I will describe the results of two projects on the construction of Calabi-Yau threefolds and certain real Lagrangian submanifolds. The first concerns the construction of a novel dataset of Calabi-Yau threefolds via an application of the Gross-Siebert algorithm to a reducible union of toric varieties obtained by degenerating anti-canonical hypersurfaces in a class of (around 1.5 million) Gorenstein toric Fano fourfolds. Many of these constructions correspond to smoothing such a hypersurface; in contrast to the famous construction of Batyrev-Borisov which exploits crepant resolutions of such hypersurfaces. A central ingredient here is the construction of a certain 'integral affine structure with singularities' on the boundary of a class of polytopes from which one can form a topological model, due to Gross, of the corresponding Calabi-Yau threefold X. In general, such topological models carry a canonical (anti-symplectic) involution i and in the second project, which is joint work with H. Argüz, we describe the fixed point locus of this involution. In particular, we prove that the map i*-1 on graded pieces of a Leray filtration of H^3(X,Z2) can be identified with the map D -> D^2, where D is an element of H^2(X',Z2) and X' is mirror-dual to X. We use this to compute the Z2 cohomology group of the fixed locus, answering a question of Castaño-Bernard--Matessi.
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- Geometry and Analysis Seminar