We prove an asymptotic formula for the eighth moment of Dirichlet L-functions averaged over primitive characters χ modulo q, over all moduli $q\le Q$, and with a short average on the critical line. Previously the same result was shown conditionally on the generalized Riemann hypothesis by the first two authors.

Let C be a smooth projective curve of genus $g\ge 2$, and let N be the moduli space of stable vector bundles on C of rank 2 and fixed determinant of odd degree. We construct a semiorthogonal decomposition of $D^b(N)$ conjectured by Narasimhan and by Belmans, Galkin, and Mukhopadhyay. It has two blocks for each ith symmetric power of C for $i=0,\dots ,g-2$ and one block for the $(g-1)$st symmetric power. We conjecture that the subcategory generated by our blocks has a trivial semiorthogonal complement, proving the full BGMN conjecture. Our proof is based on an analysis of wall-crossing between moduli spaces of stable pairs, combining classical vector bundles techniques with the method of windows.

We construct new complete Calabi–Yau metrics on the complement of an anticanonical divisor D in a Fano manifold of dimension at least three, when D consists of two transversely intersecting smooth divisors. The asymptotic geometry is modeled on a generalization of the Calabi ansatz, related to the non-Archimedean Monge–Ampère equation.

We prove that every variety with log-terminal singularities admits a crepant resolution by a smooth Artin stack. We additionally prove new McKay correspondences for resolutions by Artin stacks, expressing stringy invariants of $\mathbb{Q}$-Gorenstein varieties in terms of motivic integrals on arc spaces of smooth stacks. In the crepant case, these McKay correspondences are particularly simple, demonstrating one example of the utility of crepant resolutions by Artin stacks.