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

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.

An initial investigation into the geometrical meaning of the (pseudo-) inverses of the line matrices for the edges of platonic polyhedra

It is well known that there are five regular (Platonic) polyhedra: the tetrahedron, the hexahedron (cube), the octahedron, the icosahedron and the dodecahedron. Each of these polyhedra has an associated dual polyhedron which is also Platonic. By considering the Platonic polyhedra to be constructed from lines, and then representing the lines in terms of both ray and axis coordinates, a further aspect of this duality is exposed. This is the duality of poles and polars associated with projective configurations of points, lines and planes. This paper shows that a line matrix may be constructed for any regular polyhedron, in such a way that its columns represent the normalized ray coordinates of the edges of the polyhedron. The (pseudo-) inverse of this line matrix may then be constructed, the rows of which represent the normalized axis coordinates of the corresponding dual polyhedron.