Ramification of the volume function close to asymptotic hyperplanes. Monodromy of boundary singularities

Abstract

It is shown that the discriminants of boundary singularities introduced by V.~I.~Arnold naturally arise in integral geometry. They represent the irregularity sets of the volume function. This function maps an affine hyperplane to the volume of the part of a domain in the Euclidean space cut off by this hyperplane. For isolated boundary singularities, the role of the boundary is shown to be related to the hyperplane at infinity in the projective space. Versions of the Milnor fiber bundle adapted to the present problem are defined and studied. Analogs of the local even Petrovsky class are introduced and computed. It is proved that the volume function is locally finite-sheeted only if these classes are trivial. Such components of the complements to the discriminants (quasilacunas) are listed for simple boundary singularities --- those having no moduli in the classification. The obtained results establish new obstructions to the algebraic integrability of domains in a Euclidean space.