Frobenius manifolds: Isomonodromic deformations and infinitesimal period mappings. (English) Zbl 0904.32009
A Frobenius structure on a complex analytic manifold \(M\) consists of the data of two objects on the tangent bundle \(TM\): a symmetric nondegenerate bilinear form which is flat, and a commutative associative product with unit.
The contents of these notes mainly consists in a “mise au point” and a reformulation of known results: to B. Malgrange after the work of M. Jimbo, T. Miwa and K. Ueno; to Dubrovin and the approach given by N. Hitchin; to M. Saito and to P. Deligne.
Contents. 1. Families of vector bundles on \({\mathbf P^1}\) with an integrable meromorphic connection. 2. The Riemann-Hilbert-Birkhoff problem. 3. Frobenius manifolds. 4. Saito structures and Frobenius manifolds. 5. Examples of Frobenius manifolds. Appendix A. The Riemann-Hilbert problem and its variants. Appendix B. Proofs of Theorem 3.1.3.
The contents of these notes mainly consists in a “mise au point” and a reformulation of known results: to B. Malgrange after the work of M. Jimbo, T. Miwa and K. Ueno; to Dubrovin and the approach given by N. Hitchin; to M. Saito and to P. Deligne.
Contents. 1. Families of vector bundles on \({\mathbf P^1}\) with an integrable meromorphic connection. 2. The Riemann-Hilbert-Birkhoff problem. 3. Frobenius manifolds. 4. Saito structures and Frobenius manifolds. 5. Examples of Frobenius manifolds. Appendix A. The Riemann-Hilbert problem and its variants. Appendix B. Proofs of Theorem 3.1.3.
Reviewer: V.V.Chueshev (Kemerovo)
MSC:
| 32C18 | Topology of analytic spaces |
| 55R10 | Fiber bundles in algebraic topology |
| 34A26 | Geometric methods in ordinary differential equations |
| 34M50 | Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 32L05 | Holomorphic bundles and generalizations |
