Geodesic
The~differential geometry of blow-ups
Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 2, pp. 313-328

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We discuss the local geometry in the vicinity of a sphere $\mathbb P^1$ embedded with a negative normal bundle. We show that the behavior of the Kähler potential near a sphere embedded with a given normal bundle can be determined using the adjunction formula. As a by-product, we construct (asymptotically locally complex-hyperbolic) Kähler–Einstein metrics on the total spaces of the line bundles $\mathcal O(-m)$, $m\ge3$, over $\mathbb P^1$.
Keywords: blow-up, adjunction formula, Kähler–Einstein metric. 
@article{TMF_2015_185_2_a4,
     author = {D. V. Bykov},
     title = {The~differential geometry of blow-ups},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {313--328},
     publisher = {mathdoc},
     volume = {185},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2015_185_2_a4/}
}
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D. V. Bykov. The~differential geometry of blow-ups. Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 2, pp. 313-328. http://geodesic.mathdoc.fr/item/TMF_2015_185_2_a4/