'Vanishing Cycles in Holomorphic Foliations by Curves and Foliated Shells'

By Serguei Ivashkovich - June 16, 2023 - Session 3
We shall give in our lectures an introduction to the theory of holomorphic foliations. Any preliminary knowledge of foliation theory is not required but some knowledge of complex analysis of several complex variables is assumed. The supposed length of the course is 6 lectures. The material will be organized in the following three chapters:
Chapter 1. Foliations on Real Manifolds.
1.1. Poincar´e–Bendixson Theorem. 1.2. Real analytic foliations and Haefliger theorem. 1.3. Vanishing cycles and foliated currents. 1.4. Compact leaves and Novikov’s theorem.
Chapter 2. Singular Holomorphic Foliations.
2.1. Smooth and singular holomorphic foliations. 2.2. Reduction of singularities. 2.3. Baum–Bott index and formula. 2.4. Non-existence of exceptional minimal sets in codimension one holomorphic foliations on P n for n ≥ 3. 2.5. Relation to Levi flat hypersurfaces.
Chapter 3. Vanishing Cycles and Foliated Shells.
3.1. Simultaneous uniformization and Poincar´e domains. 3.2. Vanishing ends and completed leaves. 3.3. Extension of meromorphic mappings after a reparametrization and foliated shells. 3.4. Imbedded cycles and shells. 3.5 Examples.