In this article we prove the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Gorenstein smoothable, K-polystable $\mathbb{Q}$-Fano varieties, and we show how these metrics behave, in the Gromov–Hausdorff sense, under $\mathbb{Q}$-Gorenstein smoothings.

In this paper, we construct growing modes of the linearized Navier–Stokes equations about generic stationary shear flows of the boundary layer type in a regime of a sufficiently large Reynolds number: $R\to \infty$. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. The formal construction of approximate modes is well documented in physics literature, going back to the work of Heisenberg, C. C. Lin, Tollmien, Drazin, and Reid, but a rigorous construction requires delicate mathematical details, involving, for instance, a treatment of primitive Airy functions and singular solutions. Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of $e^{t/\sqrt{R}}$. The proof follows the general iterative approach introduced in our companion paper, avoiding having to deal with matching inner and outer asymptotic expansions, but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators. Unlike in the channel flows, the spatial domain in the boundary layers is unbounded and the iterative scheme is likely to diverge due to the linear growth in the vertical variable. We introduce a new iterative scheme to simultaneously treat the singularity near critical layers and the asymptotic behavior of solutions at infinity. The instability of generic boundary layers obtained in this paper is linked to the emergence of Tollmien–Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.

A numerical equivalence class of $k$-cycles is said to be big if it lies in the interior of the closed cone generated by effective classes. We construct analogues for arbitrary cycle classes of the volume function for divisors which distinguishes big classes from boundary classes.

Let $X$ be a curve of genus $g\ge 2$ over a number field $F$ of degree $d=[F:\mathbf{Q}]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X\left(F\right)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known by the work of Caporaso, Harris, and Mazur to follow from the Bombieri–Lang conjecture. A related conjecture posits the existence of a uniform bound $N_{\mathrm{tors},†}(g,d)$ on the number of geometric torsion points of the Jacobian $J$ of $X$ which lie on the image of $X$ under an Abel–Jacobi map. For fixed $X$, the finiteness of this quantity is the Manin–Mumford conjecture, which was proved by Raynaud.

We give an explicit uniform bound on $\#X\left(F\right)$ when $X$ has Mordell–Weil rank $r\le g-3$. This generalizes recent work of Stoll on uniform bounds for hyperelliptic curves of small rank to arbitrary curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of $F$-rational torsion points of $J$ lying on the image of $X$ under an Abel–Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of $J$ lying on $X$ when the reduction type of $X$ is highly degenerate.

Our methods combine Chabauty–Coleman’s $p$-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.