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ABSTRACT: In this paper, we prove that if the area functional of a surface $\Sigma^2$
in a symplectic manifold $(M^{2n},\bar{\omega})$ has a critical point or has a
compatible stable point in the same cohomology class, then it must be
$J$-holomorphic. Inspired by a classical result of Lawson-Simons, we show how
various restrictions of the stability assumption to variations of metrics in
the space "projectively induced" metrics are enough to give the desired
conclusion.
11/2012;
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ABSTRACT: We compute the Szego kernel of the unit circle bundle of a negative line
bundle dual to a regular quantum line bundle over a compact Kaehler manifold.
As a corollary we provide an infinite family of smoothly bounded strictly
pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge
manifolds) for which the log-terms in the Fefferman expansion of the Szego
kernel vanish and which are not locally CR-equivalent to the sphere. We also
give a proof of the fact that, for homogeneous Hodge manifolds, the existence
of a locally spherical CR-structure on the unit circle bundle alone implies
that the manifold is biholomorphic to a projective space. Our results
generalize those obtained by M. Englis and G. Zhang for Hermitian symmetric
spaces of compact type.
07/2012;
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ABSTRACT: This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow-ups at finitely many points
of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization
of isolated quotient singularities of compact orbifolds which carry constant scalar curvature Kähler metrics.
Acta Mathematica 04/2012; 196(2):179-228. · 3.33 Impact Factor
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ABSTRACT: In this paper we study the set of balanced metrics (in Donaldson's
terminology) on a compact complex manifold M which are homothetic to a given
balanced one. This question is related to various properties of the
Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this
set is finite when $M$ admits a non-positive Kahler-Einstein metric, in the
case of non-homogenous toric Kaehler-Einstein manifolds of dimension $\leq 4$
and in the case of Arezzo-Pacard constant scalar curvature metrics.
05/2011;
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ABSTRACT: In this note we clarify the structure of the moduli space of constant scalar curvature Kaehler metrics as one approaches the boundary of the Kaehler cone on cscK manifolds blown up at finite set of points, in the spirit of the previous work arXiv:math/0504115. Results about which Kaehler classes can be reached and about the position of the blown up points are given.
07/2007;
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ABSTRACT: In this paper we study the first eigenvalue of the Laplacian on a compact Kaehler manifold using stable bundles and balanced bases.
12/2005;
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ABSTRACT: In this paper we continue our study about the existence of Kaehler metrics of constant scalar curvature (Kcsc) on blow ups at points of compact manifolds with Kcsc metrics started in math.DG/0411522. In this second part we deal with the case of base manifolds with holomorphic vector fields and we give sufficient conditions for the position of points to be blown up to preserve the Kcsc property. These results are obtained by a gluing procedure. We give explicit examples of our construction, in particular getting optimal results in the cases of P^n and P^n x M, where M is a Kcsc compact manifold with discrete automorphism group.
05/2005;
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ABSTRACT: Building on Donaldsons work on constant scalar curvature metrics, we study the space of regular Khler metrics E, i.e. those for which deformation quantization has been defined by Cahen, Gutt and Rawnsley. After giving, in Sects. 2 and 3 a review of Donaldsons moment map approach, we study the essential uniqueness of balanced basis (i.e. of coherent states) in a more general setting (Theorem 2.5). We then study the space E in Sect.4 and we show in Sect.5 how all the tools needed can be defined also in the case of non-compact manifolds.
Communications in Mathematical Physics 03/2004; 246(3):543-559. · 1.94 Impact Factor
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Communications on Pure and Applied Mathematics 01/2003; 56(3):283 - 327. · 2.58 Impact Factor
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Communications on Pure and Applied Mathematics - COMMUN PURE APPL MATH. 01/2003; 56(3):283-327.
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ABSTRACT: In this paper we prove the existence of families of n-dimensional complete embedded minimal submanifolds of C^n with a prescribed configuration of k>1 asymptotic planes. These submanifolds are obtained by desingularizing the intersection of the asymptotes, using a gluing theorem applied to a generalization of a special lagrangian "hyperbola" found by Lawlor.
04/2001;
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ABSTRACT: In this paper we prove that, under natural assumptions, the scalar curvature of a Kaehler-Einstein metric on a compactification of C^n is strictly positive.
04/2001;
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ABSTRACT: In this paper we study the link between the asymptotic expansion of Tian–Yau–Zelditch [J. Diff. Geom. 32 (1990) 99] and the quantization of compact Kähler manifolds carried out in [J. Geophys. 7 (1990) 45; Trans. Am. Math. Soc. 337 (1993) 73].
Journal of Geometry and Physics.
