Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917102508572672 |
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| author | Sillari, Lorenzo Tomassini, Adriano |
| author_facet | Sillari, Lorenzo Tomassini, Adriano |
| contents | In this paper we investigate the Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class. First, we prove that, if the almost complex structure is also tamed, the only possible values for the Kodaira dimension are $0$ or $-\infty$. This is done by developing the theory of pseudoholomorphic structures on vector bundles. In arbitrary dimension, we study infinitesimal deformations of structures with pseudoholomorphically torsion canonical bundle. We compute their tangent space and, under suitable assumptions, we prove an unobstructedness theorem in the spirit of Bogomolov--Tian--Todorov. Together, our results allow to fully describe non-integrable infinitesimal deformations of complex structures on $K3$ and Enriques surfaces in terms of their Kodaira dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20124 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class Sillari, Lorenzo Tomassini, Adriano Differential Geometry 32Q60, 32L05 (Primary) 53C15, 32G07 (Secondary) In this paper we investigate the Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class. First, we prove that, if the almost complex structure is also tamed, the only possible values for the Kodaira dimension are $0$ or $-\infty$. This is done by developing the theory of pseudoholomorphic structures on vector bundles. In arbitrary dimension, we study infinitesimal deformations of structures with pseudoholomorphically torsion canonical bundle. We compute their tangent space and, under suitable assumptions, we prove an unobstructedness theorem in the spirit of Bogomolov--Tian--Todorov. Together, our results allow to fully describe non-integrable infinitesimal deformations of complex structures on $K3$ and Enriques surfaces in terms of their Kodaira dimension. |
| title | Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class |
| topic | Differential Geometry 32Q60, 32L05 (Primary) 53C15, 32G07 (Secondary) |
| url | https://arxiv.org/abs/2511.20124 |