22 septembre 2020

Tony Yue Yu (IMO)
Secondary fan, theta functions and moduli of Calabi-Yau pairs

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Résumé : We conjecture that any connected component Q of the moduli space of triples (X, E=E_1+...+E_n, Theta) where X is a smooth projective variety, E is a normal crossing anti-canonical divisor with a 0-stratum, every E is smooth, and Theta is an ample divisor not containing any 0-stratum of E, is *unirational*. More precisely : note that Q has a natural embedding into the Kollár-Shepherd-Barron-Alexeev moduli space of stable pairs, we conjecture that its closure admits a finite cover by a complete toric variety. We construct the associated complete toric fan, generalizing the Gelfand-Kapranov-Zelevinski secondary fan for reflexive polytopes. Inspired by mirror symmetry, we speculate a synthetic construction of the universal family over this toric variety, as the Proj of a sheaf of graded algebras with a canonical basis, whose structure constants are given by counts of non-archimedean analytic disks. In the Fano case and under the assumption that the mirror contains a Zariski open torus, we construct the conjectural universal family, generalizing the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. In the case of del Pezzo surfaces with an anti-canonical cycle of (-1)-curves, we prove the full conjecture. The reference is arXiv:2008.02299 joint with Hacking and Keel.

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