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Balanced Metrics and Noncommutative Kähler Geometry
Symmetry, integrability and geometry: methods and applications, Tome 6 (2010) Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we show how Einstein metrics are naturally described using the quantization of the algebra of functions $C^\infty(M)$ on a Kähler manifold $M$. In this setup one interprets $M$ as the phase space itself, equipped with the Poisson brackets inherited from the Kähler 2-form. We compare the geometric quantization framework with several deformation quantization approaches. We find that the balanced metrics appear naturally as a result of requiring the vacuum energy to be the constant function on the moduli space of semiclassical vacua. In the classical limit these metrics become Kähler–Einstein (when $M$ admits such metrics). Finally, we sketch several applications of this formalism, such as explicit constructions of special Lagrangian submanifolds in compact Calabi–Yau manifolds.
Keywords: balanced metrics; geometric quantization; Kähler–Einstein. 
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     author = {Sergio Luki\'c},
     title = {Balanced {Metrics} and {Noncommutative} {K\"ahler} {Geometry}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a68/}
}
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Sergio Lukić. Balanced Metrics and Noncommutative Kähler Geometry. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a68/

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