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


TANABE S.

ANALELE STIINTIFICE ALE UNIVERSITATII OVIDIUS CONSTANTA-SERIA MATEMATICA, vol.25, no.1, pp.207-231, 2017 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 25 Issue: 1
  • Publication Date: 2017
  • Doi Number: 10.1515/auom-2017-0016
  • Title of Journal : ANALELE STIINTIFICE ALE UNIVERSITATII OVIDIUS CONSTANTA-SERIA MATEMATICA
  • Page Numbers: pp.207-231

Abstract

We study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curved with bi-degree (2,2) in a product of projective lines P-1 x P-1. 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 Id with respect to the Euler form.