Algebraic Geometry Seminar: "On the rationality of MUMs and 2-functions"

Abstract: Points of maximal unipotent monodromy in Calabi-Yau moduli space play a central role in mirror symmetry, and also harbor some interesting arithmetic. In the classic examples, suitable expansion coefficients of the (all-genus) prepotential (in polylogarithms) under the mirror map are integers with an enumerative interpretation on the mirror manifold. This correspondence should be expected to extend to periods relative to algebraic cycles capturing the enumerative geometry relative to Lagrangian submanifolds. This expectation is challenged, however, when the mixed degeneration is not defined over Q. After musing about compatibility with mirror symmetry, I will discuss two recent results that sharpen these questions further: The first is a theorem proven by Felipe Müller which states that the coefficients of rational 2-functions are necessarily contained in an abelian number field. (As defined in the talk, 2-functions are formal power series whose coefficients satisfy a natural Hodge theoretic supercongruence.) The second are examples worked out in collaboration with Bönisch, Klemm, and van Straten, of MUMs that are themselves not defined over Q.

Host: Matt Kerr