Motivated by the Bloch–Beilinson conjectures, Voisin has formulated a conjecture about $0$-cycles on self-products of surfaces of geometric genus one. We verify Voisin’s conjecture for the family of Todorov surfaces with $K^2=2$ and fundamental group $\mathbb{Z}/2\mathbb{Z}$. As a by-product, we prove that certain Todorov surfaces have finite-dimensional motive.

We study $3$-dimensional divisorial contractions to $cDV$ points with discrepancy greater than $1$ which are of exceptional type. We show that every $3$-dimensional divisorial contraction is obtained as a weighted blowup.

We consider the Cauchy problem in $L^2$ for first-order systems. A necessary condition is that the system must be uniformly diagonalizable or, equivalently, that it admits a bounded symmetrizer. A sufficient condition is that it admits a smooth (Lipschitz) symmetrizer, which is true when the system is diagonalizable with eigenvalues of constant multiplicities. Counterexamples show that uniform diagonalizability is not sufficient in general for systems with variable coefficients, and they indicate that the symplectic properties of the set $\Sigma$ of the singular points of the characteristic variety are important. In this article, we give a new class of systems for which the Cauchy problem is well-posed in $L^2$. The main assumption is that $\Sigma$ is a smooth involutive manifold and the system is transversally strictly hyperbolic.

We investigate the position of amenable subalgebras in arbitrary amalga- mated free product von Neumann algebras $M=M_1{*}_B{M}_2$. Our main result states that, under natural analytic assumptions, any amenable subalgebra of $M$ that has a large intersection with $M_1$ is actually contained in $M_1$. The proof does not rely on Popa’s asymptotic orthogonality property but on the study of nonnormal conditional expectations.

We find a decomposition formula of the local Bayer–Macrì map for the nef line bundle theory on the Bridgeland moduli space over a surface. If there is a global Bayer–Macrì map, then such a decomposition gives a precise correspondence from Bridgeland walls to Mori walls. As an application, we compute the nef cone of the Hilbert scheme $S^{\left[n\right]}$ of $n$-points over special kinds of a fibered surface $S$ of Picard rank $2$.

We give a description of the local Jacquet–Langlands correspondence for simple supercuspidal representations via type theory. As a consequence, we show that the endoclasses for such representations are invariant under the local Jacquet–Langlands correspondence.

We obtain various characterizations of commutative Noetherian local rings $(R,\mathfrak{m})$ in terms of homological dimensions of certain finitely generated modules. Our argument has a series of consequences in different directions. For example, we establish that $R$ is Gorenstein if the Gorenstein injective dimension of the maximal ideal $\mathfrak{m}$ of $R$ is finite. Moreover, we prove that $R$ must be regular if a single ${\mathsf{Ext}}_R^n(I,J)$ vanishes for some integrally closed $\mathfrak{m}$-primary ideals $I$ and $J$ of $R$ and for some positive integer $n$.

We investigate the explicit Galois structures of Bloch–Kato Selmer groups of $p$-adic realizations of critical motives. We show in particular that, under natural and relatively mild hypotheses, the Krull–Schmidt decompositions of the $p$-adic lattices arising from such Selmer groups are dominated by very simple indecomposable modules (even when the ranks are very large).