Calabi–Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky, Cecotti, Ooguri, and Vafa (BCOV), it is expected that genus 1 curve-counting on a Calabi–Yau manifold is related to a conjectured invariant, only depending on the complex structure of the mirror, and built from Ray–Singer holomorphic analytic torsions. To this end, extending work of Fang, Lu, and Yoshikawa in dimension 3, we introduce and study the BCOV invariant of Calabi–Yau manifolds of arbitrary dimension. To determine it, knowledge of its behavior at the boundary of moduli spaces is imperative. To address this problem, we prove general results on degenerations of $L^2$-metrics on Hodge bundles and their determinants, refining the work of Schmid. We express the singularities of these metrics in terms of limiting Hodge structures and derive consequences for the dominant and subdominant singular terms of the BCOV invariant.

We characterize the rectifiability (both uniform and not) of an Ahlfors regular set E of arbitrary codimension by the behavior of a regularized distance function in the complement of that set. In particular, we establish a certain version of the Riesz transform characterization of rectifiability for lower-dimensional sets. We also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of E. This allows us to precisely compute the harmonic measure of those sets associated to this degenerate operator and prove that, in sharp contrast with the usual setting of codimension 1, a converse to Dahlberg’s theorem must be false on lower-dimensional boundaries without additional assumptions.

Let $\Gamma <{PSL}_2(\mathbb{C})$ be discrete, of cofinite volume, and noncocompact. We prove that, for all $K>1$, there is a subgroup $H<\Gamma$ that is K-quasiconformally conjugate to a discrete cocompact subgroup of ${PSL}_2(\mathbb{R})$. Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a nearly Fuchsian surface subgroup.

We compute the integral p-adic étale cohomology of Drinfeld symmetric spaces of any dimension. This refines the computation of the rational p-adic étale cohomology from our recent work on Stein spaces. The main tools are: the computation of the integral de Rham cohomology from that work and, as a new tool, the integral p-adic comparison theorems of Bhatt–Morrow–Scholze and Česnavičius–Koshikawa which replace the quasi-integral comparison theorem of Tsuji. Along the way, we compute the $A_{\inf }$-cohomology of Drinfeld symmetric spaces.