Matrix representation of Picard--Lefschetz--Pham theory near the real plane in $\mathbb{C}^2$

Abstract

A matrix formalism is proposed for computations based on Picard--Lefschetz theory in a 2D case. The formalism is essentially equivalent to the computation of the intersection indices necessary for the Picard--Lefschetz formula and enables one to prove non-trivial topological identities for integrals depending on parameters. We introduce the universal Riemann domain $\tilde U$, i.e. a sort of ``compactification'' of the universal covering space $\tilde U_2$ over a small tubular neighborhood $N\mathbb{R}^2$ of $\mathbb{R}^2\backslash\sigma$ in $\mathbb{B}\subset\mathbb{C}^2$, where $\mathbb{B}\subset\mathbb{C}^2$ is a big ball, and $\sigma$ is a one-dimensional complex analytic set (the set of singularities). We compute the Picard-Lefschetz monodromy of the relative homology group of the space $\tilde U$ modulo the singularities and the boundary for the standard local degenerations of type $P_1 ,P_2,P_3$ in Pham's [1] notations and for more complicated configurations in $\mathbb{C}^2$. We consider this homology group as a module over the group ring of the $\pi_1((N\mathbb{R}^2 \cap \mathbb{B})\backslash\sigma)$ over $\mathbb{Z}$. The results of the computations are presented in the form of a matrix of the monodromy operator calculated in a certain natural basis. We prove an ``inflation'' theorem, which states that the integration surfaces of interest (i.e.\ the elements of the homology group $H_2(\tilde U_2,\tilde{\partial \mathbb{B}})$) (the surfaces in the branched space possibly passing through singularities) are injectively mapped to the group $H_2(\tilde U,\tilde U'\cup\tilde{\partial \mathbb{B}})$ (the surfaces avoiding the singularities). The matrix formalism obtained describes the behaviour of integrals depending on parameters and can be applied to the study of Wiener-Hopf method in two complex variables.