# Analgebraic Geometry

It so happened that at the beginning of 2016 I gave a talk on the conference “Geometric aspects of modern dynamics” in Porto, delivered a minicourse at Journées Louis Antoine in Rennes and wrote an expository paper for the European Mathematical Society Newsletter, all devoted to the same subject. The subject, provisionally dubbed as “Analgebraic geometry”, deals with algebraic-like properties (especially from the point of view of intersection theory) of real and complex analytic varieties defined by ordinary and Pfaffian differential equations with polynomial right hand sides. Thus

analgebraic = un-algebraic + analytic + algebraic (background) + weak algebraicity-like properties.

It turns out that this analgebraic geometry has very intimate connections with classical problems like Hilbert 16th problem, properties of periods of algebraic varieties, analytic number theory and arithmetic geometry.

For more details see the presentation prepared for the minicourse (or the shorter version of the talk) and the draft of the paper.

Any remarks and comments will be highly appreciated.

# Geometric and global theory of linear ordinary differential equations

1. Global theory of linear equations. Jet bundles, Cartan distribution. Meromorphic connexion associated with a linear equation.
2. “Natural bundle” for a globally Fuchsian equation. Sum of characteristic exponents.
3. Riemann–Hilbert problem for Fuchsian equations. Hypergeometric equation.

# Linear ordinary differential equations of order n

1. Construction of the Weyl algebra (noncommutative “differential polynomials of one independent variable”). Division with remainder, factorization, solutions.
2. Reconstruction of differential equations from their solutions. Riemann theorem.
3. Regular and Fuchsian operators. Complete local reducibility. Fuchs theorem (local regularity $\iff$ local Fuchs property) and its reformulations.

Recommended reading: Section 19 from the book (printing disabled)

# Bolibruch Impossibility Theorem

Revealing an obstruction for realization of a matrix group as the monodromy of a Fuchsian system on $\mathbb C P^1$.

1. Degree (Chern class) of a complex bundle vs. that of a subbundle. The total trace of residues of a meromorphic connexion.
2. Linear algebra: Monoblock operators and their invariant subspaces.
3. Local theory revisited: local invariant subbundles of a (resonant) Fuchsian singularity in the Poincaré–Dulac–Levelt normal form.
4. Bolibruch connexions on the trivial bundle: theorem on the spectra of residues.
5. Three Matrices $4\times 4$: the Bolibruch Counterexample.

Reading: Section 18E from the book (printing disabled).

Refresh your memory: Sections 16C16D (local theory), 17E-17I (degree of bundles)

## Tuesday, February 26, 2008

### Lecture 1 (Feb 27, 2008)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 9:55
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# Systems of Linear ODEs with complex time

1. Total recall: on differential equations in the complex domain (for the newcomers, if any) and foliations.
2. Linear systems: vector, matrix and Pfaffian form. Fyndamental solutions. Linearity of the transport maps.
3. Holomorphic (gauge) equivalence of linear systems. Monodromy group.
4. Linear systems with isolated singularities. Euler system and its properties.

Reading material: Section 15 from the Book (printing disabled)

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