Logarithmic singularities
- De Rham division lemma (and its generalization)
- Definition of a logarithmic pole: (scalar case). Residues.
- Logarithmic complex: principal lemma on Λ-closedness.
- Principal example: logarithmic complex for the normal crossings. Saito theorem.
- Closed logarithmic 1-forms: complete description. Darbouxian foliations.
- Matrix casse. Conjugacy of the residues along the polar locus. Residues on the normal crossings.
- Schlesinger system: flat connexions with logarithmic poles along the diagonal.
- Flat connexions with first order poles are almost always logarithmic, yet resonances may spoil the pattern.
Recommended reading: the same notes, sect. 3-4.
Meromorphic flat connexions on holomorphic manifolds: Integrability, monodromy, classification
- Pfaffian systems and their integrability
- From local to global solutions: monodromy
- Geometric language: covariant derivative and its curvature
- Meromorphic functions, meromorphic forms
- Example: multidimensional Euler system
- Regular singularities
- Flat connexions vs. isomonodromic deformations
Recommended reading: D. Novikov & S.Y., Lectures on meromorphic flat connexions, sect. 1-2.
No classes today, as 50% of the students are speaking on a conference elsewhere.
Stokes phenomenon for irregular singularities of linear systems
- Irregular singularities: total recall. Formal diagonalizability of non-resonant systems.
- Sectorial gauge equivalence: formal, holomorphic, asymptotic series.
- Separation rays. Sibuya theorem on sectorial normalization (statement).
- Sectorial authomorphisms. Rigidity of the normal form in large sectors.
- Stokes matrix cochain and Stokes matrix multipliers as complete invariants of holomorphic classification of irregular singularities.
- Stokes phenomenon. Realization theorem (Birkhoff). Generic divergence of the formal gauge normalizing transformations.
Recommended reading: Sections 20F-20I from the Book
