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Extending Nirenberg-Spencer’s question on holomorphic embeddings to families of holomorphic embeddings. (English) Zbl 1477.14059

In the paper under review, one considers a family of free submanifolds of a complex manifold, that is, a submanifold \(A\) of a complex manifold \(X\) with the normal bundle \(N_{A/ X}\) being generated by global sections of the bundle and all these sections being un-obstracted in \(X\).
Let \(\mathcal{K}\) and \(\mathcal{K}'\) be subfamilies of free submanifolds in complex manifolds \(X\) and \(X'\), respectively. We say that the families \(\mathcal{K}\) and \(\mathcal{K}'\) are germ-equivalent (resp. iso-equivalent up to order \(l\)) if there exists a bi-holomorphic map \(f:\mathcal{K}\rightarrow\mathcal{K}'\) such that for any pair \(([A]\in\mathcal{K},\,[A']\in\mathcal{K}')\) with \(f([A])=[A']\), the Euclidean (resp. the \(k\)-th infinitesimal) nighbourhood of \(A\) in \(X\) and that of \(A'\) in \(X'\) are equivalent.
It is easily seen that if \(\mathcal{K}\) and \(\mathcal{K}'\) are germ-equivalent, then, they are iso-equivalent up to order \(k\) for any non-negative integer \(k\).
The problem considered in the paper is when the converse of this fact is true.
The main theorem (Theorem 3.1) gives a criterion for the subfamilies \(\mathcal{K}\) and \(\mathcal{K}'\) to be germ-equivalent. More precisely, it is proved that if \(\mathcal{K}\) contains a member \(A\subset X\) with \(H^0(A,\,T_A)=0\), and if \(\mathcal{K}\) is iso-equivalent up to order \(1\) to any other subfamily \(\mathcal{K}'\) of free submanifolds, then, they are germ-equivalent.
As an application of the main theorem, together with some well-known facts on \(K3\) surfaces, it is proved that linearly-normal \(K3\) surfaces \(X\) and \(X'\) in \(\mathbb{P}^g\) are projectively isomorphic if subfamilies \(\mathcal{K}\) and \(\mathcal{K}'\) of free compact submanifolds in \(X\) and \(X'\) are iso-equivalent up to order \(1\).

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14J45 Fano varieties
32C22 Embedding of analytic spaces
58A15 Exterior differential systems (Cartan theory)
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References:

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