# Isomonodromic Deformations and Frobenius Manifolds: An by Claude Sabbah

By Claude Sabbah

In response to a chain of graduate lectures, this booklet presents an advent to algebraic geometric tools within the conception of complicated linear differential equations. ranging from simple notions in complicated algebraic geometry, it develops the various classical difficulties of linear differential equations. It ends with purposes to contemporary learn questions regarding reflect symmetry. the basic software used is that of a vector package deal with connection. The e-book comprises whole proofs, and functions to contemporary learn questions. geared toward graduate scholars and researchers, the publication assumes a few familiarity with simple advanced algebraic geometry.

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Example text

2) Prove that L(E, E )∇ can be identiﬁed with the subsheaf of the sheaf H omOM (E , E ) consisting of the homomorphisms which commute to the action of ∇ and ∇ (we also say that they are morphisms of bundles with connection), or also which send E ∇ in E ∇ . 12 Exercise. c, show that, if ∇ is integrable, then so is its pullback by f on the manifold M . 13 Corollary (Analytic extension). The homomorphisms of bundles with connection satisfy the principle of analytic extension: if V ⊂ U is the inclusion of connected open sets of M which induces an isomorphism between the fundamental groups of V and U , and if ϕ : E|V → E|V is a homomorphism of bundles with connection, then ϕ can be extended in a unique way as a homomorphism of bundles with connection E|U → E|U .

We will say that E is a logarithmic lattice of the meromorphic bundle with connection (M , ∇) if 1 log Z ⊗ E ∇(E ) ⊂ ΩM OM and more generally that it is a lattice of order r if 1 ∇(E ) ⊂ ΩM (r + 1) log Z ⊗ E OM and of order r if moreover it does not have order r − 1 (logarithmic = order 0). Therefore, E is a logarithmic lattice if, in any local basis of E , the connection matrix of ∇ has forms with logarithmic poles along Z as entries. 2 Exercise (Behaviour of the order by operations on lattices).

If E is a bundle, one sets E (kZ) = E ⊗OM OM (kZ) and one deﬁnes similarly E (k1 Z1 + · · · + kp Zp ). 3 Deﬁnition (Meromorphic bundles, lattices). A meromorphic bundle on M with poles along Z is a locally free sheaf of OM (∗Z)-modules of ﬁnite rank. A lattice of this meromorphic bundle is a locally free OM -submodule of this meromorphic bundle, which has the same rank. In particular, a lattice E of a meromorphic bundle M coincides with M when restricted to M Z. Moreover, we have M = OM (∗Z) ⊗ E .