Volume 2 (2002), Number 3. Abstracts

V. Batyrev and E. Materov. Toric Residues and Mirror Symmetry [PDF]

We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture, which has close relations to toric mirror symmetry. Our conjecture, we call it the toric residue mirror conjecture, is that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidence for that. The proposed conjecture suggests a simple method for computing Yukawa couplings for toric mirror Calabi—Yau hypersurfaces without solving systems of differential equations. We make several explicit computations for Calabi—Yau hypersurfaces in weighted projective spaces and in products of projective spaces.

Keywords. Residues, toric varieties, intersection numbers, mirror symmetry.

2000 Mathematics Subject Classification. Primary: 14M25.

A. Beilinson, S. Bloch, and H. Esnault. ε-factors for Gauß—Manin Determinants [PDF]

We define ε-factors in the de Rham setting and calculate the determinant of the Gauß—Manin connection for a family of (affine) curves and a vector bundle equipped with a flat connection.

Keywords. ε-factors, cohomology determinant, D-modules.

2000 Mathematics Subject Classification. Primary: 14C40, 19E20, 14C99.

A. Braverman and D. Kazhdan. Normalized Intertwining Operators [PDF]

Let F be a local non-archimedean field and G be a split reductive group over F whose derived group is simply connected. Set G=G(F). Let also ψ: F → C^× be a nontrivial additive character of F. For two parabolic subgroups P, Q in G with the same Levi component M, we construct an explicit unitary isomorphism F_P,Q,ψ: L²(G/[P,P])\simto L²(G/[Q,Q]) commuting with the natural actions of the group G × M/[M,M] on both sides. In some special cases, F_P,Q,ψ is the standard Fourier transform. The crucial ingredient in the definition is the action of the principal sl₂-subalgebra in the Langlands dual Lie algebra m^∨ on the nilpotent radical u_p^∨ of the Langlands dual parabolic.

For M as above, we use the operators F_P,Q,ψ to define a Schwartz space S(G,M). This space contains the space C_c(G/[P,P]) of locally constant compactly supported functions on G/[P,P] for every P for which M is a Levi component (but does not depend on P). We compute the space of spherical vectors in S(G,M) and study its global analogue.

Finally, we apply the above results in order to give an alternative treatment of automorphic L-functions associated with standard representations of classical groups.

Keywords. Intertwining operators, principal nilpotent, automorphic L-functions.

2000 Mathematics Subject Classification. 22E50, 22E55.

P. Etingof and V. Ginzburg. On m-quasi-invariants of a Coxeter Group [PDF]

Let W be a finite Coxeter group in a Euclidean vector space V, and let m be a W-invariant Z₊-valued function on the set of reflections in W. Chalykh and Veselov introduced an interesting algebra Q_m, called the algebra of m-quasi-invariants for W, such that C[V]^W ⊆ Q_m ⊆ C[V], Q₀=C[V], and Q_m ⊇ Q_m' whenever m ≤ m'. Namely, Q_m is the algebra of quantum integrals of the rational Calogero—Moser system with coupling constant m. Feigin and Veselov proposed a number of interesting conjectures concerning the structure of Q_m and verified them for dihedral groups and constant functions m. Our objective is to prove some of these conjectures in the general case.

Keywords. Calogero—Moser system, Coxeter groups, m-quasi-invariants.

2000 Mathematics Subject Classification. 81Rxx, 14-xx.

B. Feigin and E. Feigin. Q-characters of the Tensor Products in sl₂-case [PDF]

Let π₁,...,π_n be irreducible finite-dimensional sl₂-modules. Using the theory of representations of current algebras, we introduce several ways to construct a q-grading on π₁⊗...⊗π_n. We study the corresponding graded modules and prove that they are essentially the same.

Keywords. Universal enveloping algebra, representation theory, current algebra, Gordon's formula.

2000 Mathematics Subject Classification. Primary 05A30; Secondary 17B35.

S. Ghorpade and G. Lachaud. Singular Varieties over Finite Fields [PDF]

We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang—Weil inequality. Moreover, we prove the Lang—Weil inequality for affine, as well as projective, varieties with an explicit description and a bound for the constant appearing therein. We also prove a conjecture of Lang and Weil concerning the Picard varieties and étale cohomology spaces of projective varieties. The general inequality for complete intersections may be viewed as a more precise version of the estimates given by Hooley and Katz. The proof is primarily based on a suitable generalization of the Weak Lefschetz Theorem to singular varieties together with some Bertini-type arguments and the Grothendieck—Lefschetz Trace Formula. We also describe some auxiliary results concerning the étale cohomology spaces and Betti numbers of projective varieties over finite fields, and a conjecture along with some partial results concerning the number of points of projective algebraic sets over finite fields.

Keywords. Étale cohomology, varieties over finite fields, complete intersections, Trace Formula, Betti numbers, zeta functions, Weak Lefschetz Theorems, hyperplane sections, motives, Lang—Weil inquality, Albanese variety.

2000 Mathematics Subject Classification. 11G25, 14F20, 14G15, 14M10.

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