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 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.

We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and Păun. We define what it means for such a metric to be positively and negatively curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the curvature Θ^{$h$}as a matrix of currents. We then proceed to show that such metrics can be regularised in such a way that the corresponding curvature tensors converge weakly to Θ^{$h$}. Finally we define what it means for$h$to be strictly negatively curved in the sense of Nakano and show that it is possible to regularise such metrics with a sequence of smooth, strictly Nakano negative metrics.

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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.