A List of Integral Representations for the Diagonal of Power Series of a Rational Function

In this paper we present integral representations for the diagonals of power series. Such representations are obtained by lowering the multiplicity of integration for the previously known integral representation. The procedure for reducing the order of integration is carried out in the framework of the Leray theory of multidimensional residues. The concept of the amoeba of a complex analytic hypersurface plays a special role in the construction of new integral representations

On Propagation of Sphericity of Real Analytic Hypersurfaces across Levi Degenerate Loci

A connected real analytic hypersurface M⊂Cn+1 whose Levi form is nondegenerate in at least one point—hence at every point of some Zariski-dense open subset—is locally biholomorphic to the model Heisenberg quadric pseudosphere of signature (k,n-k) in one point if and only if, at every other Levi nondegenerate point, it is also locally biholomorphic to some Heisenberg pseudosphere, possibly having a different signature (l,n-l). Up to signature, pseudosphericity then jumps across the Levi degenerate locus and in particular across the nonminimal locus, if there exists any.

ON THE MULTIPLICITIES OF FAMILIES OF COMPLEX HYPERSURFACE-GERMS WITH CONSTANT MILNOR NUMBER

We show that the possible drop in multiplicity in an analytic family F(z, t) of complex analytic hypersurface singularities with constant Milnor number is controlled by the powers of t. We prove equimultiplicity of μ-constant families of the form f + tg + t2h if the singular set of the tangent cone of {f = 0} is not contained in the tangent cone of {h = 0}.

Invariant analytic hypersurfaces in complex Lie groups

Suppose G is a complex Lie group and H is a closed complex subgroup of G. Let G′ denote the commutator subgroup of G. If there are no nonconstant holomorphic functions on G/H and H is not contained in any proper parabolic subgroup of G, then Akhiezer [2] asked whether every analytic hypersurface in G which is invariant under the right action of H is also invariant under the right action of G′. In this paper we answer a related question in two settings. Under the assumptions stated above we show that the orbits of the radical of G in G/H cannot be Cousin groups, provided G/H is Kähler. We also introduce an intermediate fibration of G/H induced by the holomorphic reduction of the radical orbits and resolve the related question in a situation arising from this fibration.