Some contributions on periods of Kontsevich-Zagier and on logarithmic vector fields of line arrangements.

2015年   12月 17日 10時 30分 ~   12時 00分
Juan Viu Sos (Univ. of Pau)
We are interested in the study of certain interactions between number theory, algebraic geometry and dynamical systems. This talk is composed by two different parts: a first one about {\bf periods of Kontsevich-Zagier} and another one about {\bf logarithmic vector fields on line arrangements}.

Introduced by M.~Kontsevich and D.~Zagier in 2001, \emph{periods} are complex numbers expressed as values of integrals of a special form, where both the domain and the integrand are expressed using polynomials with rational coefficients. The {\bf Kontsevich-Zagier period conjecture} affirms that any polynomial relation between periods can be obtained by linear relations between their integral representations, expressed by classical rules of integral calculus. We present a \emph{semi-canonical reduction} for periods, which allows us to develop a {\bf geometrical approach} for periods and its related problems.

Logarithmic vector fields are an algebraic-analytic tool used to study sub-varieties and germs of analytic manifolds. We are concerned with the case of {\bf line arrangements} in the affine or projective space. One is interested to study how the combinatorial data of the arrangement determines relations between its associated logarithmic vector fields: this problem is known as the {\bf Terao conjecture}. Following the spirit of some classical problems of polynomial differential systems, we give a first study of a {\bf dynamical approach} of the module of logarithmic vector fields of an affine line arrangement.