将全纯嵌入的Nirenberg-Spencer问题推广到全纯嵌入族
Extending Nirenberg-Spencer's question on holomorphic embeddings to families of holomorphic embeddings
摘要：
Nirenberg和Spencer提出了一个问题：当法丛充分正时，复流形中的紧致复子流形的芽是否由它的有穷级无穷小邻域决定。为了研究一个更大的子流形类的问题，包括自由有理曲线，我们重新制定的子流形和他们的无穷小邻域的家庭设置的问题。当子流形没有非零向量场时，我们证明了只考虑一阶邻域就足以对重新表述的问题有肯定的回答。当子流形具有非零向量场时，在子流形具有一定的形变性质的条件下，得到了一个肯定的答案，该结果适用于自由有理曲线.作为应用，我们得到了Picard数为1的Fano流形的Cartan-Fubini型扩张定理的一个更强的版本。我们还提出了射影K3曲面超平面部分的潜在应用。
英文摘要：
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.
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