Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica (Jan 2017)
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 Y with bi-degree (2,2) in a product of projective lines ℙ1 × ℙ1. We calculate two differenent monodromy representations of period integrals for the affine variety X(2,2) obtained by the dual polyhedron mirror variety construction from 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 Y with respect to the Euler form.
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