Ma 191 b Section 6: Motives and Periods

NOTE THE SCHEDULE CHANGE: starting January 13, the class will meet on MW 2:00-3:30 in Sloan 151.

Brief Course Description

References

• Yves Andre, "Une introduction aux motifs", Societe Mathematique de France, 2004.
• Jacob P. Murre, Jan Nagel, Chris A. M. Peters, "Lectures on the Theory of Pure Motives", University Lecture Series, AMS, 2013.

Additional reading material

• Maxim Kontsevich and Don Zagier, "Periods", in `Mathematics unlimited - 2001 and beyond', 771-808, Springer, 2001.
• Alexander Goncharov and Yuri Manin, "Multiple zeta-motives and moduli spaces M_{0,n}", Compos. Math. 140 (2004), no. 1, 1-14, arXiv:math/0204102.
• Francis Brown, "Mixed Tate motives over Z", Ann. of Math. (2) 175 (2012), no. 2, 949-976, arXiv:1102.1312
• Franziska Bittner, "The universal Euler characteristic for varieties of characteristic zero", Compos. Math. 140 (2004), no. 4, 1011-1032, arXiv:math/0111062.
• Michael Larsen, Valery Lunts, "Motivic measures and stable birational geometry", Mosc. Math. J. 3 (2003), no. 1, 85-95, 259, arXiv:math/0110255.
• Matilde Marcolli, Goncalo Tabuada, "Noncommutative motives and their applications", arXiv:1311.2867.

Some suggested reading material for presentations

• Scholl, A. J. "Motives for modular forms", Invent. Math. 100 (1990), no. 2, 419-430.
• Kahn, Bruno; Sebastian, Ronnie, "Smash-nilpotent cycles on abelian 3-folds", Math. Res. Lett. 16 (2009), no. 6, 1007-1010.
• Kahn, Bruno, "Zeta functions and motives", Pure Appl. Math. Q. 5 (2009), no. 1, 507-570.
• Manin, Yuri I. "Moduli, motives, mirrors", European Congress of Mathematics, Vol. I (Barcelona, 2000), 53-73, Birkhauser, 2001.
• Manin, Yuri "Lectures on zeta functions and motives (according to Deninger and Kurokawa)", Asterisque, 228 (1995) N.4, 121-163.
• Deninger, Christopher, "On the Gamma-factors attached to motives", Invent. Math. (1991) N.2, 245-261.
• Ramachandran, Niranjan, "Values of zeta functions at s=1/2", Int. Math. Res. Not. (2005) N.25, 1519-1541.

Lectures

• Tue Jan 7: Origins of the theory of motives: enumerative geometry problems, counting intersections (example of Bezout's theorem), counting of points over finite fields, zeta function, Frobenius and fixed points, algebraic and numerical relations; cohomology theories for algebraic varieties, cycles and cohomology classes, homological relation, Lefschetz tace formula and intersection numbers, Lefschetz formula, Frobenius, and points over finite fields; Galois theory, Galois-Grothendieck correspondence and its linearization; Algebraic cycles and operations: product, pullback, intersection product, pushforward, cycles as correspondences.
• Thu Jan 9: Equivalence relations on algebraic cycles: axioms of good equivaence relations; rational equivalence, properties of Chow groups; algebraic equivalence and relation to rational equivalence; smash-nilpotence equivalence and relation to other equivalence relations; axioms of Weil cohomology; homological equivalence; numerical equivalence; relations between equivalences; nilpotence conjecture.
• Mon Jan 13: Outline of the proof of the Voisin-Voevodsky theorem on algebraic and smash nilpotence equivalence; construction of the category of pure motives: correspondences, composition, projections; first step, additive category with correspondences as morphisms; second step, effective motives (pseudo-abelian envelope); third step, motives (Tate motives, Tate twists).
• Wed Jan 15: direct sum and tensor product of motives, Lefschetz and Tate motive, motivic interpretation of Chow groups, homology groups of motives, motives of curves and abelian varieties up to isogeny, Manin's identity principle, motives of projective spaces and projective bundles, blowup formula for motives
• Mon Jan 20: holiday
• Wed Jan 22: Grothendieck's standard conjectures: Kuenneth conjecture C(X) and sign conjecture, weight graduation; Lefschetz type conjecture B(X), polarization, sl2-triplets, primitive decomposition, Lefschetz involution; standard conjecture of Hodge type I(X), Hodge and Lefschetz involutions; standard conjecture D(X) homological = numerical equivalence; implications between standard conjectures
• Mon Jan 27: Tannakian categories and the Tannakian formalism: commutative Hopf algebras and affine group schemes, monoidal, braided monoidal and tensor categories, rigid tensor categories, categorical traces, Tannakian categories and neutral Tannakian categories, Deligne's Tannakian criterion
• Wed Jan 29: Tensor ideals and quotient categories, tensor ideals and good equivalence relations on cycles, largest tensor ideal and numerical equivalence, smash-nilpotence ideal; semi-simple categories, Jannsen's semi-simplicity theorem; change of commutativity constraint under the sign conjecture and Tannakian category of numerical motives; conjecture D and Weil cohomologies as fiber functors; motivic Galois groups.
• Mon Feb 3: Kimura-O'Sullivan finite dimensionality: group ring, permutations and projectors, Schur functors, symmetric and alternating powers of motives, image under Weil cohomologies, evenly/oddly finite dimensional and finite dimensional motives, dimensions, Kimura-O'Sullivan conjecture and relation to other conjectures, motives of curves and finite dimensionality, sums and products and finite dimensionality: Young tableaux, Kimura vanishing lemma, finite dimensionality of tensor products; Kimura-O'Sullivan categories.
• Wed Feb 5: Grothendieck ring of varieties: smooth quasi-projective or arbitrary generators, inclusion-exclusion relation, role of resolution of singularities and weak factorization, smooth projective varieties and Bittner relations; Grothendieck ring of motives, motivic Euler characteristic with compact support, motivic Euler characteristic without compact support, relative version of the Grothendieck ring, relations (excision, Gysin, exactness), duality and motivic Euler characteristics; the Grothendieck ring of numerical motives
• Mon Feb 10: Motivic Euler characteristics and the weight complex of Gillet-Soule'; L-functions, zeta functions, and motivic zeta functions, motivic measures, zeta functions as group homomorphisms, Kapranov's motivic zeta function, rationality for curves, zeta functions on the Grothendieck ring of motives and Kimura-O'Sullivan conjecture.
• Wed Feb 12: Grothendieck ring of varieties and stable birational equivalence classes, kernel as ideal generated by the Lefschetz motive, motivic measure and Hodge numbers, counterexample to rationality for surfaces (after Larsen-Lunts)
• Mon Feb 17: holiday
• Wed Feb 19: The Grothendieck ring is not a domain (after Poonen); motivic measures based on Poincare' polynomials and virtual Hodge numbers; Cohomology for more general algebraic varieties: homotopy invariance, Mayer-Vietoris sequences; Bloch-Ogus cohomologies; category of smooth varieties with finite correspondences; bounded complexes and complexes up to homotopy; Mayer-Vietoris complex, Nisnevich's complexes, homotopy complexes; triangulated categories; sub-triangulated categories and quotients; triangulated category of effective mixed motives (Voevodsky); mixed motives with coefficients
• Mon Feb 24: reduced motive; Mayer-Vietoris triangles and homotopy; distinguished squares and distinguished triangles; the Tate motive as a mixed motive; Tate twists and the category of mixed motives; motivic cohomology; homotopy invariance, long exact sequences for distinguished squares, cup product; mixed motives of projective spaces, of projective bundles and of blowups; Gysin distinguished triangles; Relation between pure Chow motives and mixed motives, Friedlander-Voevodsky "moving lemma"; duality and motives with compact support
• Wed Feb 26: Gyson sequence and relative motives, boundary motives and relative motives; relative motives, normal crossings divisors, and simplicial resolutions; Mixed Tate Motives: abelian categories inside triangulated categories and extensions; Beilinson-Soule' vanishing conjecture and abelian category of mixed Tate motives; Tannakian category of mixed Tate motives over a number field, weight filtration and fiber functor; mixed Tate criterion via Mayer-Vietoris triangles; Kummer motives; conjectural abelian category of mixed motives and relation to pure motives; t-structures in triangulated categories and hearts of t-structures
• Mon Mar 3: Periods, period isomorphism, period matrix; examples of periods of curves; periods of motives, motivic Galois groups and torsor of periods; Grothendieck's period conjecture; periods of mixed motives via torsor of periods; identities between periods (additivity, change of variables, Stokes theorem); periods and differential equations (Picard Fuchs equations, Gauss-Manin systems), examples from families of elliptic curves; periods and special values of L-functions, Beilinson conjecture; algebra of abstract periods (Kontsevich), conjectures about relations
• Wed Mar 5: Nori's Tannakian theorem: quiver representations; abelian category associated to a quiver representation in K-mod; universal property for representations in abelian categories; monoidal quivers, monoidal representations in monoidal categories; universal property for monoidal representations; duality, rigid tensor categories and Hopf algebra associated to a quiver representation, Tannakian category.
• Mon Mar 10: Multiple zeta values, polylogarithm functions, multiple polylogarithms, multiple zetas as iterated integrals, Goncharov-Manin's realization of MZVs as periods of moduli spaces of pointed rational curves, stuffle and shuffle products, regularized double melange relations, regularizations and Ihara-Kaneko relation, Drinfeld associator and MZV relations
• Wed Mar 12: Conjectures about relations between MZVs, Goncharov's category of mixed Tate motives over Z, Ext groups, Lie algebra and motivic Galois group, Deligne-Goncharov's category of mixed Tate motives over Z, motivic Galois groups and torsor of periods, relation to conjectures on MZVs, motivic interpretation of melange relations

Student Presentations

• Wed Mar 12: Emad Nasrollahpoursamami, "Chow-Kunneth Decomposition: the Picard and Albanese Motives"
• Wed Mar 12: Xiang Ni, "Groebner bases for operads"