15 October 2021 Extending Nirenberg–Spencer’s question on holomorphic embeddings to families of holomorphic embeddings
Jun-Muk Hwang
Author Affiliations +
Duke Math. J. 170(15): 3237-3265 (15 October 2021). DOI: 10.1215/00127094-2021-0044


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.


Download Citation

Jun-Muk Hwang. "Extending Nirenberg–Spencer’s question on holomorphic embeddings to families of holomorphic embeddings." Duke Math. J. 170 (15) 3237 - 3265, 15 October 2021. https://doi.org/10.1215/00127094-2021-0044


Received: 9 October 2019; Revised: 9 December 2020; Published: 15 October 2021
First available in Project Euclid: 23 September 2021

MathSciNet: MR4324178
zbMATH: 1477.14059
Digital Object Identifier: 10.1215/00127094-2021-0044

Primary: 14J28
Secondary: 14J28 , 14J45 , 32C22 , 58A15

Keywords: Cartan’s equivalence method , free rational curves , Infinitesimal neighborhood , K3 surfaces

Rights: Copyright © 2021 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.170 • No. 15 • 15 October 2021
Back to Top