Related papers: On the Kaehler rank of compact complex surfaces

Using results by Donaldson and Auroux on pseudo-holomorphic curves as well as Duval's rational convexity construction, the paper investigates the existence of smooth Lagrangian surfaces representing 2-dimensional homology classes in complex…

We classify indefinite simply connected hyper-Kaehler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m,4m). We establish a natural 1-1 correspondence between simply connected…

In this short note we prove an estimate on the rank a.e. of the tangent (p,p) vector to a a positive closed current of bidimension (p,p) in CP^k, in terms of the dimension of its trace measure.

We consider generalisations of the elliptic Calogero--Moser systems associated to complex crystallographic groups in accordance to [1]. In our previous work [2], we proposed these systems as candidates for Seiberg--Witten integrable systems…

Inspired by recent work of Kopparty-Moshkovitz-Zuiddam and motivated by problems in combinatorics and hypergraphs, we introduce the notion of the symmetric geometric rank of a symmetric tensor. This quantity is equal to the codimension of…

Taking a compact K\"{a}hler manifold as playground, we explore the powerfulness of Hodge index theorem. A main object is the Lorentzian classes on a compact K\"{a}hler manifold, behind which the characterization via Lorentzian polynomials…

Let f be a holomorphic automorphism of a compact Kahler manifold (X,\omega) of dimension k>1. We study the convex cones of positive closed (p,p)-currents T_p, which satisfy a functional relation $f^*(T_p)=\lambda T_p, \lambda>1,$ and some…

We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood. Without…

We prove that any two Kahler potentials on a compact Kahler manifold can be connected by a geodesic segment of C^{1,1} regularity. This follows from an a priori interior real Hessian bound for solutions of the nondegenerate complex…

We consider 6-dimensional strict nearly Kaehler manifolds acted on by a compact, cohomogeneity one automorphism group G. We classify the compact manifolds of this class up to G-diffeomorphisms. We also prove that the manifold has constant…

We characterize compact locally conformally K\"ahler (l.c.K.) manifolds under the assumption of a purely conformal, holomorphic circle action. As an application, we determine the structure of the compact l.c.K. manifolds with parallel Lee…

In this text, we recall some basics and results about complex geometry and currents in the complex scenario. Most of the results are classic and their evidence is not given here. On the other hand, we describe in detail some notions to help…

This paper concerns the explicit construction of extremal Kaehler metrics on total spaces of projective bundles, which have been studied in many places. We present a unified approach, motivated by the theory of hamiltonian 2-forms (as…

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are…

Under broad hypotheses we derive a scalar reduction of the generalized K\"ahler-Ricci soliton system. We realize solutions as critical points of a functional analogous to the classical Aubin energy defined on the orbit of a natural…

We develop new techniques for the construction and classification of representations of row-finite and locally convex higher-rank graph C*-algebras O. This class includes Cuntz--Krieger algebras associated to row-finite directed graphs. Our…

For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations…

In the late 1990s, B. Y. Chen introduced the notion of special slant surfaces in K\"{a}hler surfaces and classified non-minimal proper special slant surfaces with constant mean curvature in $2$-dimensional complex space forms. In this…

Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in arXiv:1802.00170. This construction also reveals that these solitons are generalized K\"ahler in two distinct…

We follow our study of non-K\"ahler complex structures on $R^4$ that we defined in a previous paper. We prove that these complex surfaces do not admit any smooth complex compactification. Moreover, we give an explicit description of their…