Generalized Tanaka prolongation and convergence of formal equivalence between embeddings
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917761294270464 |
|---|---|
| author | Hong, Jaehyun Hwang, Jun-Muk |
| author_facet | Hong, Jaehyun Hwang, Jun-Muk |
| contents | The works of Commichau--Grauert and Hirschowitz showed that a formal equivalence between embeddings of a compact complex manifold is convergent, if the embeddings have sufficiently positive normal bundles in a suitable sense. We show that the convergence still holds under the weaker assumption of semi-positive normal bundles if some geometric conditions are satisfied. Our result can be applied to many examples of general minimal rational curves, including general lines on a smooth hypersurface of degree less than $n$ in the $(n+1)$-dimensional projective space. As a key ingredient of our arguments, we formulate and prove a generalized version of Tanaka's prolongation procedure for geometric structures subordinate to vector distributions, a result of independent interest. When applied to the universal family of the deformations of the compact submanifolds satisfying our geometric conditions, the generalized Tanaka prolongation gives a natural absolute parallelism on a suitable fiber space. A formal equivalence of embeddings must preserve these absolute parallelisms, which implies its convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15537 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Tanaka prolongation and convergence of formal equivalence between embeddings Hong, Jaehyun Hwang, Jun-Muk Differential Geometry Complex Variables 32K07, 58A30, 32C22 The works of Commichau--Grauert and Hirschowitz showed that a formal equivalence between embeddings of a compact complex manifold is convergent, if the embeddings have sufficiently positive normal bundles in a suitable sense. We show that the convergence still holds under the weaker assumption of semi-positive normal bundles if some geometric conditions are satisfied. Our result can be applied to many examples of general minimal rational curves, including general lines on a smooth hypersurface of degree less than $n$ in the $(n+1)$-dimensional projective space. As a key ingredient of our arguments, we formulate and prove a generalized version of Tanaka's prolongation procedure for geometric structures subordinate to vector distributions, a result of independent interest. When applied to the universal family of the deformations of the compact submanifolds satisfying our geometric conditions, the generalized Tanaka prolongation gives a natural absolute parallelism on a suitable fiber space. A formal equivalence of embeddings must preserve these absolute parallelisms, which implies its convergence. |
| title | Generalized Tanaka prolongation and convergence of formal equivalence between embeddings |
| topic | Differential Geometry Complex Variables 32K07, 58A30, 32C22 |
| url | https://arxiv.org/abs/2408.15537 |