# 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.

## Riemann-Hilbert problem

The Riemann-Hilbert problem consists in “constructing a Fuchsian system with a prescribed monodromy”.

More precisely, let $M_1,M_2,\dots,M_d$ be nondegenerate matrices such that their product is an identical matrix, and $a_0,a_1,\dots, a_d\in\mathbb C$ are distinct points, such that the segments $[a_0,a_k]\subset\mathbb C,\ k=1,\dots,d$ are all disjoint except for the point $a_0$ itself.

The problem is to construct a linear system of equations

$\displaystyle \dot X=A(t)X,\quad A(t)=\sum_{k=1}^d \frac{A_k}{t-a_k},\quad \sum_{k=1}^d A_k=0$,

such that the monodromy operator along the path “$\gamma_k=$segment $[a_0,a_k]+$ small loop around $a_k+$segment $[a_k,a_0]$” is equal to $M_k$.

The modern strategy of solving this problem is surgery. One can easily construct a local solution, a differential system on a neighborhood $U_k$ of the segment $[a_0,a_k]$, which has the specified monodromy. The phase space of this system is the cylinder $U_k\times\mathbb C^n$, and without loss of generality one can assume that together the neighborhoods $U_k$ cover the whole Riemann sphere $\mathbb CP^1=\mathbb C\cup\{\infty\}$. Patching together these local solutions, one can construct a linear system with the specified monodromy, but it will be defined not on $\mathbb C P^1\times\mathbb C^n$, as required, but on a more general object, holomorphic vector bundle over $\mathbb C P^1$.

Description of different vector bundles is of an independent interest and is well known. It turns out (Birkhoff), that each holomorphic vector bundle in dimension $n$ is completely determined by a(n unordered) tuple of integer numbers $d_1,\dots,d_n\in\mathbb Z$, and the bundle is trivial if and only if $d_1=\cdots=d_n=0$.

However, the strategy of solving the Riemann-Hilbert problem by construction of the bundle and determining its holomorphic type is complicated by two facts:

1. Determination of the holomorphic type of a bundle is a transcendental problem;
2. The local realization of the monodromy is by no means unique: in the non-resonant case one can realize any matrix $M_k$ by an Euler system with the eigenvalues which can be arbitrarily shifted by integers; in the resonant case one should add to this freedom also non-Euler systems. This freedom can change the holomorphic type of the vector bundle in a very broad range.

It turns out that the fundamental role in solvability of the Riemann-Hilbert problem plays the (ir)reducibility of the linear group generated by the matrices $M_1,\dots,M_k$.

Theorem (Bolibruch, Kostov). If the group is irredicible, i.e., there is no invariant subspace in $\mathbb C^n$ common for all operators $M_k$, then one can choose the local realizations in such a way that the resulting bundle is trivial and thus yields solution to the Riemann-Hilbert problem.

The proof is achieved as follows: one constructs a possibly nontrivial bundle realizing the given monodromy, and then this bundle is brutally trivialized by a transformation that is only meromorphic at one of the singularities. The result will be a system with all but one singularities being Fuchsian, and the problem reduces to bringing to the Fuchsian form the last point (assumed to be at infinity) by transformations of the form $X\mapsto P(t)X$ with $P$ being a matrix polynomial with a constant nonzero determinant.  The group of such transformations is considerably more subtle, but ultimately the freedom in construction of the initial bundle can be used to guarantee that the last point is also “Fuchsianizible”.

All the way around, if the monodromy group is reducible, then there is an obstruction of the torsion type exists for trivializing the bundle. This obstruction was first discovered by A. Bolibruch, and its description can be found in the textbook by Yu. Ilyashenko and SY (sections 16G and 18).

## Families of Fuchsian equations

A Fuchsian equation on $\mathbb C P^1$ with only quasiunipotent singularities admits an upper bound for the number of complex roots of its solutions, which depends on the equation, in particular, in the “magnitude” (slope), but also on the relative position of its singularities.

We are interested in finding conditions ensuring that this bound does not “explode”. The easiest way to formulate this is to consider parametric families of Fuchsian equations.

We will assume that the parametric family has the form

$L_\lambda u=0,\qquad L_\lambda=\sum_{k=0}^n p_k(t,\lambda)\partial^k,\quad p_k\in\mathbb C[t,\lambda]\qquad (*)$

with the coefficients $p_k$ polynomial in $t$ and rationally depending on the parameters $\lambda\in\mathbb P^m$ (one can consider them as homogeneous polynomials of the same degree on $\mathbb C^{m+1}$). For some values of $\lambda$ the operator $L_\lambda$ may degenerate (the leading coefficient vanishes identically, not excluding the degeneracy $L_\lambda\equiv0$). Such values, however, should constitute a proper algebraic subvariety $\Lambda\subset\mathbb P^m$.

Note that, because of the semicontinuity, it is sufficient to establish the global uniform bound for the number of isolated roots only for $\lambda\notin\Lambda$: complex roots cannot disappear in the blue sky…

We will impose the following qualitative conditions, imposed on the family (*).

1. Isomonodromy: when parameters change, the monodromy group remains “the same”.
2. Tameness (regularity): solutions $u_\lambda(t)$ of the equations grow at most polynomially when $\lambda\to\Lambda$.
3. Quasiunipotence: all singular points always have quasiunipotent monodromy.

The last condition is the “regularity” with respect to the parameters rather than with respect to the independent variable $t$. All conditions need to be accurately formulated, but one can give a simple example producing such families.

Consider a rational matrix-valued 1-form $\Omega$ on $\mathbb P^1\times\mathbb P^m$ with the polar locus $\varSigma\subset \mathbb P^1\times\mathbb P^m$ which is an algebraic divisor (singular hypersurface). Assume that the linear system $\mathrm dX=\Omega X$ is locally solvable and regular on $\varSigma$. Then for any fixed $\lambda$ the first row components of the (multivalued) matrix function $X(t,\lambda)$ satisfy a linear Fuchsian equation $L_\lambda u=0$ rationally depending on $\lambda$. This way we get the family of equations automatically satisfying the first two conditions above. It turns out that the third condition is sufficient to verify only for a generic equation of the family.

(Kashiwara theorem follows).

## Boundedness of the slope

In the arbitrary family (*) the slope $\angle L_\lambda$ is a semialgebraic function of the parameter $\lambda\notin\Lambda$, eventually undefined on the locus $\Lambda$ itself, and may well be unbounded.

However, in the isomonodromic regular family this is impossible.

(Grigoriev theorem follows)

Corollary: conformal pseudoinvariance of the slope.

## Wednesday, October 22, 2008

### Coming out of the closet

Filed under: conference — Sergei Yakovenko @ 5:24
Tags: , ,

A couple of weeks ago some twothirds of the conspirators coworkers attended the workshop Equations aux dérivées partielles et théorie de Galois différentielle dit Malgrangefest in Luminy and delivered a talk on their work.

Slides from this talk are now available (static pdf, $\approx$2 Mb) for everybody to see.

## Friday, August 22, 2008

### Infinitesimal Hilbert 16th Problem

The number of limit cycles that can be born from periodic solutions of a polynomial Hamiltonian planar system $\frac{dx}{dt}=\frac{\partial H}{\partial y}(x,y),~~\frac{dy}{dt}=-\frac{\partial H}{\partial x}(x,y)$ by a small polynomial perturbation

$\frac{dx}{dt}=\frac{\partial H}{\partial y}(x,y)+\varepsilon P(x,y),~~~~~~~~\frac{dy}{dt}=-\frac{\partial H}{\partial x}(x,y)-\varepsilon Q(x,y)$

not increasing the degree $n=\text{deg}H$, is explicitly bounded by a double exponent $2^{2^{\text{Poly}(n)}}$, where $\text{Poly}(n)$ is an explicit polynomial in $n$ of degree not exceeding 60 (fine tuning of the proof gives a better value around 5 or so, which hypothetically could be reduced to just 2). For hyperelliptic Hamiltonians of the form $H(x,y)=y^2+x^{n+1}+a_1 x^{n-1}+\cdots+a_{n-1}x+a_n$ the bound can be improved to $2^{2^{O(n)}}$ with an explicit constant in the term $O(n)$. This assertion constitutes an explicit constructive solution of the so called “Infinitesimal” Hilbert 16th Problem which first implicitly appeared in the works of Petrovskii and Landis in the 1950-s. Since mid-1960-s the problem was repeatedly formulated in many sources (starting with Arnold’s problems and as recently as in Ilyashenko’s 2008 list) as the natural step towards a still evasive solution of the complete Hilbert 16th Problem.

J’ai Nous (i.e., Gal Binyamini, Dmitry Novikov et moi-même) avons trouvé une merveilleuse démonstration de cette proposition, mais je ne peux l’écrire dans cette marge car elle est trop longue.”

La démonstration is indeed a bit too long to be reproduced here: the complete exposition is available on arXiv (50+ pages) and strongly uses another paper of 30+ pages which establishes non-uniform explicit double exponential upper bound on the number of isolated complex zeros of functions satisfying linear systems of Fuchsian differential equations, provided that all residue matrices have only real eigenvalues.
Our proof is based solely on the fact that Abelian integrals of polynomial 1-forms along cycles on complexified level curves of the Hamiltonian, satisfy an integrable system of regular Pfaffian differential equations defined over $\mathbb Q$ with quasiunipotent monodromy along all small loops.

Bookmark this page, as it will display the most up-to-date version of the text of both papers. Any comments, suggestions and spotted typos will be accepted with warmest gratitude.

# Logarithmic singularities

1. De Rham division lemma (and its generalization)
2. Definition of a logarithmic pole: (scalar case). Residues.
3. Logarithmic complex: principal lemma on Λ-closedness.
4. Principal example: logarithmic complex for the normal crossings. Saito theorem.
5. Closed logarithmic 1-forms: complete description. Darbouxian foliations.
6. Matrix casse. Conjugacy of the residues along the polar locus. Residues on the normal crossings.
7. Schlesinger system: flat connexions with logarithmic poles along the diagonal.
8. Flat connexions with first order poles are almost always logarithmic, yet resonances may spoil the pattern.

Recommended reading: the same notes, sect. 3-4.

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