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Stability under deformations of extremal almost-K\"ahler metrics in dimension 4
TLDR
In this article, the existence of a smooth family of extremal almost-Kahler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path, was proved.Abstract:
Given a path of almost-Kahler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-Kahler metric is an extremal Kahler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-Kahler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path.read more
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On the J-anti-invariant cohomology of almost complex 4-manifolds
TL;DR: For a compact almost complex 4-manifold (M,J) consisting of cohomology classes representable by invariant and anti-invariant 2-forms, the subgroups of the subgroup $H^-_J$ were studied in this article.
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Toric K\"ahler-Einstein metrics and convex compact polytopes
TL;DR: In this article, it was shown that any compact convex simple lattice polytope is the moment polyTope of a K\"ahler-Einstein orbifold, unique up to orbifolds covering and homothety.
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Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds
TL;DR: In this article, the existence of extremal Kaehler metrics on toric 4-orbifolds with second Betti number b2(M)=2 was shown to be solvable if and only if the rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense.
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Integrability theorems and conformally constant Chern scalar curvature metrics in almost Hermitian geometry
Mehdi Lejmi,Markus Upmeier +1 more
TL;DR: In this article, the existence of almost Kahler metrics with conformally constant Chern scalar curvatures is studied on an almost Hermitian manifold, in particular with respect to conformal variations.
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On cohomology of almost complex 4-manifolds
TL;DR: In this paper, the authors further investigated properties of the dimension of a closed almost Hermitian 4-manifold using metric compatible almost complex structures and proved that the dimension h_J^-=0.
References
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Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II
Book
Compact Complex Surfaces
TL;DR: In this article, the authors describe the topology and algebraic properties of complex surfaces, including the following properties: 1. The Projective Plane, 2. The Jacobian Fibration, 3. Hodge Theory on Surfaces, 4. Inequahties for Hodge Numbers, 5. Holomorphic Vector Bundles, Serre Duality and Riemann-Roch Theorem.
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Introduction to symplectic topology
Dusa McDuff,Dietmar Salamon +1 more
TL;DR: In this article, the authors present a survey of the history of classical and modern manifold geometry, from classical to modern, including linear and almost complex structures, and the Arnold conjecture of the group of symplectomorphisms.
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Differential analysis on complex manifolds
TL;DR: In this paper, the authors present an introduction to the basics of analysis and geometry on compact complex manifolds and provide tools which are the building blocks of many mathematical developments over the past 30 years.
Journal ArticleDOI
Scalar Curvature and Stability of Toric Varieties
TL;DR: In this paper, a stability condition for a polarised algebraic variety is defined and a conjecture relating this to the existence of a Kahler metric of constant scalar curvature.