23 research outputs found
On the non-persistence of Hamiltonian identity cycles
We study the leading term of the holonomy map of a perturbed plane polynomial
Hamiltonian foliation. The non-vanishing of this term implies the
non-persistence of the corresponding Hamiltonian identity cycle. We prove that
this does happen for generic perturbations and cycles, as well for cycles which
are commutators in Hamiltonian foliations of degree two. Our approach relies on
the Chen's theory of iterated path integrals which we briefly resume.Comment: 17 page
Polynomial Structure of Topological String Partition Functions
We review the polynomial structure of the topological string partition
functions as solutions to the holomorphic anomaly equations. We also explain
the connection between the ring of propagators defined from special K\"ahler
geometry and the ring of almost-holomorphic modular forms defined on modular
curves.Comment: version 2: references fixe
A course in Hodge theory : with emphasis on multiple integrals
Hodge theory—one of the pillars of modern algebraic geometry—is a deep theory with many applications and open problems, the most enigmatic of which is the Hodge conjecture, one of the Clay Institute’s seven Millennium Prize Problems. Hodge theory is also famously difficult to learn, requiring training in many different branches of mathematics. The present volume begins with an examination of the precursors of Hodge theory: first, the studies of elliptic and abelian integrals by Cauchy, Abel, Jacobi, and Riemann, among many others; and then the studies of two-dimensional multiple integrals by Poincaré and Picard. Thenceforth, the focus turns to the Hodge theory of affine hypersurfaces given by tame polynomials, for which many tools from singularity theory, such as Brieskorn modules and Milnor fibrations, are used. Another aspect of this volume is its computational presentation of many well-known theoretical concepts, such as the Gauss–Manin connection, homology of varieties in terms of vanishing cycles, Hodge cycles, Noether–Lefschetz, and Hodge loci. All are explained for the generalized Fermat variety, which for Hodge theory boils down to a heavy linear algebra. Most of the algorithms introduced here are implemented in Singular, a computer algebra system for polynomial computations. Finally, the author introduces a few problems, mainly for talented undergraduate students, which can be understood with a basic knowledge of linear algebra. The origins of these problems may be seen in the discussions of advanced topics presented throughout this volume
Mixed Hodge structure of affine hypersurfaces
In this article we introduce the mixed Hodge structure of the Brieskorn
module of a polynomial in \C^{n+1}, where satisfies a certain
regularity condition at infinity (and hence has isolated singularities). We
give an algorithm which produces a basis of a localization of the Brieskorn
module which is compatible with its mixed Hodge structure. As an application we
show that the notion of a Hodge cycle in regular fibers of is given in
terms of the vanishing of integrals of certain polynomial -forms in
\C^{n+1} over topological -cycles on the fibers of . Since the -th
homology of a regular fiber is generated by vanishing cycles, this leads us to
study Abelian integrals over them. Our result generalizes and uses the
arguments of J. Steenbrink 1977 for quasi-homogeneous polynomials.Comment: 22 pages, minor corrections, To appear in Ann. Ins. Fourie