Related papers: Holomorphic vector fields and perturbed extremal K…
We study left-invariant locally conformally K\"ahler structures on Lie groups, or equivalently, on Lie algebras. We give some properties of these structures in general, and then we consider the special cases when its complex structure is…
Given a compact complex $n$-fold $X$ satisfying the $\partial\bar\partial$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-K\"ahler) Hermitian metric $\omega$, we introduce the concept of…
We present a substantial generalisation of a classical result by Lie on integrability by quadratures. Namely, we prove that all vector fields in a finite-dimensional transitive and solvable Lie algebra of vector fields on a manifold can be…
A Lie algebra structure on variation vector fields along an immersed curve in a $2$-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure…
It is shown that a compact $n$-dimensional K\"ahler manifold with $\frac{n}{2}$-positive Calabi curvature operator has the rational cohomology of complex projective space. For even $n,$ this is sharp in the sense that the complex quadric…
We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact K\"ahler manifolds.
The purpose of this note is to establish the following theorem: Let N be a Kahler manifold, L be a compact oriented immersed minimal Lagrangian submanifold in N and V be a holomorphic vector field in a neighbourhood of L in N. Let div(V) be…
The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally K\"ahler…
We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi-Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault-Hodge theory and its description in terms of the cohomology of the…
We prove that any $4$-dimensional almost-K\"ahler Lie algebra of constant Hermitian holomorphic sectional curvature with respect to the canonical Hermitian connection is K\"ahler.
Based on recent work of S. K. Donaldson and T. Mabuchi, we prove that any extremal Kaehler metric in the sense of E. Calabi, defined on the product of polarized compact complex projective manifolds is the product of extremal Kaehler metrics…
We construct the deformation functor associated to a couple of morphisms of differential graded Lie algebras, and use it to study the infinitesimal deformations of a holomorphic map of compact complex manifolds. In particular, in the case…
We study Lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U,H) where U is a complex…
We show that, under the definiteness of holomorphic sectional curvature, the spaces of some holomorphic tensor fields on compact Chern-K\"{a}hler-like Hermitian manifolds are trivial. These can be viewed as counterparts to Bochner's…
The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) K\"ahler structure, famously used to realize the group's irreducible representations in holomorphic sections of appropriate line bundles (Borel-Weil…
Consider a fibred compact K\"ahler manifold X endowed with a relatively ample line bundle, such that each fibre admits a constant scalar curvature K\"ahler metric and has discrete automorphism group. Assuming the base of the fibration…
We examine Hermitian metrics on unimodular Lie algebras which contains a $J$-invariant abelian ideal of codimension two, and give a classification for all Bismut K\"ahler-like and all Bismut torsion-parallel metrics on such Lie algebras.
We introduce a moment map picture for holomorphic string algebroids where the Hamiltonian gauge action is described by means of inner automorphisms of Courant algebroids. The zero locus of our moment map is given by the solutions of the…
We show that holomorphic vector fields on (C^3,0) have separatrices provided that they are embedded in a rank 2 representation of a two-dimensional Lie algebra. In turn, this result enables us to show that the second jet of a holomorphic…
We generalize the notions of the Futaki invariant and extremal vector field of a compact K\"ahler manifold to the general almost-Kahler case and show the periodicity of the extremal vector field when the symplectic form represents an…