On the Hodge and V-filtrations of mixed Hodge modules

In this paper, we prove a Beilinson-type formula for the V-filtration of Kashiwara and Malgrange on a complex mixed Hodge module, using Hodge filtrations on the localization. Our formula expresses the V-filtration as the filtered D-module underlying a pro-mixed Hodge module. We apply this to the theory of higher multiplier and Hodge ideals. Our first result shows that higher multiplier ideals can be obtained directly from Hodge ideals by taking a suitable limit. As a corollary, we deduce that Hodge ideals are left semi-continuous if and only if they coincide with higher multiplier ideals, thereby improving results of Saito and Mustaţă--Popa and resolving a folklore question. We further prove a birational transformation formula for higher multiplier ideals, generalising the classical formula for multiplier ideals and answering a question of Schnell and the second author. Finally, we provide very quick proofs of the main vanishing theorems for Hodge ideals, and strengthen a result of B. Chen.