We construct a Rankin Selberg integral to represent the exterior cube L function L(π,Λ3,s) of an automorphic cuspidal module π of GL6(${\Bbb A}$F) (where F is a number field). We determine the poles of this L function and find period conditions for the special value L(π,Λ3,1/2). We use the Siegal Weil formula. We also state an analogue of the Gross–Prasad conjecture concerning a criterion for the nonvanishing of L(π,Λ3,1/2).

An irreducible representation of the extended affine Lie algebra of type $A_{n-1}$ coordinatized by a quantum torus of ν variables is constructed by using the Fock space for the principal vertex operator realization of the affine Lie algebra ${\widetilde{gl}_{n}$.

A symplectic fibration is a fibre bundle in the symplectic category (a bundle of symplectic fibres over a symplectic base with a symplectic structure group). We find the relation between the deformation quantization of the base and the fibre, and that of the total space. We consider Fedosov's construction of deformation quantization. We generalize the Fedosov construction to the quantization with values in a bundle of algebras. We find that the characteristic class of deformation of a symplectic fibration is the weak coupling form of Guillemin, Lerman, and Sternberg. We also prove that the classical moment map could be quantized if there exists an equivariant connection.

We study primitive theta functions, which were first introduced by Shintani, in a purely local setting. We investigate a metaplectic representation of U(1) acting on the space of local primitive theta functions and give its explicit irreducible decomposition. As a by-product, we give a new proof of epsilon dichotomy for (U(1),U}(1)).

We prove that the Hessian matrix of the real period function ψ(λ) associated with the real versal deformation fλ(x)=±x4+λ2x2+λ1x+λ0 of a singularity of type A3, is nondegenerate, provided that λ∈${\Bbb R}$3 does not belong to the discriminant set of the singularity. We explain the relation between this result and the perturbations of the spherical pendulum.

Let V be a finite-dimensional real vector space on which a root system Σ is given. Consider a meromorphic function φ on V$\Bbb C$=V+iV, the singular locus of which is a locally finite union of hyperplanes of the form {λ ∈ V$\Bbb C$[mid ]〈 λ, α 〉 = s}, α ∈ Σ, s ∈ $\Bbb R$. Assume φ is of suitable decay in the imaginary directions, so that integrals of the form ∫η +iV φ λ, dλ make sense for generic η ∈ V. A residue calculus is developed that allows shifting η. This residue calculus can be used to obtain Plancherel and Paley–Wiener theorems on semisimple symmetric spaces.

Let ${\cal S}$(g,N,p) be the Siegel modular variety of principally polarized Abelian varieties of dimension g with a Γ0(p)-level structure and a full N-level structure (where p is a prime not dividing N[ges ]3 and Γ0(p) is the inverse image of a Borel subgroup of Sp(2g,${\Bbb F}$p) in Sp(2g,${\Bbb Z}$)). This variety has a natural integral model over ${\Bbb Z}$[1/N] which is not semi-stable over the prime p if g>1. Using the theory of local models of Rapoport–Zink, we construct a semi-stable integral model of ${\cal S}$(g,N,p) over ${\Bbb Z}$[1/N] for g=2 and g=3. For g=2, our construction differs from de Jong's one though the resulting model is the same.

Working over the complex numbers, we study curves lying in a complete intersection K3 surface contained in a (nodal) complete intersection Calabi–Yau threefold. Under certain generality assumptions, we show that the linear system of curves in the surface is a connected componend of the the Hilbert scheme of the threefold. In the case of genus one, we deduce the existence of infinitesimally rigid embeddings of elliptic curves of arbitrary degree in the general complete intersection Calabi–Yau threefold.

Let X be a proper, smooth, connected curve, defined over an algebraically closed field of characteristic p>0 and of genus g ≥ 2. We show that there exists a finite solvable group G, of order prime to p, and a Galois étale cover Y → X, with Galois group G, which is not ordinary.

Let F a locally compact non-Archimedean field, of residue characteristic p, and ψ a nontrivial additive character of F. Let σ, σ′ be irreducible representations of the absolute Weil group of F, each of degree a power of p and not induced from a nontrivial unramified extension of F. We give a formula for the value at $s=½ of the ϵ-factor ϵ (σ ⊗ σ ',ψ,s)$, modulo roots of unity in ${\Bbb C}$ of order a power of p. Via the Langlands correspondence, we get an analogous formula for supercuspidal representations of ${\rm GL}_n(F)$.

Let L be an ample line bundle on a Kähler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kähler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace–Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite-dimensional. Using Jucys–Murphy elements of the Hecke algebra, the eigenvalues of the Laplace–Beltrami operator for the Hopf algebra ${\mathcal {O}}$(SLq(N)) are computed.

We prove Sturmfels' conjecture that toric varieties of codimension two have no other flat deformations than those obtained by Gröbner basis theory.

Using multiple point spaces some new examples of perverse sheaves on images of maps are described. Furthermore, suppose $f:X → Y$ is a finite and proper map of complex analytic manifolds of dimension n and $n+1$ such that every multiple point space is nonsingular and has the dimension expected of a generic map. Then we can describe the composition series for the constant sheaf on the image in the category of perverse sheaves.