Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture

The Kawaguchi–Silverman conjecture predicts that if $f: X \dashrightarrow X$ is a dominant rational-self map of a projective variety over $\overline{{\mathbb{Q}}}$, and $P$ is a $\overline{{\mathbb{Q}}}$-point of $X$ with a Zariski dense orbit, then the dynamical and arithmetic degrees of $f$ coincide: $\lambda _1(f) = \alpha _f(P)$. We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than $1$, and all endomorphisms of hyper-Kähler manifolds in any dimension. In the latter case, we construct a canonical height function associated with any automorphism $f: X \to X$ of a hyper-Kähler manifold defined over $\overline{{\mathbb{Q}}}$. We additionally obtain results on the periodic subvarieties of automorphisms for which the dynamical degrees are as large as possible subject to log concavity.

Projective manifolds modeled after hyperquadrics

We classify all projective manifolds with flat holomorphic conformal structure. The Kähler–Einstein case was treated by Kobayashi and Ochiai, there exists a very short list of possible manifolds. In the non-Kähler–Einstein case a classification was only known in small dimensions: by Kobayashi and Ochiai for complex surfaces and by the authors for projective threefolds. This paper completes the general case.

Threefolds in ℙ6 of degree 12

Linearly normal complex projective threefolds of degree 12, embedded in ℙ

PROJECTIVE THREEFOLDS WITH HOLOMORPHIC CONFORMAL STRUCTURE

The authors give a complete classification of projective threefolds admitting a holomorphic conformal structure. A corollary is the complete list of projective threefolds, whose tangent bundle is a symmetric square.