Hermitian null loci
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911821554778112 |
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| author | Dang, Quang-Tuan |
| author_facet | Dang, Quang-Tuan |
| contents | We establish a transcendental generalization of Nakamaye's theorem to compact complex manifolds when the form is not assumed to be closed. We apply the recent analytic technique developed by Collins--Tosatti to show that the non-Hermitian locus of a nef and big $(1,1)$-form, which is not necessarily closed, on a compact complex manifold equals the union of all positive-dimensional analytic subvarieties where the restriction of the form is not big (null locus). As an application, we can give an alternative proof of the Nakai--Moishezon criterion of Buchdahl and Lamari for complex surfaces and generalize this result in higher dimensions. This is also used for studying degenerate complex Monge--Ampère equations on compact Hermitian manifolds. Finally, we investigate finite time non-collapsing singularities of the Chern--Ricci flow, partially answering a question raised by Tosatti and Weinkove. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01126 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hermitian null loci Dang, Quang-Tuan Complex Variables Differential Geometry 32W20, 53E30, 32U15 We establish a transcendental generalization of Nakamaye's theorem to compact complex manifolds when the form is not assumed to be closed. We apply the recent analytic technique developed by Collins--Tosatti to show that the non-Hermitian locus of a nef and big $(1,1)$-form, which is not necessarily closed, on a compact complex manifold equals the union of all positive-dimensional analytic subvarieties where the restriction of the form is not big (null locus). As an application, we can give an alternative proof of the Nakai--Moishezon criterion of Buchdahl and Lamari for complex surfaces and generalize this result in higher dimensions. This is also used for studying degenerate complex Monge--Ampère equations on compact Hermitian manifolds. Finally, we investigate finite time non-collapsing singularities of the Chern--Ricci flow, partially answering a question raised by Tosatti and Weinkove. |
| title | Hermitian null loci |
| topic | Complex Variables Differential Geometry 32W20, 53E30, 32U15 |
| url | https://arxiv.org/abs/2404.01126 |