Codimension 2
From Mathematics
(→Schedule) |
(→Schedule) |
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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]] | ||
| − | | | + | |Vetere |
|- | |- | ||
| 4.5. | | 4.5. | ||
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| 8.6. | | 8.6. | ||
|[[#Cycles in codim 2|Cycles in codim 2, part II]] | |[[#Cycles in codim 2|Cycles in codim 2, part II]] | ||
| − | | | + | |Kelly |
|- | |- | ||
| 15.6. | | 15.6. | ||
|[[#The dilogarithm|The dilogarithm]] | |[[#The dilogarithm|The dilogarithm]] | ||
| − | | | + | |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]] | ||
| − | | | + | |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]] | ||
| − | | | + | |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]] | ||
| − | | | + | |Eberhardt |
|- | |- | ||
| 13.7. | | 13.7. | ||
|[[#The dilog in physics|The dilog in physics]] | |[[#The dilog in physics|The dilog in physics]] | ||
| − | | | + | |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]] | ||
| − | | | + | |Hörmann |
|} | |} | ||
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<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 | ||
| − | * | + | * 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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* 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 | + | * 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 | + | * 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: | ||
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<math>K_2^M(R) \cong K_2(R)</math> | <math>K_2^M(R) \cong K_2(R)</math> | ||
| − | for fields (cf. | + | for fields (cf. §11, §12<ref name="Mil71"/>). |
* extension of the localization squence to <math>K_2</math> (the tame symbol). | * extension of the localization squence to <math>K_2</math> (the tame symbol). | ||
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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 | + | <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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** 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 | ||
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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"/>. | ||
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= 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"/>. | ||
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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 = | ||
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<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> | ||
