2015/03/27 by Konstantinos Kourliouros, Kourliouros, Konstantinos
Biochemistry, Genetics and Molecular Biology · Mathematics · #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Connective tissue disorders research #FOS: Mathematics #math.AG #math.CV
paper · pdf · doi:10.48550/arxiv.1503.08021
arxiv created 2015/03/27 · openalex publication_date 2015/03/27 · arxiv updated 2015/03/30 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We study here the relative cohomology and the Gauss-Manin connections associated to an isolated singularity of a function on a manifold with boundary, i.e. with a fixed hyperplane section. We prove several relative analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange, concerning the properties of the Gauss-Manin connection as well as its relations with the Picard-Lefschetz monodromy and the asymptotics of integrals of holomorphic forms along the vanishing cycles. Finally, we give an application in isochore deformation theory, i.e. the deformation theory of boundary singularities with respect to a volume form. In particular we prove the relative analog of J. Vey's isochore Morse lemma, J. -P. Françoise's generalisation on the local normal forms of volume forms with respect to the boundary singularity-preserving diffeomorphisms, as well as M. D. Garay's theorem on the isochore version of Mather's versal unfolding theorem.