# Codimension 2

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# Motivation

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 [1]:

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

 20.4. Introduction Braeunling 27.4. Cech cohomology and the Hodge conjecture in codim 1 Vetere 4.5. Chow groups and Chern classes Bertini 11.5. Elementary algebraic K-Theory: K0 and K1 Schwald 18.5. Pentecost 25.5. Elementary algebraic K-Theory: K2 Zaccanelli 1.6. Cycles in codim 2, part I Voelkel 8.6. Cycles in codim 2, part II Kelly 15.6. The dilogarithm Bergner, Sartori 22.6. A generalization of the exponential sequence Soergel 29.6. The dilog, scissors congruences and volumes of hyperbolic spaces Lye 6.7. The dilog and zeta functions Eberhardt 13.7. The dilog in physics Scheidegger 20.7. Outlook: Mixed motives and periods Hörmann

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 δ homomorphism. Treat the analytic and Zariski topology parallel.
• Classification of vector bundles by Cech-cocycles, Theorem I.5.10[2].
• Mention also that for a projective complex algebraic variety X, complex analytic (rank n) vector bundles, classified by $H^1(X^{an}, \mathrm{GL}_n(\mathcal{O}_X^{an}))$ (analytic Cech cohomology), are the same as algebraic (rank n) vector bundles on X, classified by $H^1(X, \mathrm{GL}_n(\mathcal{O}_X))$ (Zariski Cech cohomology).
• The exponential sequence

$0 \rightarrow 2\pi i \mathbb{Z}_X \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^* \rightarrow 1$

• Proof of the Hodge conjecture for divisors using the long exact sequence associated with the exponential sequence
• Explain also Lemma 2.1[3] on a non-Abelian version of the long exact sequence. (This will be needed later)

# Chow groups and Chern classes

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 $\mathbb{C}$.
• Splitting principle: Theorem 5.19[2]
• Definition of algebraic Chern classes using the splitting prinicple [4], 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:

$Pic(X) \cong CH^1(X)$

(Maybe also isomorphism of CH^1(X) with quotient sheaf of the sheaf of meromorphic functions! I.5.12[2])

# Elementary algebraic K-Theory: K0 and K1

• The definition of K0(R) for a ring R and K0(X) for a variety X.
• Chern classes are maps from K0(X)
• Also mention that there is a map

$\{\text{codim } n \text{ cycles}\} \rightarrow K_0(X)$

which is right inverse to the n-th Chern class up to a rational factor.

• The Whitehead Lemma, 3.1[5] and the definition of K1(R)
• The localization exact sequence, if R is a Dedekind ring, K its quotient field:

$\bigoplus_{\mathfrak{p}}K_1(R/\mathfrak{p}) \rightarrow K_1(R) \rightarrow K_1(K) \rightarrow \bigoplus_{\mathfrak{p}} K_0(R/\mathfrak{p}) \rightarrow K_0(R) \rightarrow K_0(K) \rightarrow 0$

References: §1-§5[5], II, §8[2], Chapter I[6], 1.1-1.2 and 2.1-2.2[7]

# Elementary algebraic K-Theory: K2

(only for commuative rings)

• Motivation of higher K-groups by non-exactness of the localization sequence on the left
• For a ring R define the Steinberg groups Stn(R) and St(R), Definition p.40[5].
• Definition of K2(R) as the Kernel of the map $St(R) \rightarrow \mathrm{GL}(R)$, p.40[5].
• Sketch of proof Theorem 5.1[5].
• Mention the description as universal central extension.
• Mention that and how K2(R) is a module over K0(R).
• Introduce the symbol $\{a, b\} \in K_2(R)$ for a pair of elements $a, b \in R$ of a commutative ring R.

Theorem 8.8[5] interprets the symbol as bimultiplicative skew-symmetric map

$K_1(R) \times K_1(R) \rightarrow K_2(R)$

• Introduction of Milnor K-groups $K_n^M$ and give the idea of the proof of the statement

$K_2^M(R) \cong K_2(R)$

for fields (cf. §11, §12[5]).

• extension of the localization squence to K2 (the tame symbol).

References: §5-§12[5], Chapter III[2], Chapter I[6], Chapter 4[7]. A nice collection of facts about K2 can be found here: [8].

More advances references: [9], [10] .

# Cycles in codim 2

These are two talks.

• Explain how K2, and St (Steinberg group) can be sheafified
• Prove Proposition 1.13[3]
• Remind of Lemma 2.1[3] which has been discussed in the talk on Cech cohomology
• Deduce the long exact sequence associated with the sequence of sheaves

$1 \rightarrow K_2 \rightarrow St \rightarrow \mathrm{SL} \rightarrow 1$

and obtain a map

$CK_2: H^1(X, \mathrm{SL}(\mathcal{O}_X)) \rightarrow H^2(X, K_2(\mathcal{O}_X))$

of Cech cohomology groups. The purpose of the rest of the two talks is to see that

$H^2(X, K_2(\mathcal{O}_X)) \cong CH^2(X)$

in such a way that CK2 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: [3], ([11], §4)

• Bloch's construction (§3[11]) of a map

$H^2(X, K_2) \rightarrow CH^2(X)$

• Show for a vector bundle E (of trivial determinant) that this map identifies SK2(E) with the second Chern class SK2(E) defined in the third talk (Theorem 4.2'[11]).
• Explain that (via resolutions of the structure sheaf of codimension 2 subvarieties by vector bundles) one gets a map

$\{\text{codim 2 subvarieties}\} \rightarrow H^2(X, K_2)$

One would like to have that it factors through CH2(X) and establishes that the morphism $H^2(X, K_2) \rightarrow CH^2(X)$ 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 $K_n^M(\mathcal{O}_X)$ (Sheafified higher Milnor K-groups, cf. talk 5) that $H^n(X, K_n^M) \rightarrow CH^n(X)$. In the end one could say a couple of words on this (Reference: )

References: [3], ([11], §4)

# The dilogarithm

• Introduction of the dilogarithm via integral formula and power series [1]
• 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 [12] but restricted to the dilogarithm. (A theoretical, more modern explanation in terms of mixed Hodge structures can be found in [13] 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 [11] and its proof
• a good explanation for what is going on can be found in [14]
• deduce special values as presented in [1], I, 1

References: [11], [12], [15], in particular 7.4 and all what is needed from before, [1], [16]

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 [1] can be presented.

# A generalization of the exponential sequence

A survey of Bloch's article[20].

References: [20], ([11], §6)

Reference for the Dilogarithm: [1], [16]

# The dilog, scissors congruences and volumes of hyperbolic spaces

References: [21], [22], [16]

Further reading: [23], [24] especially the introduction

# The dilog and zeta functions

References: [15], ([1], I, §5), [16]

Starting point: The quantum dilogarithm ([1], 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: [25], [26], [13], [18], [16]

# References

1. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Zagier, D.; The dilogarithm function. Frontiers in number theory, physics, and geometry. II, 3–65, Springer, Berlin, 2007.
2. 2.0 2.1 2.2 2.3 2.4 Weibel, Ch. A.; The K-book. An introduction to algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. xii+618 pp.
3. 3.0 3.1 3.2 3.3 3.4 Bloch, S.; K2 and algebraic cycles. Ann. of Math. (2) 99 (1974), 349–379.
4. Grothendieck, A. La théorie des classes de Chern. (French) Bull. Soc. Math. France 86 1958 137–154.
5. 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Milnor, J., Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. xiii+184 pp.
6. 6.0 6.1 Srinivas, V.; Algebraic K-theory. Progress in Mathematics, 90. Birkhäuser Boston, Inc., Boston, MA, 1991. xvi+314 pp.
7. 7.0 7.1 Rosenberg, J.; Algebraic K-theory and its applications. Graduate Texts in Mathematics, 147. Springer-Verlag, New York, 1994. x+392 pp.
8. Dalawat, Ch. S.; Some aspects of the functor K2 of fields. arXiv:math/0311099, 2006
9. Suslin, A. A.; Homology of GLn, characteristic classes and Milnor K-theory. Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982), 357–375, Lecture Notes in Math., 1046, Springer, Berlin, 1984.
10. 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.
11. 11.0 11.1 11.2 11.3 11.4 11.5 11.6 Bloch, S.; Lectures on algebraic cycles. Second edition. New Mathematical Monographs, 16. Cambridge University Press, Cambridge, 2010. xxiv+130 pp.
12. 12.0 12.1 Ramakrishnan, D.; On the monodromy of higher logarithms. Proc. Amer. Math. Soc. 85 (1982), no. 4, 596–599.
13. 13.0 13.1 Bloch, S.; Function theory of polylogarithms. Structural properties of polylogarithms, 275–285, Math. Surveys Monogr., 37, Amer. Math. Soc., Providence, RI, 1991.
14. Suslin, A. A.; K3 of a field, and the Bloch group. (Russian) Translated in Proc. Steklov Inst. Math. 1991, no. 4, 217–239. Galois theory, rings, algebraic groups and their applications (Russian).
15. 15.0 15.1 Bloch, S.; Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM Monograph Series, 11. American Mathematical Society, Providence, RI, 2000. x+97 pp.
16. 16.0 16.1 16.2 16.3 16.4 Hain,R.M.; Classical Polylogarithms, Motives, in Motives (Seattle, WA, 1991), manual, vol. 55 (1994), pp. 3-42, American Mathematical Society
17. Bloch, S.; The dilogarithm and extensions of Lie algebras. Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), pp. 1–23, Lecture Notes in Math., 854, Springer, Berlin-New York, 1981.
18. 18.0 18.1 18.2 18.3 Goncharov, A. B.; Geometry of configurations, polylogarithms, and motivic cohomology. Adv. Math. 114 (1995), no. 2, 197–318.
19. Lewin, L.; Polylogarithms and associated functions. With a foreword by A. J. Van der Poorten. North-Holland Publishing Co., New York-Amsterdam, 1981. xvii+359 pp.
20. 20.0 20.1 Bloch, S.; Applications of the dilogarithm function in algebraic K-theory and algebraic geometry. Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), pp. 103–114, Kinokuniya Book Store, Tokyo, 1978.
21. Dupont, J. L.; Sah, C. H.; Scissors congruences. II. J. Pure Appl. Algebra, 25(2):159–195, 1982.
22. Goncharov, A.; Volumes of hyperbolic manifolds and mixed Tate motives. J. Amer. Math. Soc. 12 (1999), no. 2, 569–618.
23. Goette, S.; Zickert, Ch.; The extended Bloch group and the Cheeger-Chern-Simons class. Geom. Topol. 11 (2007), 1623–1635.
24. Hutchinson, K.; A Bloch-Wigner complex for SL2. J. K-Theory 12 (2013), no. 1, 15–68.
25. Levine, Marc; Mixed motives. Handbook of K-theory. Vol. 1, 2, 429–521, Springer, Berlin, 2005.
26. 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.