Contemporary Trends in Algebraic Geometry and Algebraic Topology

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• Local to Global

• Increase in Dimensions

• Commutative to Non-Commutative

• Linear to Non-Linear

• Geometry versus Algebra

• Techniques in Common
  • Homology theory
  • K-theory
  • Lie groups

• Finite Groups

• Impact of Physics

• Historical Summary

The following sections are included:

• Introduction

• Proof of Theorem 1.2

• Acknowledgments

• References

We study the question of when a map between the fundamental groups of algebraic curves is induced by an underlying geometric map on the curves themselves. The quotients of the group ring by powers of the augmentation ideal I of the fundamental group of algebraic varieties are known to have additional structures (mixed Hodge structures, Galois module structures or motivic structures). These structures must be respected by any map induced by geometry. We describe how this question can be reduced to the calculation of an invariant lying in a certain Ext group, and calculate it explicitly for the mixed Hodge fundamental group module I³. We show that this invariant has a K-theoretic interpretation, and give an example of a map between fundamental groups of curves module I³ which is not geometric.

The goal of this paper is to produce evidence for a connection between the work of Kuo-Tsai Chen on iterated integrals and de Rham homotopy theory on the one hand, and the work of Wei-Liang Chow on algebraic cycles on the other. Evidence for such a profound link has been emerging steadily since the early 1980s when Carlson, Clemens and Morgan [13] and Bruno Harris [40] gave examples where the periods of non-abelian iterated integrals coincide with the periods of homologically trivial algebraic cycles. Algebraic cycles and the classical Chow groups are nowadays considered in the broader arena of motives, algebraic K-theory and higher Chow groups. This putative connection is best viewed in this larger context. Examples relating iterated integrals and motives go back to Bloch's work on the dilogarithm and regulators [9] in the mid 1970s, which was developed further by Beilinson [7] and Deligne (unpublished). Further evidence to support a connection between de Rham homotopy theory and iterated integrals includes [48,4,30,57,50,22,37,59,58,25,38,51,55,19,60,20]. Chen would have been delighted by these developments, as he believed iterated integrals and loopspaces contain non-trivial geometric information and would one day become a useful mathematical tool outside topology.

The following sections are included:

• Introduction

• Chen's Iterated Integrals on Compact Riemann Surfaces and Generalizations
  • Iterated integrals involving two differential forms
  • Real Archimedean height pairing on a compact oriented Riemannian manifold X of real dimension m
  • Iterated integrals for the diagonal embedding of S in S × S × S
  • Generalized pairing in S × S × S

• The Complex Kahler Height Pairing Using dd^c

• References

The following sections are included:

• Weak Semistable Model

• Real Oriented Blowing-up

• Weight Filtration

• Relative Log de Rham Complex

• Cohomological Mixed Hodge Complex

• References

The following sections are included:

• Statement of the Main Result and Preliminaries
  • Main result
  • Equality of potentials
  • Polarization

• Algebraic Extension of a Germ of Holomorphic Isometric Embedding
  • Statement of Proposition 2.1
  • Prof of Proposition 2.1 for n = 1
  • Proof of Proposition 2.1 for general n

• Proof of Theorem 1.1

• Acknowledgement

• References

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 Hodge-Tate structures (MHS) related to multiple logarithms and some other multiple polylogarithms of lower weights. Following Deligne and Beilinson we define the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known 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 single-valued version of multiple polylogarithms. The main results of this paper with complete proofs will appear elsewhere.

The following sections are included:

• What is a Real Variety?

• Complex Manifolds which are K(π, 1)'s and Real Curves of Genus 1
  • Real curves of genus one
  • The description of the moduli space for the case υ = 1

• Real Hyperelliptic Surfaces on the Scene

• Tori and Curve Times a Torus

• The Blanchard–Calabi Threefolds

• Moduli Spaces of Surfaces of General Type

• Some Non Real Triangle Curves

• The Examples of Surfaces Isogenous to a Product

• Acknowledgments

• References

After a long illness Wei-Liang died on August 10, 1995. He and I first met in Hamburg, Germany, in October 1934, when I had just come from China as an entering student while he was on his way from Göttingen to Leipzig in order to work with van der Waerden…

The following sections are included:

• Chow Coordinates

• Abelian Varieties and Group Varieties

• Homogeneous Spaces

• TheChow Ring

• Algebraic Geometry over Rings

• Algebraicity of Analytic Objects

• Other Works in Algebraic Geometry

• PDE

• References

Kuo-Tsai Chen was born on July 15, 1923 in Chekiang, China. He earned a Bachelor of Science degree in mathematics from Southwest Associated University in Kungming in 1946. He then moved to Shanghai, where he became an Assistant at the Mathematics Institute of the Academia Sinica in 1946–47. On the recommendation of its director, Shing-Shen Chern, he went to work with Samuel Eilenberg at Indiana University. After one year there, he followed Eilenberg to Columbia University in New York, where he received his doctorate in 1950. During his graduate studies, he was a mathematics instructor at the National Bible Institute in New York from 1948 to 1950, and an Assistant at Columbia University in 1949–50. After being awarded his Ph.D. degree, he went first to Princeton University as an instructor in 1950–51, and then to the University of Illinois as a Research Associate in 1951–52…