We show that the blowup of an extremal Kähler manifold at a relatively stable point in the sense of GIT admits an extremal metric in Kähler classes that make the exceptional divisor sufficiently small, extending a result of Arezzo, Pacard, and Singer. We also study the K-polystability of these blowups, sharpening a result of Stoppa in this case. As an application we show that the blowup of a Kähler–Einstein manifold at a point admits a constant scalar curvature Kähler metric in classes that make the exceptional divisor small, if it is K-polystable with respect to these classes.

We introduce a notion of volume of a normal isolated singularity that generalizes Wahl’s characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw several consequences regarding the existence of noninvertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a noninvertible polarized endomorphism is pseudoeffective.

Our techniques build on Shokurov’s $b$-divisors. We define the notions of nef Weil $b$-divisors and of nef envelopes of $b$-divisors. We relate the latter to the pullback of Weil divisors introduced by de Fernex and Hacon. Using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by Takagi, we carry over to the isolated singularity case the intersection theory of nef Weil $b$-divisors formerly developed by Boucksom, Favre, and Jonsson in the smooth case.

Under mild hypotheses, we prove that if $F$ is a totally real field, and $\overline{\rho }:G_F\to {GL}_2\left({\overline{\mathbb{F}}}_l\right)$ is irreducible and modular, then there is a finite solvable totally real extension $F\prime /F$ such that $\overline{\rho }{\mid }_{G_{F^{\prime }}}$ has a modular lift which is ordinary at each place dividing $l$. We deduce a similar result for $\overline{\rho }$ itself, under the assumption that at places $v|l$ the representation $\overline{\rho }{\mid }_{G_{F_v}}$ is reducible. This allows us to deduce improvements to results in the literature on modularity lifting theorems for potentially Barsotti–Tate representations and the Buzzard–Diamond–Jarvis conjecture. The proof makes use of a novel lifting technique, going via rank $4$ unitary groups.

We prove global well-posedness in $H^1\left({\mathbb{T}}^3\right)$ for the energy-critical defocusing initial-value problem $$(i{\partial }_t+\Delta )u=u|u{|}^4,u(0)=\varphi .$$