<abstract xmlns="http://www.w3.org/1999/xhtml"> 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. </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.

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

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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...

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

Published Online: Sep 21, 2017

Page range: 207 - 231

Received: May 01, 2016

Accepted: Jul 01, 2016

DOI: https://doi.org/10.1515/auom-2017-0016

Keywords
period integrals, monodromy, Mellin-Barnes integrals, homological mirror symmetry conjecture

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

Published Online: Sep 21, 2017

Page range: 207 - 231

Received: May 01, 2016

Accepted: Jul 01, 2016

DOI: https://doi.org/10.1515/auom-2017-0016

Keywordsperiod integrals, monodromy, Mellin-Barnes integrals, homological mirror symmetry conjecture

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
period integrals, monodromy, Mellin-Barnes integrals, homological mirror symmetry conjecture