Band 2022, Heft 786

We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show – under additional congruence assumptions on the CM order – that the reductions are surjective (and even become equidistributed) on the product of supersingular loci when the discriminant of the order becomes large. This variant of the equidistribution theorems of Duke and Cornut–Vatsal is an(other) application of the recent work of Einsiedler and Lindenstrauss on the classification of joinings of higher-rank diagonalizable actions.

We show that almost every entry in the character table of S N {S_{N}} is divisible by any fixed prime as N → ∞ {N\to\infty} . This proves a conjecture of Miller.

We prove that the K-moduli space of cubic fourfolds is identical to their GIT moduli space. More precisely, the K-(semi/poly)stability of cubic fourfolds coincide to the corresponding GIT stabilities, which was studied in detail by Laza. In particular, this implies that all smooth cubic fourfolds admit Kähler–Einstein metrics. Key ingredients are local volume estimates in dimension three due to Liu and Xu, and Ambro–Kawamata’s non-vanishing theorem for Fano fourfolds.

We construct a one-parameter family of singly periodic translating solutions to mean curvature flow that converge as the period tends to 0 to the union of a grim reaper surface and a plane that bisects it lengthwise. The surfaces are semigraphical: they are properly embedded, and, after removing a discrete collection of vertical lines, they are graphs. We also provide a nearly complete classification of semigraphical translators.

For a smooth algebraic variety X , a monodromic D -module on X × ℂ {X\times\mathbb{C}} is decomposed into a direct sum of some D -modules on X . We show that the Hodge filtration of a mixed Hodge module on X × ℂ {X\times\mathbb{C}} whose underlying D -module is monodromic is also decomposed. Moreover, we show that there is an equivalence of categories between the category of monodromic mixed Hodge modules on X × ℂ {X\times\mathbb{C}} and the category of “gluing data”. As an application, we endow the Fourier–Laplace transformation of the underlying D -module of a monodromic mixed Hodge module with a mixed Hodge module structure.

We prove that a closed embedded minimal surface in the round three-sphere which satisfies the symmetries of a Lawson surface and has the same genus is congruent to the Lawson surface.

We provide qualitative bounds on the area and topology of separating constant mean curvature (CMC) surfaces of bounded (Morse) index. We also develop a suitable bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular, we show that convergence always occurs with multiplicity one, which implies that the minimal blow-ups (bubbles) are all catenoids.

We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery’s function F ⁢ ( α , T ) {F(\alpha,T)} in bounded intervals, the second is an integral introduced by Selberg related to estimating the variance of primes in short intervals, and the last is the second moment of the logarithmic derivative of the Riemann zeta-function near the critical line. The conjectured asymptotic for any of these three integrals is equivalent to Montgomery’s pair correlation conjecture. Assuming the Riemann hypothesis, we substantially improve the known upper and lower bounds for these integrals by introducing new connections to certain extremal problems in Fourier analysis. In an appendix, we study the intriguing problem of establishing the sharp form of an embedding between two Hilbert spaces of entire functions naturally connected to Montgomery’s pair correlation conjecture.

Let X be a compact Kähler space with klt singularities and vanishing first Chern class. We prove the Bochner principle for holomorphic tensors on the smooth locus of X : any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-étale cover X splits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf of X according to its holonomy representation. In particular, we classify those X which have strongly stable tangent sheaf: up to quasi-étale covers, these are either irreducible Calabi–Yau or irreducible holomorphic symplectic. As an application of these results, we show that if X has dimension four, then it satisfies Campana’s Abelianity Conjecture.