Iterated Integrals and Algebraic Cycles: Examples and Prospects

Abstract

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work on the dilogarithm and regulators [9] in the mid 1970s, which was developed further by Beilinson [7] and Deligne (unpublished).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 paper is largely expository, beginning with an introduction to iterated integrals and Chen's de Rham theorems for loop spaces and fundamental groups. It does contain some novelties, such as the de Rham theorem for fundamental groups of smooth algebraic curves in terms of "meromorphic iterated integrals of the second kind," and the treatment of the Hodge and weight filtrations of the algebraic de Rham cohomology of loop spaces of algebraic varieties in characteristic zero. A generalization of the theorem of Carlson-Clemens-Morgan in Section 10 is presented, although the proof is not complete for lack of a rigorous theory of iterated integrals of currents. Even though there is no rigorous theory, iterated integrals of currents are a useful heuristic tool which illuminate the combinatorial and geometric content of iterated integrals. The development of this theory should be extremely useful for applications of de Rham theory to the study of algebraic cycles. The heuristic theory is discussed in Section 6.A major limitation of iterated integrals and rational homotopy theory of nonsimply connected spaces is that they usually only give information about nilpotent completions of topological invariants. This is particularly limiting in many cases, such as when studying knots and moduli spaces of curves. By using iterated integrals of twisted differential forms or certain convergent infinite sums of iterated integrals, one may get beyond nilpotence. Non-nilpotent iterated integrals and their Hodge theory should emerge when studying the periods of extensions of variations Date

Higher order Poincaré-Pontryagin functions and iterated path integrals

“…Some basic properties of iterated path integrals are summarized below. The missing proofs (and much more) may be found in R. Hain [12,13].…”

Signatures of paths transformed by polynomial maps

“…In stochastic analysis, the study of the signatures of paths arises in the theory of rough paths, where [FV10,FH14] are textbook references. Iterated integrals and the noncommutative series that encode them have also arisen in a variety of contexts in geometry and arithmetic, including the work of R. Hain in [Hai02], M. Kapranov in [Kap09], and J. Balakrishnan in [Bal13]. The results we derive in this paper have the potential for future applications in all of these contexts.…”

Iterated integrals, diagonal cycles and rational points on elliptic curves

“…The definition of the Hodge filtration on M dR,C was given by R. Hain in [Hain2], and rests on the explicit description of M dR,C in terms of K.-T. Chen's iterated integrals of smooth differential forms on path spaces. We now briefly recall this description, referring the reader to [Chen], [Hain3] and the references therein for further details. The path space on Y based at o, denoted P(Y ; o), is the set of piecewise smooth paths…”

“…The Z-module M B is in fact a prototypical example of an integral mixed Hodge structure. (See [Hain3,Cor. 9.3] and [Hain1] for a discussion in a more general setting.)…”

“…Let (Y, o) be a pointed algebraic variety over the complex numbers. Its pro-unipotent fundamental group π 1 (Y ; o) is a rich source of links between the topology, geometry and arithmetic of the variety: see for example the works of Bloch [Bl77], Chen [Chen], Deligne [De], Drinfeld [Dr], Hain [Hain3] and Kim [Ki] and the references therein. Among these developments, the contributions of Carlson, Clemens and Morgan [CCM], Harris [Har] and Pulte [Pu] in the eighties exhibited examples where the values of Chen's iterated integrals -which provide an explicit description of the underlying mixed Hodge structure of the de Rham realization of π 1 (Y ; o)-coincide with the Abel-Jacobi images of a suitable nullhomologous cycle on Y or on some other variety that one may naturally construct out of Y .…”