International Journal of Mathematics

The classical Airy function has been generalized by Kontsevich to a function of a matrix argument, which is an integral over the space of skew-hermitian matrices of a unitary-invariant exponential kernel. In this paper, the Kontsevich integral is further generalized to integrals over the Lie algebra of an arbitrary connected compact Lie group, using exponential kernels invariant under the group. The (real) polynomial defining this kernel is said to have the Airy property if the integral defines a function of moderate growth. A very general sufficient criterion for a polynomial to have the Airy property is given. It is shown that an invariant polynomial on the Lie algebra has the Airy property if its restriction to a Cartan subalgebra has the Airy property. This result is used to evaluate these invariant integrals completely and explicitly on the hermitian matrices, obtaining formulae that contain those of Kontsevich as special cases.

The stratified vector bundles on a smooth variety defined over an algebraically closed field k form a neutral Tannakian category over k. We investigate the affine group-scheme corresponding to this neutral Tannakian category.

In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces from certain holomorphic 1-forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we prove that the Gauss bundle of a harmonic map of finite type into a Grassmannian is also of finite type.

We study a residual form of a real analytic Siegel–Eisenstein series, which generates a certain derived functor module occurring in a degenerate principal series representation. We compute its Mellin transforms twisted by various Maass wave forms to get explicit formulas as our results. We apply them to prove meromorphic continuations together with functional equations which are satisfied by those twisted Mellin transforms.

In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This, together with the characterization of the points of the image of the Krichever map, allows us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provides us with a method for explicitly computing KP-type equations that describe the moduli space of Higgs pairs. Finally, for the case where the spectral cover is totally ramified at a fixed point of the curve, these equations are given in terms of the characteristic coefficients of the Higgs field.

Consider a fixed connected, finite graph Γ and equip its vertices with weights p_i which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically Gorenstein surface singularity having Γ as the dual graph of the minimal resolution, the weights p_i of the vertices being the arithmetic genera of the corresponding irreducible components. As a consequence we get that if Γ is not a cycle, then there is a finite number of possibilities of self-intersection numbers which one can attach to the vertices which are of valency ≥ 2, such that one gets the dual graph of the minimal resolution of a numerically Gorenstein surface singularity. Moreover, we describe precisely the situations when there exists an infinite number of possibilities for the self-intersections of the component corresponding to some fixed vertex of Γ.

A Riemann surface X of genus g > 2 is elliptic-hyperelliptic if it admits a conformal involution h such that the orbit space X/〈h〉 has genus one. This elliptic-hyperelliptic involution h is unique for g > 5 [1]. In a previous article [3], we established the non-connectedness of the subspace of real elliptic-hyperelliptic algebraic curves in the moduli space of Riemann surfaces of genus g, when g is even and > 5. In this paper we improve this result and give a complete answer to the connectedness problem of the space of real elliptic-hyperelliptic surfaces of genus > 5: we show that is connected if g is odd and has exactly two connected components if g is even; in both cases the closure of in the compactified moduli space is connected.