Results 1  10
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19
Bloch invariants of hyperbolic 3manifolds
"... We define an invariant β(M) of a finite volume hyperbolic 3manifold M in the Bloch group B(C) and show it is determined by the simplex parameters of any degree one ideal triangulation of M. We show β(M) lies in a subgroup of B(C) of finite Qrank determined by the invariant trace field of M. Moreo ..."
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Cited by 32 (6 self)
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We define an invariant β(M) of a finite volume hyperbolic 3manifold M in the Bloch group B(C) and show it is determined by the simplex parameters of any degree one ideal triangulation of M. We show β(M) lies in a subgroup of B(C) of finite Qrank determined by the invariant trace field of M. Moreover, the ChernSimons invariant of M is determined modulo rationals by β(M). This leads to a simplicial formula and rationality results for the Chern Simons invariant which appear elsewhere. Generalizations of β(M) are also described, as well as several interesting examples. An appendix describes a scissors congruence interpretation of B(C).
The existence of higher logarithms
, 1993
"... Abstract. In this paper we establish the existence of all higher logarithms as Deligne cohomology classes in a sense slightly weaker than that of [13, §12], but in a sense that should be strong enough for defining Chern classes on the algebraic Ktheory of complex algebraic varieties. In particular, ..."
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Cited by 18 (3 self)
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Abstract. In this paper we establish the existence of all higher logarithms as Deligne cohomology classes in a sense slightly weaker than that of [13, §12], but in a sense that should be strong enough for defining Chern classes on the algebraic Ktheory of complex algebraic varieties. In particular, for each integer p ≥ 1, we construct a multivalued holomorphic function on a Zariski open subset of the self dual grassmannian of pplanes in C 2p which satisfies a canonical 2p + 1 term functional equation. The key new technical ingredient is the construction of a topology on the generic part of each Grassmannian which is coarser than the Zariski topology and where each open contains another which is both a K(π,1) and a rational K(π,1). 1.
Multiple polylogarithms: analytic continuation, monodromy, and variations of mixed Hodge structure, Contemporary trends in algebraic geometry and algebraic topology
 Department of Mathematics, Duke University
, 2002
"... Abstract. In this exposition we shall describe a new way to analytically continue the multiple polylogarithms by using Chen’s theory of iterated path integrals. Then we explicitly determine the good unipotent variations of mixed HodgeTate structures (MHS) related to multiple logarithms and some oth ..."
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Cited by 8 (4 self)
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Abstract. In this exposition we shall describe a new way to analytically continue the multiple polylogarithms by using Chen’s theory of iterated path integrals. Then we explicitly determine the good unipotent variations of mixed HodgeTate structures (MHS) related to multiple logarithms and some other multiple polylogarithms of lower weights. Following Deligne and Beilinson we define the singlevalued real analytic version of the multiple polylogarithms which generalizes the wellknown result of Zagier on classical polylogarithms. At the end, motivated by Zagier’s conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the singlevalued version of multiple polylogarithms. The main results of this paper with complete proofs will appear elsewhere.
STRING DUALITY AND ENUMERATION OF CURVES BY JACOBI FORMS
, 1998
"... For a CalabiYau threefold admitting both a K3 fibration and an elliptic fibration (with some extra conditions) we discuss candidate asymptotic expressions of the genus 0 and 1 GromovWitten potentials in the limit (possibly corresponding to the perturbative regime of a heterotic string) where the a ..."
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Cited by 8 (0 self)
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For a CalabiYau threefold admitting both a K3 fibration and an elliptic fibration (with some extra conditions) we discuss candidate asymptotic expressions of the genus 0 and 1 GromovWitten potentials in the limit (possibly corresponding to the perturbative regime of a heterotic string) where the area of the base of the K3 fibration is very large. The expressions are constructed by lifting procedures using nearly holomorphic Weylinvariant Jacobi forms. The method we use is similar to the one introduced by Borcherds for the constructions of automorphic forms on type IV domains as infinite products and employs in an essential way the elliptic polylogarithms of Beilinson and Levin. In particular, if we take a further limit where the base of the elliptic fibration decompactifies, the GromovWitten potentials are expressed simply by these elliptic polylogarithms. The theta correspondence considered by Harvey and Moore which they used to extract the expression for the perturbative prepotential is closely related to the EisensteinKronecker double series and hence the real versions of elliptic polylogarithms introduced by Zagier.
Rationality problems for Ktheory and ChernSimons invariants of hyperbolic 3manifolds
"... This paper makes certain observations regarding some conjectures of Milnor and Ramakrishnan in hyperbolic geometry and algebraic Ktheory. As a consequence of our observations, we obtain new results and conjectures regarding the rationality and irrationality of ChernSimons invariants of hyperbolic ..."
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Cited by 7 (1 self)
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This paper makes certain observations regarding some conjectures of Milnor and Ramakrishnan in hyperbolic geometry and algebraic Ktheory. As a consequence of our observations, we obtain new results and conjectures regarding the rationality and irrationality of ChernSimons invariants of hyperbolic 3manifolds. In this paper, by a hyperbolic 3manifold, we shall mean a complete, oriented hyperbolic 3manifold with finite volume. So a hyperbolic 3manifold is compact or has finitely many cusps as its ends (see e.g. [20]). By a cusped manifold we shall mean a noncompact hyperbolic 3manifold. A hyperbolic 3manifold M is a quotient of the hyperbolic 3space H3 by a discrete subgroup Γ of P SL2(C) with finite covolume. The isometry class of M determines the discrete subgroup up to conjugation. To each subgroup Γ of P SL2(C), we can associate the trace field of Γ, that is, the subfield of C generated by traces of all elements in Γ. The trace field clearly depends only on the conjugacy class of Γ, so one can define it to be the trace field of the hyperbolic 3manifold. However, the trace field is not an invariant of commensurability class of Γ, although it is not
RESURGENCE OF THE FRACTIONAL POLYLOGARITHMS
"... Abstract. The fractional polylogarithms, depending on a complex parameter α, are defined by a series which is analytic inside the unit disk. After an elementary conversion of the series into an integral presentation, we show that the fractional polylogarithms are multivalued analytic functions in th ..."
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Cited by 5 (1 self)
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Abstract. The fractional polylogarithms, depending on a complex parameter α, are defined by a series which is analytic inside the unit disk. After an elementary conversion of the series into an integral presentation, we show that the fractional polylogarithms are multivalued analytic functions in the complex plane minus 0 and 1. For noninteger values of α, we prove the analytic continuation, compute the monodromy around 0 and 1, give a MittagLeffler decomposition and computer the asymptotic behavior for large values of the complex variable. The fractional polylogarithms are building blocks of resurgent functions that are used in proving that certain power series associated with knotted objects are resurgent. This is explained in a separate publication [CG3]. The motivic or physical interpretation of the monodromy of the fractional
Vector Braids
, 1994
"... In this paper we define a new family of groups which generalize the classical braid groups on C . We denote this family by fB n g nm+1 where n; m 2 N. The family fB n g n2N is the set of classical braid groups on n strings. ..."
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Cited by 5 (0 self)
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In this paper we define a new family of groups which generalize the classical braid groups on C . We denote this family by fB n g nm+1 where n; m 2 N. The family fB n g n2N is the set of classical braid groups on n strings.
Evaluation of a class of double Lvalues
 Proc. Amer. Math. Soc
"... Abstract. An analytic proof of an evaluation theorem for the “convolution”type double Lvalues of nonprincipal characters is given. Along the way, Dirichlet character analogues of generalized single and double polylogarithms are defined. The monodromies of these functions play a pivotal role. 1. ..."
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Cited by 3 (1 self)
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Abstract. An analytic proof of an evaluation theorem for the “convolution”type double Lvalues of nonprincipal characters is given. Along the way, Dirichlet character analogues of generalized single and double polylogarithms are defined. The monodromies of these functions play a pivotal role. 1.
Real Grassmann polylogarithms and Chern classes
, 1996
"... In this paper we define and prove the existence of real Grassmann polylogarithms which are the real singlevalued analogues of the Grassmann polylogarithms defined in [24] and constructed in [24, 25, 26, 23]. We prove that if ηX is the generic point of a complex algebraic variety X, then the mth suc ..."
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Cited by 3 (2 self)
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In this paper we define and prove the existence of real Grassmann polylogarithms which are the real singlevalued analogues of the Grassmann polylogarithms defined in [24] and constructed in [24, 25, 26, 23]. We prove that if ηX is the generic point of a complex algebraic variety X, then the mth such polylogarithm represents the