arXiv subjects

A. A. Glutsuk

Publications and source records attributed to A. A. Glutsuk.

4 recordsLinked to original sources

arXiv2000

Nonuniformizable skew cylinders. A counterexample to the simultaneous uniformization problem

At the end of 1960-ths Yu.S.Ilyashenko stated the problem: is it true that for any one-dimensional holomorphic foliation with singularities on a Stein manifold leaves intersecting a transversal disc can be uniformized so that the uniformization function would depend holomorphically on the transversal parameter? In the present paper we construct a two-dimensional counterexample. This together with the previous result of Yu.S.Ilyashenko (Lemma 1) implies existence of a counterexample given by a foliation on affine (projective) algebraic surface by level curves of a rational function with singularities deleted. This implies also that Bers' simultaneous uniformization theorem for topologically trivial holomorphic foliations by compact Riemann surfaces does not extend to the general fibrations by compact Riemann surfaces with singularities.

math.CV↗

arXiv2000

An explicit formula for the determinant of the Abelian integral matrix

We consider a polynomial h(x,y) in two complex variables of degree n+1>1 with a generic higher homogeneous part. The rank of the first homology group of its nonsingular level curve h(x,y)=t is n*n. To each 1- form in the variable space and a generator of the homology group one associates the (Abelian) integral of the form along the generator. The Abelian integral is a multivalued function in t. For a fixed canonic tuple of n*n monomial 1- forms we consider the multivalued square matrix function in t whose elements are the Abelian integrals of the forms along the generators. Its determinant does not depend on the choice of the generators in the homology group (up to change of sign, which corresponds to change of generator system that reverses orientation). In 1999 Yu.S.Ilyashenko proved that the determinant of the Abelian integral matrix is a polynomial in t of degree n*n whose zeroes are the critical values of h. We give an explicit formula for the determinant.

math.DS↗

arXiv2000

Smoothness of the uniformization of two-dimensional linear foliation on torus with nonstandard metric

Consider a parallel plane foliation on real finite-dimensional linear vector space. It induces a foliation on the torus obtained by factorization of the space by the integer lattice (let us denote the latter foliation by F). Let g be arbitrary metric on the torus. It induces a complex structure on each leaf of F such that all the leaves are parabolic: each individual leaf admits a conformal flat complete metric. We show that for any foliation F as above and arbitrary smooth metric g the correspondent complete conformal flat metric on the leaves, which is smooth on individual leaf, can be chosen to be smooth in the whole torus (in particular, in transversal parameter). We show that under appropriate Diophantine condition on the foliation for arbitrary smooth metric g there exists a smooth Euclidean metric on the torus conformal on the leaves such that the latters are totally geodesic; in other terms, the correspondent triple consisting of the torus, the foliation and the complex structure on the leaves is isomorphic to a triple correspondent to a linear foliation on another torus equipped with the standard complex structure on the leaves. This Diophantine condition is exact. We state and prove the analytic versions of these statements.

math.DS↗

arXiv1999

Nonlinear Stokes phenomena analytic classification invariants via generic perturbation

For a generic deformation of a two-dimensional holomorphic vector field with elementary degenerate singular point (saddle-node) we express the Martinet - Ramis orbital analytic classification invariants of the nonperturbed field in terms of the limit transitions between the linearizing charts of singularities of the perturbed field. In the case, when the multiplicity of the singular point of the nonperturbed field is equal to 2, we show that the Martinet - Ramis invariants are limits of transition functions that compare appropriately normalized canonic first integrals of the perturbed field in the linearizing charts. We prove a generalization of this statement for higher multiplicity singularities. For a generic deformation of higher-dimensional holomorphic vector field with saddle-node singularity we show that appropriate sectorial central manifolds of the nonperturbed field are limits of appropriate separatrices of singularities of the perturbed field. We prove the analogues of the two first results for Ecalle - Voronin analytic classification invariants of one-dimensional conformal maps tangent to identity.

math.DS↗