AAG Seminar: Archimedean zeta function and Hodge theory

Abstract: The minimal exponent of a holomorphic function f, i.e. the negative of the largest root of the reduced Bernstein-Sato polynomial, is an invariant refining the log canonical threshold and plays an important role in the recent study of higher rational/Du Bois singularities. I will talk about a recent joint work with Dougal Davis and Andras Lőrincz, where we prove that the minimal exponent is the (negative of) largest nontrivial pole of the Archimedean zeta function. This is a natural generalization of the analytic description of the log canonical threshold and solves a question of Mustață-Popa. As a byproduct, we obtain analytic characterization of V-filtration, higher multiplier ideals and Hodge ideals, addressing another question of Mustață-Popa. A key ingredient is the positivity property of polarizations of complex mixed Hodge modules of Sabbah and Schnell, which ultimately boils down to the Hodge-Riemann bilinear relations and Schmid’s norm estimates.

Host: Roya Beheshti