GT Seminar with Renato Vianna (Federal university of Rio de Janeiro)

Abstract: This is joint work with Santiago Achig-Andrango. Given a lattice polytope Q in R^n, we can consider its cone C(Q), in R^n+1, given by taking the rays from the origin passing through Qx{1} in R^nxR.

Being a rational polyhedral cone, we associate an affine toric algebraic variety Y, which is often singular. Altaman shows how to construct deformations of Y from a Minkowiski decomposition Q = M_1 + … + M_k of Q. Gross constructs a special Lagrangian fibration out of this data, but with general Lagrangian fibre equals to T^n x R (rather than T^n+1). Auroux describes in his work how to obtain a singular Lagrangian fibration  with a generic fibre a torus T^n out of a auxiliary complex (symplectic) fibration with a fibrewise preserving T^n-1 action. 

In his work, Achig-Andrango imposes conditions in the Minkowiski decomposition to ensure that Altman’s deformation gives a smoothing of Y and constructs a complex fibration in the generic fibre Y_e, with genral fibre (C^*)^n and singular fibres with local model associated to each term M_i of the Minkowiski decomposition. Using that as an auxiliary complex fibration, Achig-Angrango constructs a singula Lagrangian fibration, studies its monodromies and show it can be represented by a convex base diagram with cuts, whose image in R^n+1 is the dual of the cone C(Q). Moreover, the fibration contains a one parameter family of monotone Lagrangian fibres, whose superpotential can be obtained under wall-crossing formulas and described in terms of the Minkowiski decomposition of Q. 

We will present Achig-Andrango’s work and time permiting, discuss some further developments. 

See upcoming seminars in our seminar series on Geometry and Topology (GT).