We show that if X is a cocompact G-CW-complex such that each isotropy subgroup Gσ is L(2)-good over an arbitrary commutative ring k, then X satisfies some fixed-point formula which is an L(2)-analogue of Brown’s formula in 1982. Using this result we present a fixed point formula for a cocompact proper G-CW-complex which relates the equivariant L(2)-Euler characteristic of a fixed point CW-complex Xs and the Euler characteristic of X/G. As corollaries, we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji-Mislin in 2009. We also show that if X is a free G-CW-complex such that C∗(X) is chain homotopy equivalent to a chain complex of finitely generated projective Zπ1(X)-modules of finite length and X satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula, then χ(X/G)=χ(2)(X). As an application, we prove that the weak Bass conjecture holds for any finitely presented group G satisfying the following condition: for any finitely dominated CW-complex Y with π1(Y )=G, Y satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula.

We study singular Hermitian metrics on vector bundles. There are two main results in this paper. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections. As an application, we show the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose standard approximations are not Nakano semipositive. The aim of the second main result is to determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for example, certain rank 2 bundles on elliptic curves.

We study Nakai-Moishezon type question and Donaldson’s “tamed to compatible” question for almost complex structures on rational four manifolds. By extending Taubes’ subvarieties-currentform technique to J-nef genus 0 classes, we give affirmative answers of these two questions for all tamed almost complex structures on S^2 bundles over S^2 as well as for many geometrically interesting tamed almost complex structures on other rational four manifolds, including the del Pezzo ones.

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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.

Given a hypersurface coamoeba of a Laurent polynomial$f$, it is an open problem to describe the structure of the set of connected components of its complement. In this paper we approach this problem by introducing the lopsided coamoeba. We show that the closed lopsided coamoeba comes naturally equipped with an order map, i.e. a map from the set of connected components of its complement to a translated lattice inside the zonotope of a Gale dual of the point configuration $\operatorname{supp}(f)$ . Under a natural assumption, this map is a bijection. Finally we use this map to obtain new results concerning coamoebas of polynomials of small codimension.

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite-dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite-dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic family.

We discuss applications of an improvement on the Riemann mapping theorem which replaces the unit disc by another “double quadrature domain,” i.e., a domain that is a quadrature domain with respect to both area and boundary arc length measure. Unlike the classic Riemann mapping theorem, the improved theorem allows the original domain to be finitely connected, and if the original domain has nice boundary, the biholomorphic map can be taken to be close to the identity, and consequently, the double quadrature domain is close to the original domain. We explore some of the parallels between this new theorem and the classic theorem, and some of the similarities between the unit disc and the double quadrature domains that arise here. The new results shed light on the complexity of many of the objects of potential theory in multiply connected domains.

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 that the symmetrized bidisc may be exhausted by strongly linearly convex domains. It shows in particular the existence of a strongly linearly convex domain that cannot be exhausted by domains biholomorphic to convex ones.