Speaker: Sergey Galkin

Title: The miraculous eta-product formula and prime Fano 3-folds

Date: Feb 10, 2010, 13:30 - 15:00

Place: IPMU, Balcony A on the 4th floor of the new building

I will review the Golyshev's conjecture (math/0510287). Consider one of 10 smooth prime Fano threefolds V_{2g-2} of index 1, where g is 2,3,4,5,6,7,8,9,10 or 12. Take N = g-1, then Atkin-Lehner curve X_0(N)+ is of genus zero. Let T be the inverse of the Hauptmodulus (Conway-Norton uniformizer) H for X_0(N)+ shifted by the constant c_N: T = (H + c_N)-1. Let I(t) be the fundamental solution of the regularized quantum differential equation for V. Consider I(T(q)) as power-series in q.

Miraculous Eta-Product Formula states that:

I(T(q)) = eta(q)^2 eta(q^N)^2

Although it has been proven ad hoc, no conceptual uniform explanation found so far. Also I'll discuss what other geometries (2-folds, non-prime Fano 3-folds) we can find if we change or drop some of the conditions above, especially weaken the absolutely non-explained miraculous eta-product formula to ordinary modularity.