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Holomorphic vector fields tangent to foliations in dimension three.
1Universidade Federal de Itajubá, Departamento de Matemática, Rua Irmã Ivone Drumond, 200, 35903-087 Itabira, MG, Brazil.
Anais Da Academia Brasileira De Ciencias
|September 8, 2021
Summary
This study examines singular holomorphic vector fields and their tangency to holomorphic foliations in C3. Researchers explore desingularization techniques and analyze vector fields tangent to multiple independent foliations.
Area of Science:
- Complex Geometry
- Differential Geometry
- Complex Analysis
Background:
- Singular holomorphic vector fields and foliations are fundamental objects in complex geometry.
- Understanding their interactions, particularly tangency, is crucial for classifying geometric structures in complex spaces.
Purpose of the Study:
- To investigate the relationship between desingularized models of singular holomorphic vector fields and singular holomorphic foliations.
- To analyze the specific case of a vector field being tangent to three independent foliations.
Main Methods:
- Consideration of a singular holomorphic vector field in a neighborhood of 0 in C3.
- Analysis of a singular holomorphic foliation of codimension one to which the vector field is tangent.
- Exploration of desingularization techniques for both the vector field and the foliation.
- Examination of the scenario involving tangency to three independent foliations.
Main Results:
- The study establishes a framework for relating the desingularized models of singular vector fields and foliations.
- It provides insights into the geometric conditions under which a vector field can be tangent to multiple independent foliations.
Conclusions:
- The desingularization process offers a way to compare and relate complex geometric objects.
- The analysis of tangency to multiple foliations reveals specific structural properties of singular vector fields in C3.
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