Lectures on Hodge Theory and Algebraic Cycles
James D. Lewis
Communications in Mathematics and Statistics ›› 2016, Vol. 4 ›› Issue (2) : 93 -188.
Lectures on Hodge Theory and Algebraic Cycles
Notes for a mini course at the University of Science and Technology of China in Hefei, China, June 23–July 12, 2014.
Chow group / Hodge theory / Algebraic cycle / Regulator / Deligne cohomology / Beilinson–Hodge conjecture / Abel–Jacobi map / Bloch–Beilinson filtration
| [1] |
|
| [2] |
|
| [3] |
Angel, P.L., Lewis, J.D. et al.: A going-up theorem for $K$-theory and normal functions. Manuscript in progress |
| [4] |
Asakura, M.: Motives, de Rham, algebraic: cohomology. In: Gordon, B., Lewis, J., Müller-Stach, S., Saito, S., Yui, N. (eds.) The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, 7–19 June, 1998, Banff, Alberta. NATO Science Series, vol. 548. Kluwer Academic Publishers, Berlin (2000) |
| [5] |
Asakura, M.: On $d\log $ of elliptic surface minus singular fibers, preprint (2006). arXiv:math/0511190v4 |
| [6] |
Asakura, M., Saito, S.: Beilinson’s Hodge conjecture with coefficients. In: Nagel, J., Peters, C. (eds.) Algebraic Cycles and Motives. LNS, vol. 344, pp. 3–37. London Mathematical Society, London (2007) |
| [7] |
Bass, H., Tate, J.: The Milnor ring of a global field. In: Algebraic $K$-theory II. Lecture Notes in Math, vol. 342, pp. 349–446. Springer-Verlag, New York (1972) |
| [8] |
Beilinson, A.: Notes on absolute Hodge cohomology. In: Applications of Algebraic $K$-Theory to Algebraic Geometry and Number Theory. Contemporary Mathematics 55, part 1, pp. 35–68. AMS, Providence (1986) |
| [9] |
|
| [10] |
|
| [11] |
|
| [12] |
|
| [13] |
|
| [14] |
Bloch, S., Ogus, A.: Gersten’s conjecture and the homology of schemes. Annales scientifiques de l’É.N.S. $4^e$ série, tome 7(2), 181–201 (1974) |
| [15] |
Kerr, M., Lewis, J.D., Lopatto, P.: Simplicial Abel-Jacobi maps and reciprocity laws, preprint (2015). arXiv:1502.05459 |
| [16] |
|
| [17] |
|
| [18] |
|
| [19] |
|
| [20] |
|
| [21] |
|
| [22] |
|
| [23] |
|
| [24] |
Collino, A.: Indecomposable motivic cohomology classes on quartic surfaces and on cubic fourfolds. Algebraic $K$-Theory and Its Applications (Trieste, 1997), pp. 370–402. World Scientific Publishing, River Edge (1999) |
| [25] |
|
| [26] |
|
| [27] |
de Jeu, R., Lewis, J.D (with an appendix by M. Asakura): Beilinson’s Hodge conjecture for smooth varieties. J K-Theory 11(2), 243–282 (2013) |
| [28] |
de Jeu, R., Lewis, J.D., Patel, D.: A relative version of the Beilinson-Hodge conjecture. In: Kerr, M., Pearlstein, G. (eds.) Recent Advances in Hodge Theory: Period Domains, Algebraic Cycles, and Arithmetic. Cambridge University Press, Cambridge (to appear) |
| [29] |
|
| [30] |
|
| [31] |
|
| [32] |
|
| [33] |
|
| [34] |
Gordon, B.B., Lewis, J.D.: Indecomposable higher Chow cycles. In: The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), vol. 548, pp. 193–224. Nato Science Series C: - Mathematical and Physical Sciences, vol. 548. Kluwer Academic Publication, Dordrecht (2000) |
| [35] |
Green, M., Griffiths, P.: The regulator map for a general curve. In: Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), pp. 117–127, Contemporary Mathematics, vol. 312. American Mathematical Society, Providence (2002) |
| [36] |
|
| [37] |
|
| [38] |
|
| [39] |
|
| [40] |
|
| [41] |
Jannsen, U. Deligne cohomology. In: Rapoport, M., Schappacher, N., Schneider, P. (eds.) Hodge-${{\cal D}}$-Functions. Perspective Mathematics, vol. 4, pp. 305–372. Academic Press, New York (1988) |
| [42] |
Jannsen, U.: Motives, Mixed, Algebraic $K$-Theory. Lecture Notes in Math, vol. 1000. Springer-Verlag, Berlin (1990) |
| [43] |
Jannsen, U.: Equivalence relations on algebraic cycles. In: Gordon, B., Lewis, J., Müller-Stach, S., Saito, S., Yui), N. (eds.) The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, 7–19 June, 1998, Banff, Alberta, Canada. NATO Science Series, vol. 548, pp. 225–260. Kluwer Academic Publishers, Berlin (2000) |
| [44] |
Kahn, B.: Groupe de Brauer et $(2,1)$-cycles indecomposables. Preprint (2011) |
| [45] |
|
| [46] |
|
| [47] |
|
| [48] |
|
| [49] |
|
| [50] |
Kang, S-J., Lewis, J. D.: Beilinson’s Hodge conjecture for $K_1$ revisited, Cycles, motives and Shimura varieties, 197-215, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2010 |
| [51] |
|
| [52] |
Lewis, J.D.: A Survey of the Hodge Conjecture. Appendix B by B. Brent Gordon. CRM Monograph Series, vol. 10, 2nd edn. American Mathematical Society, Providence (1999) |
| [53] |
Lewis, J.D.: Real regulators on Milnor complexes. $K$-Theory 25(3), 277–298 (2002) |
| [54] |
Lewis, J.D.: Regulators of Chow cycles on Calabi-Yau varieties. In: Calabi-Yau Varieties and Mirror Symmetry (Toronto, ON, 2001). Fields Institute Communications, vol. 38, pp. 87–117. American Mathematical Society, Providence (2003) |
| [55] |
|
| [56] |
|
| [57] |
|
| [58] |
|
| [59] |
|
| [60] |
Lewis, J. D.: Cycles on varieties over subfields of ${\mathbb{C}}$ and cubic equivalence. In: Motives and Algebraic Cycles, pp. 233–247. Fields Institute Communications, vol. 56. American Mathematical Society, Providence (2009) |
| [61] |
Lewis, J. D.: Hodge type conjectures and the Bloch-Kato theorem. In: Hodge Theory, Complex Geometry, and Representation Theory, pp. 235–258. Contemporary Mathathematics, vol. 608. American Mathematical Society, Providence (2014) |
| [62] |
Lewis, J. D.: Transcendental methods in the study of algebraic cycles with a special emphasis on Calabi-Yau varieties. In: Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds. Fields Institute Communications, vol. 67, 29-69, Springer, New York (2013) |
| [63] |
|
| [64] |
Milne, J.: Étale Cohomology. Princeton Mathematical Series, vol. 33. Princeton University Press, Princeton (1980) |
| [65] |
|
| [66] |
Müller-Stach, S.: Algebraic cycle complexes. In: Saito, S. (ed.) Proceedings of the NATO Advanced Study Institute on the Arithmetic and Geometry of Algebraic Cycles 548, (Lewis, Yui, Gordon, Müller-Stach, pp. 285–305. Kluwer Academic Publishers, Dordrecht (2000) |
| [67] |
|
| [68] |
|
| [69] |
Saito, M.: Direct image of logarithmic complexes and infinitesimal invariants of cycles. In: Algebraic Cycles and Motives, vol. 2, pp. 304–318. London Mathematical Society Lecture Note Series, vol. 344. Cambridge University Press, Cambridge (2007) |
| [70] |
|
| [71] |
Saito, S.: Beilinson’s Hodge and Tate conjectures. In: Müller-Stach, S., Peters, C. (eds.) Transcendental Aspects of Algebraic Cycles. LNS, vol. 313, pp. 276–290. London Mathematical Society, London (2004) |
| [72] |
Steenbrink, J.H.M.: A summary of mixed Hodge theory, Motives (Seattle, WA, 1991), pp. 31–41. In: Proceedings of Symposia in Pure Mathematics, vol. 55, Part 1. American Mathematical Society, Providence (1994) |
| [73] |
|
| [74] |
|
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