We define relative Donaldson–Thomas invariants of local log Calabi–Yau 4-folds, under the assumption that the derived Hilbert scheme on the anti-canonical divisor is a derived critical locus. We prove a degeneration formula and apply it to compute zero-dimensional invariants on $\mathbb{C}^4$ and on any local curve. This verifies a conjecture proposed by the first author and Kool.

Let k be a perfect field of characteristic p and $\Gamma$ an infinite, first countable pro-p group. We study the behavior of the p-primary part of the ‘motivic class group’, i.e. the full p-divisible group of the Jacobian, in any $\Gamma$-tower of function fields over k that is unramified outside a finite (possibly empty) set of places $\Sigma$, and totally ramified at every place of $\Sigma$. When $\Sigma=\emptyset$ and $\Gamma$ is a torsion-free p-adic Lie group, we obtain asymptotic formulae which show that the p-torsion class group schemes grow in a remarkably regular manner. In the ramified setting $\Sigma\neq\emptyset$, we obtain a similar asymptotic formula for the p-torsion in ‘physical class groups’, i.e. the k-rational points of the Jacobian, which generalizes the work of Mazur and Wiles, who studied the case $\Gamma=\mathbf Z_p$.

We prove a conjecture of Manin and Orlov: the moduli space $\overline{\mathcal{M}}_{0,n}$ of stable rational curves with n marked points admits a full exceptional collection that is equivariant under the action of the symmetric group $S_n$ permuting the marked points. In particular, its $K_0$-group with integral coefficients is a permutation $S_n$-lattice.

In this paper, we prove the abundance conjecture for threefolds over a perfect field k of characteristic $p \gt 3$ in the case where the numerical dimension is equal to 2. More precisely, we show that, if (X,B) is a projective log canonical pair of dimension 3 over k such that $K_{X}+B$ is nef and $\nu(K_{X}+B)=2$, then $K_{X}+B$ is semi-ample.

We develop a unified approach for identifying spaces of stability conditions of triangulated categories arising from weighted marked surfaces with moduli spaces of quadratic differentials. This identification is based on the use of perverse schobers (perverse sheaves of triangulated categories) and a notion of positive arc system kit on a perverse schober $\mathcal F$, which provides a systematic way of assigning to a graded curve on the surface a global section of $\mathcal F$. This assignment allows us to identify mixed angulations and their flips with finite-length hearts and their tilts. As an application we obtain a generalization of the results of Bridgeland and Smith to quadratic differentials with arbitrary singularity type (zero/pole/exponential).