Codimension 2

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(Elementary algebraic K-Theory: K_0 and K_1)
(Schedule)
 
Line 26: Line 26:
 
| 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.
Line 42: Line 42:
 
| 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|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|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
 
|}
 
|}
  
Line 102: Line 102:
 
<math> 0 \rightarrow 2\pi i \mathbb{Z}_X \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^* \rightarrow 1</math>
 
<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
 
* Proof of the Hodge conjecture for divisors using the long exact sequence associated with the exponential sequence
* Need also Lemma 2.1<ref name="Blo74"/> on a non-Abelian version of the long exact 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 =
Line 110: Line 110:
 
* Definition of the Chow groups of a smooth variety X
 
* Definition of the Chow groups of a smooth variety X
 
* A survey over its properties, including the intersection pairing
 
* A survey over its properties, including the intersection pairing
* Cycle class map to Cohomology for varieties over <math>\mathbb{C}</math>.  
+
* Cycle class map to singular cohomology for varieties over <math>\mathbb{C}</math>.  
 
* Splitting principle: Theorem 5.19<ref name="Wei13"/>  
 
* Splitting principle: Theorem 5.19<ref name="Wei13"/>  
 
* Definition of algebraic Chern classes using the splitting prinicple <ref name="Gro58"/>, axiomatic characterization
 
* Definition of algebraic Chern classes using the splitting prinicple <ref name="Gro58"/>, axiomatic characterization
* Mention also the compatibility (via the cycle class map to sigular cohomology) with the complex analytic construction of Chern classes using metrics and connections
+
* 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:
 
* The isomorphism:
  
Line 123: Line 123:
  
 
* 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>.
 
* 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>.
 +
* Chern classes are maps from <math>K_0(X)</math>
 +
* Also mention that there is a map
 +
 +
<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 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:
 
* The localization exact sequence, if <math>R</math> is a Dedekind ring, <math>K</math> its quotient field:
Line 151: Line 157:
 
<math>K_2^M(R) \cong K_2(R)</math>
 
<math>K_2^M(R) \cong K_2(R)</math>
  
for fields (cf. §17, §18<ref name="Mil71"/>).
+
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"/>.
 
References: §5-§12<ref name="Mil71"/>, Chapter III<ref name="Wei13"/>, Chapter I<ref name="Sri91"/>, Chapter 4<ref name="Ros94"/>.
Line 180: Line 188:
 
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.  
Line 207: Line 215:
 
* 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
Line 213: Line 221:
 
** 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="Blo10"/>, <ref name="Ram82"/>, <ref name="Blo00"/>, in particular 7.4 and all what is needed from before, <ref name="Zag07"/>
+
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
Line 229: Line 237:
 
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"/>.
 
More advanced reference: <ref name="Gon95"/>.
Line 241: Line 249:
 
= 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"/>.
 
More advanced reference: <ref name="Gon95"/>.
Line 256: Line 264:
 
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"/>, <ref name="Gon95"/>
+
References: <ref name="Lev05"/>, <ref name="BD94"/>, <ref name="Blo91"/>, <ref name="Gon95"/>, <ref name="Hai94"/>
  
 
= References =
 
= References =
Line 326: Line 334:
 
<ref name="Gro58">Grothendieck, A.
 
<ref name="Gro58">Grothendieck, A.
 
''La théorie des classes de Chern''. (French)
 
''La théorie des classes de Chern''. (French)
Bull. Soc. Math. France 86 1958 137–154.
+
Bull. Soc. Math. France 86 1958 137–154. </ref>
</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 17:58, 25 July 2016

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