We will introduce a quantity which measures the singularity of a plurisubharmonic function$φ$relative to another plurisubharmonic function$ψ$, at a point$a$. We denote this quantity by$ν$_{$a$,$ψ$}($φ$). It can be seen as a generalization of the classical Lelong number in a natural way: if$ψ$=($n$−1)log| ⋅ −$a$|, where$n$is the dimension of the set where$φ$is defined, then$ν$_{$a$,$ψ$}($φ$) coincides with the classical Lelong number of$φ$at the point$a$. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form {$z$:$ν$_{$z$,$ψ$}($φ$)≥$c$} where$c$>0, are in fact analytic sets, provided that the$weight$$ψ$satisfies some additional conditions.

We show the intersection of a compact almost complex subvariety of dimension 4 and a compact almost complex submanifold of codimension 2 is a J-holomorphic curve. This is a generalization of positivity of intersections for J-holomorphic curves in almost complex 4-manifolds to higher dimensions. As an application, we discuss pseudoholomorphic sections of a complex line bundle. We introduce a method to produce J-holomorphic curves using the differential geometry of almost Hermitian manifolds. When our main result is applied to pseudoholomorphic maps, we prove the singularity subset of a pseudoholomorphic map between almost complex 4-manifolds is J-holomorphic. Building on this, we show degree one pseudoholomorphic maps between almost complex 4-manifolds are actually birational morphisms in pseudoholomorphic category.

In the moduli space $$ \mathcal{M} $$ _{$g$}of genus-$g$Riemann surfaces, consider the locus $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ of Riemann surfaces whose Jacobians have real multiplication by the order $$ \mathcal{O} $$ in a totally real number field$F$of degree$g$. If$g$= 3, we compute the closure of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of $$ \mathcal{M} $$ _{$g$}and the closure of the locus of eigenforms over $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ . Boundary strata of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ are parameterized by configurations of elements of the field$F$satisfying a strong geometry of numbers type restriction.

In this paper, we prove some results on bi-Hölder extensions not only for biholomorphisms but also for more general Kobayashi metric quasi-isometries between the domains. Furthermore, we establish the Gehring–Hayman type theorems on certain complex domains which play an important role through the paper. Then by applying the above results, we show the bi-Hölder equivalence between the Euclidean boundary and the Gromov boundary of bounded convex domains which are Gromov hyperbolic with respect to their Kobayashi metrics.

We investigate the Chern–Ricci flow, an evolution equation of Hermitian metrics generalizing the Kähler–Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kähler–Einstein metric from the base. Some of our estimates are new even for the Kähler–Ricci flow. A consequence of our result is that, on every minimal non-Kähler surface of Kodaira dimension one, the Chern–Ricci flow converges in the sense of Gromov–Hausdorff to an orbifold Kähler–Einstein metric on a Riemann surface.