On monodromy representation of period integrals associated to an algebraic curve with bi-degree (2,2)

Abstract

We study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curve $\mathcal{Y}$ with bi-degree (2,2) in a product of projective lines ${\mathbb{P}}^1\times {\mathbb{P}}^1$. We calculate two differenent monodromy representations of period integrals for the affine variety ${\mathcal{X}}^{(2,2)}$ obtained by the dual polyhedron mirror variety construction from $\mathcal{Y}$. The first method that gives a full representation of the fundamental group of the complement to singular loci relies on the generalised Picard-Lefschetz theorem. The second method uses the analytic continuation of the Mellin-Barnes integrals that gives us a proper subgroup of the monodromy group. It turns out both representations admit a Hermitian quadratic invariant form that is given by a Gram matrix of a split generator of the derived category of coherent sheaves on on $\mathcal{Y}$ with respect to the Euler form.