Rigid local systems.

*(English)*Zbl 0864.14013
Annals of Mathematics Studies, 139. Princeton, NJ: Princeton Univ. Press. vii, 223 p. (1996).

A rank \(n\) local system \({\mathcal F}\) on \(\mathbb{P}^1_{\mathbb{C}} \backslash \{m\) points} is a sheaf of complex vector spaces which is locally isomorphic to the constant sheaf \(\mathbb{C}^n\). There is a 1-1 correspondence between (equivalence classes of) rank \(n\) local systems and (equivalence classes of) \(n\times n\) first order linear differential equations on \(\mathbb{P}^1_\mathbb{C}\) having regular singularities at the \(m\) points. An irreducible local system is called rigid if the sheaf \(\text{End} ({\mathcal F})\) of the endomorphism of \({\mathcal F}\), extended to a sheaf on \(\mathbb{P}^1_\mathbb{C}\), has a trivial first cohomology group. Under the assumption that \({\mathcal F}\) is irreducible one can easily calculate the dimension of this first cohomology group from the local data, i.e., the conjugacy classes of the local monodromies around the \(m\) missing points. The dimension turns out to be \(2\rho\).

In the general situation one can associate to \({\mathcal F}\) also an \(n\)-th order differential equation which has \(\rho\) as the number of “accessory parameters”. Thus, roughly speaking, \({\mathcal F}\) is rigid if and only if the associated \(n\)-th order equation has no accessory parameters. Known examples of rigid differential equations are: ordinary hypergeometric equations, generalized hypergeometric equations, Pochhammer equations (i.e., the irreducible ones). This clearly shows that the theme of this book is a very classical one. The basic problems are:

Irreducible recognition problem: Given the local data, determine whether they are attached to an irreducible local system.

Irreducible construction problem: Suppose that the local data of an irreducible local system are given. Construct one or all irreducible local systems with these data.

In the book these problems are solved under the condition that all data and local systems are rigid. The methods are hardly classical. Two operations on (rigid) local systems are introduced: middle convolution and middle tensor product. One makes use of: étale cohomology, perverse sheaves, Laumon’s work on \(l\)-adic Fourier transforms, Deligne’s proof of the Weil conjectures (and maybe more) to build the theory. The solution to the two problems turns out to be an algorithm, which starts with rank 1 local systems and uses the two operations above to obtain a rigid local system with the required data. If this algorithm does not produce a suitable rigid local system in a finite number of steps, then there is none.

The last chapter of the book proves that a rigid local system is, in a certain sense, related to the Gauss-Manin differential equation on the De Rham cohomology of a family of algebraic varieties. The author’s earlier proof of Grothendieck’s conjecture for Gauss-Manin differential equations is extended to rigid local systems. Maybe accidently, the only rigid local systems, given explicitly in the book, are the three families mentioned above. For these families other proofs of Grothendieck’s conjecture were already known.

It is clear that this book presents highly important new views and results on the classical theory of complex linear differential equations. Unfortunately, it is equally clear that the researchers in the field of complex linear differential equations will recognize nothing in the book (except for the hypergeometric equation on page 7).

In the general situation one can associate to \({\mathcal F}\) also an \(n\)-th order differential equation which has \(\rho\) as the number of “accessory parameters”. Thus, roughly speaking, \({\mathcal F}\) is rigid if and only if the associated \(n\)-th order equation has no accessory parameters. Known examples of rigid differential equations are: ordinary hypergeometric equations, generalized hypergeometric equations, Pochhammer equations (i.e., the irreducible ones). This clearly shows that the theme of this book is a very classical one. The basic problems are:

Irreducible recognition problem: Given the local data, determine whether they are attached to an irreducible local system.

Irreducible construction problem: Suppose that the local data of an irreducible local system are given. Construct one or all irreducible local systems with these data.

In the book these problems are solved under the condition that all data and local systems are rigid. The methods are hardly classical. Two operations on (rigid) local systems are introduced: middle convolution and middle tensor product. One makes use of: étale cohomology, perverse sheaves, Laumon’s work on \(l\)-adic Fourier transforms, Deligne’s proof of the Weil conjectures (and maybe more) to build the theory. The solution to the two problems turns out to be an algorithm, which starts with rank 1 local systems and uses the two operations above to obtain a rigid local system with the required data. If this algorithm does not produce a suitable rigid local system in a finite number of steps, then there is none.

The last chapter of the book proves that a rigid local system is, in a certain sense, related to the Gauss-Manin differential equation on the De Rham cohomology of a family of algebraic varieties. The author’s earlier proof of Grothendieck’s conjecture for Gauss-Manin differential equations is extended to rigid local systems. Maybe accidently, the only rigid local systems, given explicitly in the book, are the three families mentioned above. For these families other proofs of Grothendieck’s conjecture were already known.

It is clear that this book presents highly important new views and results on the classical theory of complex linear differential equations. Unfortunately, it is equally clear that the researchers in the field of complex linear differential equations will recognize nothing in the book (except for the hypergeometric equation on page 7).

Reviewer: M.van der Put (Groningen)

##### MSC:

14G20 | Local ground fields in algebraic geometry |

12H25 | \(p\)-adic differential equations |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

34G20 | Nonlinear differential equations in abstract spaces |