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Vestislav Apostolov

Publications and source records attributed to Vestislav Apostolov.

At least 37 records · Page 2Linked to original sources
arXiv2020

Weighted K-stability of polarized varieties and extremality of Sasaki manifolds

We use the correspondence between extremal Sasaki structures and weighted extremal Kahler metrics defined on a regular quotient of a Sasaki manifold, established by the first two authors, and Lahdili's theory of weighted K-stability in order to define a suitable notion of (relative) weighted K-stability for compact Sasaki manifolds of regular type. We show that the (relative) weighted K-stability with respect to a maximal torus is a necessary condition for the existence of a (possibly irregular) extremal Sasaki metric. We also compare weighted K-stability to the K-stability of the corresponding polarized affine cone (introduced by Collins-Szekelyhidi), and prove that they agree on the class of test configurations we consider. As a byproduct, we strengthen the obstruction to the existence of a scalar-flat Kahler cone metric from the K-semistability to the K-stability on these test configurations. We use our approach to give a characterization of the existence of a compatible extremal Sasaki structure on a principal circle bundle over an admissible ruled manifold, expressed in terms of the positivity of a single polynomial of one variable over a given interval.

math.DG↗
arXiv2019

The CR geometry of weighted extremal Kahler and Sasaki metrics

We establish an equivalence between conformally Einstein--Maxwell Kahler 4-manifolds (recently studied in many works) and extremal Kahler 4-manifolds (in the sense of Calabi) with nowhere vanishing scalar curvature. The corresponding pairs of Kahler metrics arise as transversal Kahler structures of Sasaki metrics compatible with the same CR structure and having commuting Sasaki--Reeb vector fields. This correspondence extends to higher dimensions using the notion of a weighted extremal Kahler metric, illuminating and uniting several explicit constructions in Kahler and Sasaki geometry. It also leads to new existence and non-existence results for extremal Sasaki metrics, suggesting a link between notions of relative weighted K-stability for a polarized variety, and relative K-stability of the Kahler cone corresponding to a Sasaki polarization.

math.DG↗
arXiv2018

Weighted extremal Kähler metrics and the Einstein-Maxwell geometry of projective bundles

We study the existence of weighted extremal Kähler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge Kähler manifold of constant scalar curvature. Admissible projective bundles have been defined by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman, and they include the projective line bundles (Hwang-Singer) and their blow-downs (Koiso-Sakane), thus providing a most general setting for extending the existence theory for extremal Kähler metrics pioneered by a seminal construction of Calabi. We obtain a general existence result for weighted extremal metrics on admissible manifolds, which yields many new examples of conformally Kähler, Einstein-Maxwell metrics of complex dimension $m>2$, thus extending the recent constructions of LeBrun and Koca-Tønnesen-Friedman to higher dimensions. For each admissible Kähler class on an admissible projective bundle, we associate an explicit function of one variable and show that if it is positive on the interval $(-1,1)$, then there exists a weighted extremal Kähler metric in the given class, whereas if it is strictly negative somewhere in $(-1,1)$, there is no Kähler metrics of constant weighted scalar curvature in that class. We also relate the positivity of the function to a notion of weighted K-stability, thus establishing a Yau-Tian-Donaldson type correspondence for the existence of Kähler metrics of constant weighted scalar curvature in the rational admissible Kähler classes on an admissible projective bundle. Weighted extremal orthotoric metrics are examined in an appendix.

math.DG↗
arXiv2017

Toric contact geometry in arbitrary codimension

We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

math.DG↗
arXiv2017

Extremal Kähler Poincaré type metrics on toric varieties

We develop a general theory for the existence of extremal Kähler metrics of Poincaré type in the sense of Auvray, defined on the complement of a toric divisor of a polarized toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist. Using the explicit methods of Apostolov-Calderbank-Gauduchon together with the computational approach of Sektnan, we show that on a Hirzebruch complex surface the necessary conditions are also sufficient. In particular, on such a complex surface the complement of the infinity section admits an extremal Kähler metric of Poincaré type whereas the complement of a fibre admits a complete ambitoric extremal Kähler metric which is not of Poincaré type.

math.DG↗
arXiv2017

Levi-Kahler reduction of CR structures, products of spheres, and toric geometry

We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of such quotients, and find Levi-Kahler quotients of products of 3-spheres which are extremal in a weighted sense introduced by G. Maschler and the first author.

math.DG↗
arXiv2017

Relative K-polystability of projective bundles over a curve

Let $P(E)$ be the projectivization of a holomorphic vector bundle $E$ over a compact complex curve $C$. We characterize the existence of an extremal Kähler metric on the ruled manifold $P(E)$ in terms of relative K-polystability and the fact that $E$ decomposes as a direct sum of stable bundles.

math.AG↗
arXiv2016

On the Lee classes of locally conformally symplectic complex surfaces

We prove that the deRham cohomology classes of Lee forms of locally conformally symplectic structures taming the complex structure of a compact complex surface $S$ with first Betti number equal to $1$ is either a non-empty open subset of $H^1_{dR}(S, \mathbb R)$, or a single point. In the latter case, we show that $S$ must be biholomorphic to a blow-up of an Inoue-Bombieri surface. Similarly, the deRham cohomology classes of Lee forms of locally conformally Kähler structures of a compact complex surface $S$ with first Betti number equal to $1$ is either a non-empty open subset of $H^1_{dR}(S, \mathbb R)$, a single point or the empty set. We give a characterization of Enoki surfaces in terms of the existence of a special foliation, and obtain a vanishing result for the Lichnerowicz-Novikov cohomology groups on the class ${\rm VII}$ compact complex surfaces with infinite cyclic fundamental group.

math.DG↗
arXiv2016

ALE scalar-flat Kähler metrics on non-compact weighted projective spaces

We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact type.

math.DG↗
arXiv2015

Conformally Kähler, Einstein-Maxwell Geometry

On a given compact complex manifold or orbifold $(M,J)$, we study the existence of Hermitian metrics $\tilde g$ in the conformal classes of Kähler metrics on $(M,J)$, such that the Ricci tensor of $\tilde g$ is of type $(1,1)$ with respect to the complex structure, and the scalar curvature of $\tilde g$ is constant. In real dimension $4$, such Hermitian metrics provide a Riemannian counter-part of the Einstein--Maxwell (EM) equations in general relativity, and have been recently studied in \cite{ambitoric1, LeB0, LeB, KTF}. We show how the existence problem of such Hermitian metrics (which we call in any dimension {\it conformally Kähler, EM} metrics) fits into a formal momentum map interpretation, analogous to results by Donaldson and Fujiki~\cite{donaldson, fujiki} in the cscK case. This leads to a suitable notion of a Futaki invariant which provides an obstruction to the existence of conformally Kähler, EM metrics invariant under a certain group of automorphisms which are associated to a given Kähler class, a real holomorphic vector field on $(M,J)$, and a positive normalization constant. Specializing to the toric case, we further define a suitable notion of $K$-polystability and show it provides a (stronger) necessary condition for the existence of toric, conformally Kähler, EM metrics. We use the methods of \cite{ambitoric2} to show that on a compact symplectic toric $4$-orbifold with second Betti number equal to $2$, $K$-polystability is also a sufficient condition for the existence of (toric) conformally Kähler, EM metrics, and the latter are explicitly described as ambitoric in the sense of \cite{ambitoric1}. As an application, we exhibit many new examples of conformally Kähler, EM metrics defined on compact $4$-orbifolds, and obtain a uniqueness result for the construction in \cite{LeB0}.

math.DG↗
arXiv2015

An extremal eigenvalue problem in Kähler geometry

We study Laplace eigenvalues $λ_k$ on Kähler manifolds as functionals on the space of Kähler metrics with cohomologous Kähler forms. We introduce a natural notion of a $λ_k$-extremal Kähler metric and obtain necessary and sufficient conditions for it. A particular attention is paid to the $λ_1$-extremal properties of Kähler-Einstein metrics of positive scalar curvature on manifolds with non-trivial holomorphic vector fields.

math.DG↗
arXiv2013

Ambikaehler geometry, ambitoric surfaces and Einstein 4-orbifolds

We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing local geometry depending on a quadratic polynomial and two arbitrary functions of one variable, these two functions being explicit degree 4 polynomials when the Kaehler metrics are extremal (in the sense of Calabi). One motivation for and application of this result is an explicit local description of Einstein 4-manifolds which are hermitian with respect to either orientation. This can be considered as a riemannian analogue of a result in General Relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. We discuss toric compactifications of these metrics on orbifolds and provide infinite discrete families of compact toric extremal Kaehler orbifolds. Our examples include Bach-flat Kaehler orbifolds which are conformal to complete smooth Einstein metrics on an open subset. We illustrate how these examples fit with recent conjectures relating the existence of extremal toric metrics to various notions of stability.

math.DG↗
arXiv2013

Ambitoric geometry I: Einstein metrics and extremal ambikaehler structures

We present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kaehler metrics which are toric with respect to a common 2-torus action. In the generic case, these "ambitoric" structures have an intriguing local geometry depending on a quadratic polynomial q and arbitrary functions A and B of one variable. We use this description to classify Einstein 4-metrics which are hermitian with respect to both orientations, as well a class of solutions to the Einstein-Maxwell equations including riemannian analogues of the Plebanski-Demianski metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kaehler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kaehler metrics. For both the Einstein-Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kaehler metrics on toric 4-orbifolds with second betti number b2=2.

math.DG↗
arXiv2013

Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds

We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results provide a computational test for the K-stability of labelled quadrilaterals. Extremal ambitoric structures were classified locally in Part I of this work, but herein we only use the straightforward fact that explicit Kaehler metrics obtained there are extremal, and the identification of Bach-flat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bach-flat Kaehler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset, thus extending the work of many authors.

math.DG↗
arXiv2012

A splitting theorem for extremal Kaehler metrics

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 on each factor, provided that the integral Futaki invariants of the polarized manifold vanish or its automorphism group satisfies a constraint. This extends a result of S.-T. Yau about the splitting of a Kaehler-Einstein metric on the product of compact complex manifolds to the more general setting of extremal Kaehler metrics.

math.DG↗
arXiv2010

Extremal Kähler metrics on projective bundles over a curve

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to Σ$ over a compact complex curve $Σ$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$ admits a Kähler metric of constant scalar curvature if and only if $E$ is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of $E$ as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in sufficiently small Kähler classes. The methods used to prove the above results apply to a wider class of manifolds, called {\it rigid toric bundles over a semisimple base}, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.

math.DG↗