Volume 9 (2009), Number 1. Abstracts

A. Bayer and Yu. Manin. Stability Conditions, Wall-Crossing and Weighted Gromov–Witten Invariants [PDF]

We extend B. Hassett's theory of weighted stable pointed curves to weighted stable maps. The space of stability conditions is described explicitly, and the wall-crossing phenomenon studied. This can be considered as a non-linear analog of the theory of stability conditions in abelian and triangulated categories (cf. works by A. Gorodentsev, S. Kuleshov, and A. Rudakov, T. Bridgeland, D. Joyce).

We introduce virtual fundamental classes and thus obtain weighted Gromov–Witten invariants. We show that by including gravitational descendants, one obtains an L-algebra as introduced by A. Losev and Yu. Manin as a generalization of a cohomological field theory.

Keywords. Weighted stable maps, gravitational descendants.

2000 Mathematics Subject Classification. Primary 14N35, 14D22; Secondary 53D45, 14H10, 14E99.

G. Henniart. Sur la fonctorialité, pour GL(4), donnée par le carré extérieur [PDF]

Let k be a number field. Henry H. Kim has established the exterior square transfer for GL(4), which attaches to any cuspidal automorphic representation ρ of GL(4,A_k) an automorphic representation Π of GL(6,A_k). At a finite place v of k, the local component ρ_v of ρ gives, via the Langlands correspondence, a degree 4 representation σ_v of the Weil–Deligne group of k_v. Then Π is the unique isobaric automorphic representation of GL(6,A_k) such that, whenever ρ_v is unramified, Π_v corresponds, via the Langlands correspondence, to the exterior square Λ²σ_v of σ_v. Kim proves that Π_v corresponds to Λ²σ_v even when ρ_v is ramified, except possibly if v is above 2 or 3 and ρ_v is cuspidal. We complete Kim's work in showing that Π_v corresponds to Λ²σ_v at all finite places v of k.

Keywords. Automorphic representation, functoriality, Langlands correspondence.

2000 Mathematics Subject Classification. 22E47, 22E50, 22E55.

L. Illusie. « La descente galoisienne... » [PDF]

Cohomological descent was invented by Deligne in the early 60's. I recall memories about this and discuss related developments and applications that arose since then, pertaining to cotangent complex and deformation theory, mixed Hodge theory, de Jong alterations, p-adic Hodge theory.

Keywords. Galois descent, cohomological descent, hypercovering, simplicial space, Grothendieck's six operations, cotangent complex, first order deformation, algebraic stack, mixed Hodge theory, de Jong alteration, rigid cohomology, p-adic étale cohomology, p-adic Hodge theory, independence of l.

2000 Mathematics Subject Classification. 13D03, 13D10, 14C30, 14F20, 14F30, 18G30.

N. Katz. From Clausen to Carlitz: Low-Dimensional Spin Groups and Identities among Character Sums [PDF]

We relate the classical formulas of Clausen and Schläfli for the squares of hypergeometric and Bessel functions respectively, and a 1969 formula of Carlitz for the square of a very particular Kloosterman sum, to the “accident” that for 3 &le n ≤ 6, the “spin” double cover of SO(n) is itself a classical group. We exploit this accident to obtain identities among character sums over finite fields, some but not all of which are finite field analogues of known identities among classical functions.

Keywords. Spin groups, character sums.

2000 Mathematics Subject Classification. 11F23, 11F75, 11T23, 14D05.

G. Lusztig. Notes on Character Sheaves [PDF]

In the first section we study a functor of Bezrukavnikov, Finkelberg and Ostrik defined on character sheaves; we compute it in a Grothendieck group taking weights into account. In the second section we enlarge the class of character sheaves to a larger class of simple perverse sheaves which behaves well with respect to tensor product (unlike the character sheaves themselves).

Keywords. Character sheaf, perverse sheaf, tensor product.

2000 Mathematics Subject Classification. 20G99.

J. Milne. Rational Tate Classes [PDF]

In despair, as Deligne put it, of proving the Hodge and Tate conjectures, one can try to find substitutes. For abelian varieties in characteristic zero, Deligne in his 1978–1979 IHES seminar constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of “rational Tate classes” on varieties over finite fields having the properties that the algebraic classes would have if the Hodge and Tate conjectures were known. In particular, I prove that there exists at most one “good” such theory.

Keywords. Abelian varieties, finite fields, Tate conjecture.

2000 Mathematics Subject Classification. 14C25, 14K15, 11G10.

D. Orlov. Remarks on Generators and Dimensions of Triangulated Categories [PDF]

In this paper we prove that the dimension of the bounded derived category of coherent sheaves on a smooth quasi-projective curve is equal to one. We also discuss dimension spectrums of these categories.

Keywords. Triangulated categories, derived categories, generators, dimension.

2000 Mathematics Subject Classification. 18E30, 14F05.

M. Saito. On the Hodge Filtration of Hodge Modules [PDF]

We show an estimate of the generating level of the Hodge filtration on pure Hodge modules in terms of the dimension of the support and the level of the generic variation of Hodge structure. We also prove a rather explicit formula for the Hodge filtration in the constant case if the support has only weighted-homogeneous isolated singularities. This implies a counterexample to a conjecture of Brylinski on the Hodge filtration of pure Hodge modules.

Keywords. Hodge filtration, mixed Hodge module, generating level.

2000 Mathematics Subject Classification. 32S40.

J.-P. Serre. A Minkowski-style Bound for the Orders of the Finite Subgroups of the Cremona Group of Rank 2 over an Arbitrary Field [PDF]

Let Cr(k) = Aut k(X,Y) be the Cremona group of rank 2 over a field k. We give a sharp multiplicative bound M(k) for the orders of the finite subgroups A of Cr(k) such that |A| is prime to char(k). For instance M(Q) = 120960, M(F₂) = 945 and M(F₇) = 847065600.

Keywords. Cremona group, algebraic torus, Del Pezzo surface, conic bundle.

2000 Mathematics Subject Classification. Primary: 14E07; Secondary: 14J26.

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