On the geometry of holomorphic curves and complex surface
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914211467100160 |
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| author | Falqueto, Amanda Dias Tari, Farid |
| author_facet | Falqueto, Amanda Dias Tari, Farid |
| contents | We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that measure the contact between curves or surfaces and model objects become holomorphic. This allows the application of singularity theory, yielding analogues of classical results from the real case. Our approach enables the definition of geometric invariants of curves, which we call the $C$-curvature and $C$-torsion, as well as surface invariants such as the $C$-principal curvature and $C$-Gaussian curvature. It also gives geometric meaning to the complexification of of the families measuring contact of analytic surfaces in $\mathbb R^3$ with lines, planes and spheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the geometry of holomorphic curves and complex surface Falqueto, Amanda Dias Tari, Farid Differential Geometry 57R45, 53B99 We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that measure the contact between curves or surfaces and model objects become holomorphic. This allows the application of singularity theory, yielding analogues of classical results from the real case. Our approach enables the definition of geometric invariants of curves, which we call the $C$-curvature and $C$-torsion, as well as surface invariants such as the $C$-principal curvature and $C$-Gaussian curvature. It also gives geometric meaning to the complexification of of the families measuring contact of analytic surfaces in $\mathbb R^3$ with lines, planes and spheres. |
| title | On the geometry of holomorphic curves and complex surface |
| topic | Differential Geometry 57R45, 53B99 |
| url | https://arxiv.org/abs/2505.24719 |