Resumen de Periods and line arrangements: contributions to the kontsevich-zagier period conjecture and to the terao conjecture

Juan Viu Sos

• The principal motivation of the present Ph.D subject is the study of certain interactions between number theory, algebraic geometry and dynamical systems. This thesis is composed by two different parts: a first one about periods of Kontsevich-Zagier and another one about logarithmic vector fields on line arrangements.

  Each of these subjects are dominated by a main conjecture: the Kontsevich-Zagier period conjecture and the Terao conjecture, respectively. In both cases, we introduce a new point of view making connexion with different fields of research, which allows us to propose a new understanding of these conjectures, as well as new approaches toward their resolution.

  The first part concerns a problem of number theory, for which we develop a geometrical approach based on tools coming from algebraic geometry and combinatorial geometry. Introduced by M. Kontsevich and D. Zagier in 2001, 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 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 give a geometrical approach presenting a semi-canonical reduction of a integral representing a period focusing on give constructible and algorithmic methods respecting the classical rules of integral transformations. Using resolution of singularities, we prove that any non-zero real period represented by an integral can be expressed up to sign as the volume of a compact semi-algebraic set.

  The semi-canonical reduction permit a reformulation of the Kontsevich-Zagier conjecture in terms of volume-preserving change of variables between compact semi-algebraic sets, via triangulations and methods of PL-geometry. We study the problems and obstructions of this approach, making explicit the directions for future work.

  Using the previous construction, we complete the works of J. Wan to develop a degree theory for periods based on the minimality of the ambient space needed to obtain such a compact reduction. This degree theory gives a first geometric notion of transcendence of periods. We extend this study introducing notions of geometric and arithmetic complexities for periods based in the minimal polynomial complexity of the semi-algebraic sets representing the period, and showing that there exist examples of periods of minimal geometric complexity.

  The second part deals with the understanding of particular objects coming from algebraic geometry with a strong background in combinatorial geometry. Using tools from dynamical systems theory, we develop a dynamical approach for these objects. The 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 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 Terao conjecture.

  This point of view is used in the study of affine line arrangements. We study the module of logarithmic vector fields of a line arrangement by the filtration induced by the degree of the polynomial components. We determine that there exist only two types of non-trivial polynomial vector fields fixing an infinity of lines. Then, we describe the influence of the combinatorics of the arrangement on the expected minimal degree for these kind of vector fields. We prove that the combinatorics do not determine the minimal degree of the logarithmic vector fields of an affine line arrangement, giving two pair of counter-examples, each pair corresponding to a different notion of combinatorics. We determine that the dimension of the strata follows a quadratic growth from a certain degree. Illustrated by computations over some examples, we conjecture a quadratic formula for the dimension of the strata expressed only by the combinatorial data of the arrangement. In order to study computationally these filtration, we develop a suite of functions in the mathematical software Sage.