Lectures on Hodge Theory and Algebraic Cycles
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Lectures on Hodge Theory and Algebraic Cycles James D. Lewis1
Received: 25 June 2015 / Revised: 11 December 2015 / Accepted: 16 December 2015 / Published online: 4 June 2016 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2016
Abstract Notes for a mini course at the University of Science and Technology of China in Hefei, China, June 23–July 12, 2014. Keywords Chow group · Hodge theory · Algebraic cycle · Regulator · Deligne cohomology · Beilinson–Hodge conjecture · Abel–Jacobi map · Bloch–Beilinson filtration Mathematics Subject Classification 14C35 · 14C25 · 19E15 · 14C30
1 Preface These lectures are based on a mini course of 24 lectures presented at the USTC in the summer of 2014. The intent was to present new material beyond some earlier lecture notes the author presented in Mexico in the summer of 2000 on algebraic cycles [55]. The central core of these notes is based on a course on Deligne cohomology that the author offered at the University of Alberta some years ago, but these lecture notes go well beyond that, including unedited subsections of [61,62] which may not be easily accessible for the reader to find. Some new materials are also included. In summary, there is a strong emphasis on Hodge theory, K -theory, algebraic cycles, and regulators.
Partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
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James D. Lewis [email protected] University of Alberta, 632 Central Academic Building, Edmonton, AB T6G 2G1, Canada
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The author owes deep debt of gratitude to his host Mao Sheng, as well as to Xi Chen, for working out the logistics of arranging this visit. As mathematicians, the author views both of them in the highest regard! A special thanks goes to the referee for a careful read of this paper. China is now evolving as a major center for mathematics in the world. Much of this can be attributed to the tireless effort of our friend and colleague S.-T. Yau, as a major influence in that country. This is a monumental accomplishment that the entire mathematical community can take pride in.
2 Prerequisite Reading Material These lecture notes are fairly advanced. The reader would certainly benefit by looking at the [36,52] for the complex analytic side of the subject, and to [13,55] for that part pertaining to algebraic cycles.
3 Outline of Lecture Notes and Preliminary Material 3.1 Motivation In the (now) classical literature, there are two cycle class maps used to detect (generalized, viz., motivic) algebraic cycles CHr (X, m) of codimension r , (see [11]), on a projective algebraic manifold X . If ξ is such a cycle, the fundamental class determines an element of Betti cohomology [ξ ] ∈ H 2r −m (X, Z(r )). Failing this (viz., if [ξ ] = 0), then we say that ξ is null-homologous, and we attempt to detect it via a secondary cycle class map, which takes values in a particular complex torus, A J (ξ ) ∈ J r,m (X ), where A J is the so-called Abel–Jacobi map. Deligne cohomol
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