Special Year Research Seminar

In this talk, I will discuss some effective computations for variations of integral Hodge structures.

Several years ago, with Ren and Javanpeykar-Kühne, I conjectured (in the Shimura setting) that a variation has only finitely many "non-factor" special subvarieties of degree at most d. Urbanik established this conjecture and, in fact, gave an explicit algorithm.  More recently, Klingler-Otwinowska-Urbanik showed that, when the variation is defined over the algebraic numbers, so too are the non-factor special subvarieties.

In work in progress with Binyamini and Lerer, we show that there is an effectively computable polynomial p such that any non-factor special subvariety of degree d is defined by polynomials of complexity at most p(d) — complexity is the maximum of the degree, the log-height of the coefficients, and the degree of the field of definition. Using this result, we give a separation bound for non-factor special subvarieties in the period domain, also of polynomial strength, thus nearly resolving a conjecture of Lairez-Sertoz.