Sergei Yakovenko's blog: on Math and Teaching

Sunday, November 30, 2014

Lecture 9 (Mon., Dec. 1, 2014)

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.

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