Codimension 2

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= Motivation =
 
= Motivation =
  
−
<ref name="Blo81"/>
 
−
<ref name="Goe07"/>
 
−
<ref name="Gon95"/>
 
−
<ref name="Lew81"/>
 
−
<ref name="Sus84"/>
 
−
<ref name="Zic15"/>
 
−
<ref name="Sus87"/>
 
−
 
−
Not: <ref name="Hut13"/> but has an informative introduction...
 
  
 
In the  1970's a beautiful story began, when mathematicians, with Spencer Bloch leading the way, began to discover a
 
In the  1970's a beautiful story began, when mathematicians, with Spencer Bloch leading the way, began to discover a
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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
 
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 beyound the
+
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).  
 
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).  
+
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"/>:
 
Let me quote Don Zagier <ref name="Zag07"/>:
  
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| 27.4.
 
| 27.4.
 
|[[#Cech cohomology and the Hodge conjecture in codim 1|Cech cohomology and the Hodge conjecture in codim 1]]
 
|[[#Cech cohomology and the Hodge conjecture in codim 1|Cech cohomology and the Hodge conjecture in codim 1]]
−
|N.N.
+
|Vetere
 
|-
 
|-
 
| 4.5.
 
| 4.5.
 
|[[#Chow groups and Chern classes|Chow groups and Chern classes]]
 
|[[#Chow groups and Chern classes|Chow groups and Chern classes]]
−
|N.N.
+
|Bertini
 
|-
 
|-
 
| 11.5.
 
| 11.5.
 
|[[#Elementary algebraic K-Theory: K0 and K1|Elementary algebraic K-Theory: K0 and K1]]
 
|[[#Elementary algebraic K-Theory: K0 and K1|Elementary algebraic K-Theory: K0 and K1]]
−
|N.N.
+
|Schwald
 
|-
 
|-
 
| 18.5.
 
| 18.5.
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| 25.5.
 
| 25.5.
 
|[[#Elementary algebraic K-Theory: K2|Elementary algebraic K-Theory: K2]]
 
|[[#Elementary algebraic K-Theory: K2|Elementary algebraic K-Theory: K2]]
−
|N.N.
+
|Zaccanelli
 
|-
 
|-
 
| 1.6.
 
| 1.6.
−
|[[#Cycles in codim 2, part I|Cycles in codim 2, part I]]
+
|[[#Cycles in codim 2|Cycles in codim 2, part I]]
−
|N.N.
+
|Voelkel
 
|-
 
|-
 
| 8.6.
 
| 8.6.
−
|[[#Cycles in codim 2, part II|Cycles in codim 2, part II]]
+
|[[#Cycles in codim 2|Cycles in codim 2, part II]]
−
|N.N.
+
|Kelly
 
|-
 
|-
 
| 15.6.
 
| 15.6.
 
|[[#The dilogarithm|The dilogarithm]]
 
|[[#The dilogarithm|The dilogarithm]]
−
|N.N.
+
|Bergner, Sartori
 
|-
 
|-
 
| 22.6.
 
| 22.6.
 
|[[#A generalization of the exponential sequence|A generalization of the exponential sequence]]
 
|[[#A generalization of the exponential sequence|A generalization of the exponential sequence]]
−
|N.N.
+
|Soergel
 
|-
 
|-
 
| 29.6.
 
| 29.6.
 
|[[#The dilog, scissors congruences and volumes of hyperbolic spaces|The dilog, scissors congruences and volumes of hyperbolic spaces]]
 
|[[#The dilog, scissors congruences and volumes of hyperbolic spaces|The dilog, scissors congruences and volumes of hyperbolic spaces]]
−
|N.N.
+
|Lye
 
|-
 
|-
 
| 6.7.
 
| 6.7.
 
|[[#The dilog and zeta functions|The dilog and zeta functions]]
 
|[[#The dilog and zeta functions|The dilog and zeta functions]]
−
|N.N.
+
|Eberhardt
 
|-
 
|-
 
| 13.7.
 
| 13.7.
 
|[[#The dilog in physics|The dilog in physics]]
 
|[[#The dilog in physics|The dilog in physics]]
−
|Wendland/Scheidegger ?
+
|Scheidegger
 
|-
 
|-
 
| 20.7.
 
| 20.7.
 
|[[#Outlook: Mixed motives and periods|Outlook: Mixed motives and periods]]
 
|[[#Outlook: Mixed motives and periods|Outlook: Mixed motives and periods]]
−
|Huber/Hörmann
+
|Hörmann
 
|}
 
|}
 +
 +
 +
== 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 =
 
= Introduction =
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= Cech cohomology and the Hodge conjecture in codim 1 =
 
= Cech cohomology and the Hodge conjecture in codim 1 =
  
−
Definition of vector bundles by Cech cocycles - Weibel, p. 54 ff. I.5.10
+
* 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"/>.
−
* Need also Lemma 2.1<ref name="Blo74"/> on a non-Abelian version of the long exact sequence.
+
* 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 =
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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.
  
−
= Elementary algebraic K-Theory: K0 and K1 =
+
* 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:
  
−
Also the relation of algebraic K_0 with the gamma-filtration on K_0 ?
+
<math>Pic(X) \cong CH^1(X)</math>
−
+
−
References: <ref name="Mil71"/>
+
−
<ref name="Wei13"/>
+
  
−
Weibel II, §8
+
(Maybe also isomorphism of CH^1(X) with quotient sheaf of the sheaf of meromorphic functions! I.5.12<ref name="Wei13"/>)
  
−
= Elementary algebraic K-Theory: K2 =
+
= Elementary algebraic K-Theory: <math>K_0</math> and <math>K_1</math> =
  
−
References: <ref name="Mil71"/>
+
* 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>.
−
<ref name="Wei13"/>
+
* Chern classes are maps from <math>K_0(X)</math>
 +
* Also mention that there is a map
  
−
Weibel, III
+
<math>\{\text{codim } n \text{ cycles}\} \rightarrow K_0(X) </math>
  
−
= Cycles in codim 2, part I =
+
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
 
* Explain how <math>K_2</math>, and <math>St</math> (Steinberg group) can be sheafified
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of Cech cohomology groups. The purpose of the rest of the two talks is to see that  
 
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>
+
<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.  
 
in such a way that <math>CK_2</math> becomes identified with the second Chern class as explained in the third talk.  
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References: <ref name="Blo74"/>, (<ref name="Blo10"/>, §4)
 
References: <ref name="Blo74"/>, (<ref name="Blo10"/>, §4)
−
 
−
= Cycles in codim 2, part II =
 
  
 
* Bloch's construction (§3<ref name="Blo10"/>) of a map
 
* Bloch's construction (§3<ref name="Blo10"/>) of a map
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* Explain that (via resolutions of the structure sheaf of codimension 2 subvarieties by vector bundles) one gets a map
 
* 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)
 
References: <ref name="Blo74"/>, (<ref name="Blo10"/>, §4)
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* Introduction of the dilogarithm via integral formula and power series <ref name="Zag07"/>
 
* 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).)  
 
* 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 non-analytic variant of the dilog)
+
* Introduce the Bloch-Wigner dilogarithm (a single valued real-analytic variant of the dilog)
 
* The five term relation and special values
 
* The five term relation and special values
 
** Present Theorem 7.4.4 from <ref name="Blo10"/> and its proof
 
** Present Theorem 7.4.4 from <ref name="Blo10"/> and its proof
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** deduce special values as presented in <ref name="Zag07"/>, I, 1
 
** deduce special values as presented in <ref name="Zag07"/>, I, 1
  
−
References: <ref name="Blo00"/>, in particular 7.4 and all what is needed from before
+
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
 
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 times is left some more motivations from <ref name="Zag07"/> can be presented.
+
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 generalization of the exponential sequence =
  
−
A survey of Bloch's article '78
+
A survey of Bloch's article<ref name="Blo78"/>.
  
 
References: <ref name="Blo78"/>, (<ref name="Blo10"/>, §6)
 
References: <ref name="Blo78"/>, (<ref name="Blo10"/>, §6)
  
−
Reference for the Dilogarithm: <ref name="Zag07"/>
+
Reference for the Dilogarithm: <ref name="Zag07"/>, <ref name="Hai94"/>
  
 
= The dilog, scissors congruences and volumes of hyperbolic spaces =
 
= The dilog, scissors congruences and volumes of hyperbolic spaces =
  
−
References: <ref name="DS83"/>, <ref name="Gon99"/>
+
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 =
 
= The dilog and zeta functions =
  
−
References: <ref name="Blo00"/>, (<ref name="Zag07"/>, I, §5)
+
References: <ref name="Blo00"/>, (<ref name="Zag07"/>, I, §5), <ref name="Hai94"/>
 +
 
 +
More advanced reference: <ref name="Gon95"/>.
  
 
= The dilog in physics =
 
= The dilog in physics =
  
−
Starting point: The quantum dilogarithm (<ref name="Zag07"/>, I D, § 3)
+
Starting point: The quantum dilogarithm (<ref name="Zag07"/>, II, 2. D)
  
 
= Outlook: Mixed motives and periods =
 
= Outlook: Mixed motives and periods =
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Historical overview about the developments in the last 35 years...:
 
Historical overview about the developments in the last 35 years...:
 
Mixed motives via their realizations and comparison encoding periods.
 
Mixed motives via their realizations and comparison encoding periods.
−
Explain that the values of the dilog appears as periods. Relation between Extensions of Tate-motives and K-theory, Regulators.
+
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.
 
Connection with the topics discussed in the seminar.
  
−
References: <ref name="Lev05"/>, <ref name="BD94"/>, <ref name="Blo91"/>
+
References: <ref name="Lev05"/>, <ref name="BD94"/>, <ref name="Blo91"/>, <ref name="Gon95"/>, <ref name="Hai94"/>
  
 
= References =
 
= References =
 +
 +
 
<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="Zic15">Zickert, Ch. K.; ''The extended Bloch group and algebraic K-theory''. J. Reine Angew. Math. 704 (2015), 21–54. </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>
Line 257: Line 321:
 
''On the monodromy of higher logarithms''.
 
''On the monodromy of higher logarithms''.
 
Proc. Amer. Math. Soc. 85 (1982), no. 4, 596–599. </ref>
 
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>

Latest revision as of 18:58, 25 July 2016

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