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Hwang, Jun Muk
복소기하학 연구단
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EXTENDING NIRENBERG-SPENCER'S QUESTION ON HOLOMORPHIC EMBEDDINGS TO FAMILIES OF HOLOMORPHIC EMBEDDINGS

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Title
EXTENDING NIRENBERG-SPENCER'S QUESTION ON HOLOMORPHIC EMBEDDINGS TO FAMILIES OF HOLOMORPHIC EMBEDDINGS
Author(s)
Jun-Muk Hwang
Publication Date
2021-10-15
Journal
DUKE MATHEMATICAL JOURNAL, v.170, no.15, pp.3237 - 3265
Publisher
DUKE UNIV PRESS
Abstract
Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first- order neighborhoods to have an affirmative answer to the reformulated question. When the submanifolds do have nonzero vector fields, we obtain an affirmative answer to the question under the additional assumption that submanifolds have certain nice deformation properties, which is applicable to free rational curves. As an application, we obtain a stronger version of the Cartan-Fubini-type extension theorem for Fano manifolds of Picard number 1. We also propose a potential application on hyperplane sections of projective K3 surfaces.
URI
https://pr.ibs.re.kr/handle/8788114/10926
DOI
10.1215/00127094-2021-0044
ISSN
0012-7094
Appears in Collections:
Center for Complex Geometry (복소기하학 연구단) > 1. Journal Papers (저널논문)
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