Geodesic
Bochner--Hartogs type extension theorem for roots and logarithms of holomorphic line bundles
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and geometric issues of complex analysis, Tome 279 (2012), pp. 269-287

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We prove an extension theorem for roots and logarithms of holomorphic line bundles across strictly pseudoconcave boundaries: they extend in all cases except one, when the dimension and Morse index of a critical point is 2. In that case we give an explicit description of obstructions to the extension. 
@article{TM_2012_279_a17,
     author = {S. Ivashkovich},
     title = {Bochner--Hartogs type extension theorem for roots and logarithms of holomorphic line bundles},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {269--287},
     publisher = {mathdoc},
     volume = {279},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2012_279_a17/}
}
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S. Ivashkovich. Bochner--Hartogs type extension theorem for roots and logarithms of holomorphic line bundles. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and geometric issues of complex analysis, Tome 279 (2012), pp. 269-287. http://geodesic.mathdoc.fr/item/TM_2012_279_a17/