.8 (maybe omitting some details, and presenting concrete examples instead) | ||
| + | |||
| + | References: <ref name="Blo74"/>, (<ref name="Blo10"/>, §4) | ||
| + | |||
| + | * Bloch's construction (§3<ref name="Blo10"/>) of a map | ||
| + | |||
| + | <math>H^2(X, K_2) \rightarrow CH^2(X)</math> | ||
| + | |||
| + | * Show for a vector bundle <math>E</math> (of trivial determinant) that this map identifies <math>SK_2(E)</math> with the second Chern class <math>SK_2(E)</math> defined in the third talk (Theorem 4.2'<ref name="Blo10"/>). | ||
| + | |||
| + | * Explain that (via resolutions of the structure sheaf of codimension 2 subvarieties by vector bundles) one gets a map | ||
| + | |||
| + | <math> \{\text{codim 2 subvarieties}\} \rightarrow H^2(X, K_2)</math> | ||
| + | |||
| + | One would like to have that it factors through <math>CH^2(X) </math> and establishes that the morphism <math>H^2(X, K_2) \rightarrow CH^2(X)</math> is an ''isomorphism''. | ||
| + | However, at this point, Bloch's article is a bit out-of-date. Quillen later proved by using the '''Gersten resolution''' of the sheaf <math>K_n^M(\mathcal{O}_X)</math> (Sheafified higher Milnor K-groups, cf. talk 5) that <math>H^n(X, K_n^M) \rightarrow CH^n(X)</math>. In the end one could say a couple of words on this (Reference: ) | ||
| + | |||
| + | References: <ref name="Blo74"/>, (<ref name="Blo10"/>, §4) | ||
| + | |||
| + | = The dilogarithm = | ||
| + | |||
| + | * Introduction of the dilogarithm via integral formula and power series <ref name="Zag07"/> | ||
| + | * Multivalence (dependence of paths of integration) and how this problem is solved by considering the values as a cosets of certain unipotent matrices. Present the ideas from <ref name="Ram82"/> but restricted to the dilogarithm. (A theoretical, more modern explanation in terms of mixed Hodge structures can be found in <ref name="Blo91"/> but this should not be presented this way (yet).) | ||
| + | * Introduce the Bloch-Wigner dilogarithm (a single valued real-analytic variant of the dilog) | ||
| + | * The five term relation and special values | ||
| + | ** Present Theorem 7.4.4 from <ref name="Blo10"/> and its proof | ||
| + | ** a good explanation for what is going on can be found in <ref name="Sus91"/> | ||
| + | ** deduce special values as presented in <ref name="Zag07"/>, I, 1 | ||
| + | |||
| + | References: <ref name="Blo10"/>, <ref name="Ram82"/>, <ref name="Blo00"/>, in particular 7.4 and all what is needed from before, <ref name="Zag07"/>, <ref name="Hai94"/> | ||
| + | |||
| + | There should/could be some overlap with the following two talks; speakers should coordinate; some material can be shifted to the talk on the relation with hyperbolic geometry | ||
| + | |||
| + | If time is left some more motivations from <ref name="Zag07"/> can be presented. | ||
| + | |||
| + | More advanced references: <ref name="Blo81"/>, <ref name="Gon95"/>. | ||
| + | |||
| + | Further reading: <ref name="Lew81"/>. | ||
| + | |||
| + | = A generalization of the exponential sequence = | ||
| + | |||
| + | A survey of Bloch's article<ref name="Blo78"/>. | ||
| + | |||
| + | References: <ref name="Blo78"/>, (<ref name="Blo10"/>, §6) | ||
| + | |||
| + | Reference for the Dilogarithm: <ref name="Zag07"/>, <ref name="Hai94"/> | ||
| + | |||
| + | = The dilog, scissors congruences and volumes of hyperbolic spaces = | ||
| + | |||
| + | References: <ref name="DS83"/>, <ref name="Gon99"/>, <ref name="Hai94"/> | ||
| + | |||
| + | More advanced reference: <ref name="Gon95"/>. | ||
| + | |||
| + | Further reading: <ref name="Goe07"/>, <ref name="Hut13"/> especially the introduction | ||
| + | |||
| + | = The dilog and zeta functions = | ||
| + | |||
| + | References: <ref name="Blo00"/>, (<ref name="Zag07"/>, I, §5), <ref name="Hai94"/> | ||
| + | |||
| + | More advanced reference: <ref name="Gon95"/>. | ||
| + | |||
| + | = The dilog in physics = | ||
| + | |||
| + | Starting point: The quantum dilogarithm (<ref name="Zag07"/>, II, 2. D) | ||
| + | |||
| + | = Outlook: Mixed motives and periods = | ||
| + | |||
| + | Historical overview about the developments in the last 35 years...: | ||
| + | Mixed motives via their realizations and comparison encoding periods. | ||
| + | Explain that the values of the dilog appear as periods. Relation between Extensions of Tate-motives and K-theory, Regulators. | ||
| + | Connection with the topics discussed in the seminar. | ||
| + | |||
| + | References: <ref name="Lev05"/>, <ref name="BD94"/>, <ref name="Blo91"/>, <ref name="Gon95"/>, <ref name="Hai94"/> | ||
= References = | = References = | ||
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<references> | <references> | ||
<ref name="Blo74">Bloch, S.; ''<math>K_2</math> and algebraic cycles''. Ann. of Math. (2) 99 (1974), 349–379. </ref> | <ref name="Blo74">Bloch, S.; ''<math>K_2</math> and algebraic cycles''. Ann. of Math. (2) 99 (1974), 349–379. </ref> | ||
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<ref name="Sus87">Suslin, A. A.; | <ref name="Sus87">Suslin, A. A.; | ||
''Algebraic K-theory of fields''. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 222–244, Amer. Math. Soc., Providence, RI, 1987. </ref> | ''Algebraic K-theory of fields''. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 222–244, Amer. Math. Soc., Providence, RI, 1987. </ref> | ||
| − | |||
<ref name="Lev05">Levine, Marc; ''Mixed motives''. Handbook of K-theory. Vol. 1, 2, 429–521, Springer, Berlin, 2005.</ref> | <ref name="Lev05">Levine, Marc; ''Mixed motives''. Handbook of K-theory. Vol. 1, 2, 429–521, Springer, Berlin, 2005.</ref> | ||
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''Function theory of polylogarithms''. Structural properties of polylogarithms, 275–285, | ''Function theory of polylogarithms''. Structural properties of polylogarithms, 275–285, | ||
Math. Surveys Monogr., 37, Amer. Math. Soc., Providence, RI, 1991.</ref> | Math. Surveys Monogr., 37, Amer. Math. Soc., Providence, RI, 1991.</ref> | ||
| + | |||
| + | <ref name="DS83">Dupont, J. L.; Sah, C. H.; ''Scissors congruences. II''. J. Pure Appl. Algebra, 25(2):159–195, 1982.</ref> | ||
| + | |||
| + | <ref name="Gon99">Goncharov, A.; | ||
| + | ''Volumes of hyperbolic manifolds and mixed Tate motives''. | ||
| + | J. Amer. Math. Soc. 12 (1999), no. 2, 569–618. </ref> | ||
| + | |||
| + | <ref name="BD94">Beĭlinson, A.; Deligne, P. | ||
| + | ''Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs''. Motives (Seattle, WA, 1991), 97–121, | ||
| + | Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994. </ref> | ||
| + | |||
| + | <ref name="Ram82">Ramakrishnan, D.; | ||
| + | ''On the monodromy of higher logarithms''. | ||
| + | Proc. Amer. Math. Soc. 85 (1982), no. 4, 596–599. </ref> | ||
| + | |||
| + | <ref name="Dal06">Dalawat, Ch. S.; ''Some aspects of the functor <math>K_2</math> of fields''. arXiv:math/0311099, 2006</ref> | ||
| + | |||
| + | <ref name="Sri91">Srinivas, V.; | ||
| + | ''Algebraic K-theory''. | ||
| + | Progress in Mathematics, 90. Birkhäuser Boston, Inc., Boston, MA, 1991. xvi+314 pp.</ref> | ||
| + | |||
| + | <ref name="Ros94">Rosenberg, J.; | ||
| + | ''Algebraic K-theory and its applications''. | ||
| + | Graduate Texts in Mathematics, 147. Springer-Verlag, New York, 1994. x+392 pp.</ref> | ||
| + | |||
| + | <ref name="Gro58">Grothendieck, A. | ||
| + | ''La théorie des classes de Chern''. (French) | ||
| + | Bull. Soc. Math. France 86 1958 137–154. </ref> | ||
| + | |||
| + | <ref name="Hai94">Hain,R.M.; ''Classical Polylogarithms'', Motives, in Motives (Seattle, WA, 1991), manual, vol. 55 (1994), pp. 3-42, American Mathematical Society</ref> | ||
| + | |||
</references> | </references> | ||
