A rational function of degree at least $2$ with coefficients in an algebraically closed field is postcritically finite (PCF) if and only if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattès maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattès PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a nonarchimedean version of Fatou’s classical result that every attracting cycle of a rational function over $\mathbb{C}$ attracts a critical point.

Given a $K3$ surface $X$ over a field of characteristic $p$, Artin conjectured that if $X$ is supersingular (meaning infinite height), then its Picard rank is 22. Along with work of Nygaard–Ogus, this conjecture implies the Tate conjecture for $K3$ surfaces over finite fields with $p\ge 5$. We prove Artin’s conjecture under the additional assumption that $X$ has a polarization of degree $2d$ with $p>2d+4$. Assuming semistable reduction for surfaces in characteristic $p$, we can improve the main result to $K3$ surfaces which admit a polarization of degree prime to $p$ when $p\ge 5$.

The argument uses Borcherds’s construction of automorphic forms on $O(2,n)$ to construct ample divisors on the moduli space. We also establish finite-characteristic versions of the positivity of the Hodge bundle and the Kulikov–Pinkham–Persson classification of $K3$ degenerations. In the appendix by A. Snowden, a compatibility statement is proven between Clifford constructions and integral $p$-adic comparison functors.

We construct a family of Lagrangian submanifolds in the Landau–Ginzburg mirror to the projective plane equipped with a binodal cubic curve as anticanonical divisor. These objects correspond under mirror symmetry to the powers of the twisting sheaf $\mathcal{O}\left(1\right)$, and hence their Floer cohomology groups form an algebra isomorphic to the homogeneous coordinate ring. An interesting feature is the presence of a singular torus fibration on the mirror, of which the Lagrangians are sections. This gives rise to a distinguished basis of the Floer cohomology and the homogeneous coordinate ring parameterized by fractional integral points in the singular affine structure on the base of the torus fibration. The algebra structure on the Floer cohomology is computed using the symplectic techniques of Lefschetz fibrations and the topological quantum field theory counting sections of such fibrations. We also show that our results agree with the tropical analogue proposed by Abouzaid, Gross, and Siebert. Extensions to a restricted class of singular affine manifolds and to mirrors of the complements of components of the anticanonical divisor are discussed.

Suppose that $f:S^2\to S^2$ determines a dynamical system on the sphere which is topologically coarse expanding conformal in the sense of our previous work. We prove that if its Ahlfors regular conformal dimension $Q$ is realized by some metric $d$, then either (i) $Q=2$ and $f$ is topologically conjugate to a semihyperbolic rational map with Julia set equal to the whole sphere or (ii) $Q>2$ and $f$ is topologically conjugate to a map which lifts to an affine expanding map of a torus whose differential has distinct real eigenvalues. This is an analogue of a known result for Gromov hyperbolic groups with a two-sphere boundary.