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Equations of minimal coupling

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Indeed, any complex submanifold of the complex projective space Pn admits a K¨ahler metric - the restriction of the Fubini–Study metric is an example. Conversely, one might wonder whether a compact complex manifold that admits a K¨ahler structure can always be realized as a complex submanifold of Pn or, in other words, whether the complex structure is projective. This is obviously not the case, general complex tori Cn /Γ (n ≥ 2) and general K3 surfaces provide counter-examples. , [ω] ∈ H 2 (X, Z) (see [6, Thm.

7] S. Bosch, W. L¨ utkebohmert & M. Raynaud – N´eron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 21, Springer-Verlag, Berlin, 1990. [8] G. -L. Chai – Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 22, Springer-Verlag, Berlin, 1990, avec un appendice de D. Mumford. [9] R. , vol. 52, Springer-Verlag, New York, 1977. [10] S. Keel & S. Mori – Quotients by groupoids , Ann. of Math. (2) 145 (1997), no. 1, p. 193–213. ´ r & N. I. Shepherd-Barron – Threefolds and deformations of [11] J.

15 (2002), no. 3, p. 617–664 (electronic). [MM02b] , « Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation », Ann. of Math. (2) 155 (2002), no. 1, p. 235–280. [MM04] , « Review on blow up and asymptotic dynamics for critical and subcritical gKdV equations », in Noncompact problems at the intersection of geometry, analysis, and topology, Contemp. , vol. 350, Amer. Math. , Providence, 2004, p. 157–177. [MR03] F. Merle & P. Raphael – « Sharp upper bound on the blow-up rate for the critical nonlinear Schr¨odinger equation », Geom.

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