Algebraic and Geometric Isomonodromic Deformations

@article{Doran2001AlgebraicAG,
  title={Algebraic and Geometric Isomonodromic Deformations},
  author={Charles F. Doran},
  journal={Journal of Differential Geometry},
  year={2001},
  volume={59},
  pages={33-85}
}
  • C. Doran
  • Published 1 September 2001
  • Mathematics
  • Journal of Differential Geometry
Using the Gauss-Manin connection (Picard-Fuchs differential equation) and a result of Malgrange, a special class of algebraic solutions to isomonodromic deformation equations, the geometric isomonodromic deformations ,i s defined from “families of families” of algebraic varieties. Geometric isomonodromic deformations arise naturally from combinatorial strata in the moduli spaces of elliptic surfaces over P1. The complete list of geometric � 

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References

SHOWING 1-10 OF 78 REFERENCES
Introduction to the Theory of Isomonodromic Deformations of Linear Ordinary Differential Equations with Rational Coefficients
The general solution of a linear ordinary differential equation (ODE) with rational coefficients is generically multivalued. This property is described by a representation of the fundamental homotopy
Picard-Fuchs Uniformization: Modularity of the Mirror Map and Mirror-Moonshine
Motivated by a conjecture of Lian and Yau concerning the mirror map in string theory, we determine when the mirror map q-series of certain elliptic curve and K3 surface families are Hauptmoduln
Mixed Hodge Structures and Singularities
1. Gauss-Manin connection 2. Limit mixed Hodge structure on the vanishing cohomology of an isolated hypersurface singularity 3. The period map of a m-constant deformation of an isolated hypersurface
Groups as Galois groups : an introduction
Part 1. The Basic Rigidity Criteria: 1. Hilbert's irreducibility theorem 2. Finite Galois extensions of C (x) 3. Descent of base field and the rigidity criterion 4. Covering spaces and the
Self-dual Bianchi metrics and the Painlevé transcendents
We give two constructions that relate the self-duality condition on the Weyl curvature of a spacetime with Bianchi IX symmetry to the Painleve equations, the first by way of the self-dual Yang--Mills
Moduli spaces for covers of the Riemann sphere
Moduli spaces for covers of the Riemann sphere have been constructed in a joint work with M. Fried [FV1]. They were used to realize groups as Galois groups [FV1], [Vö1], and to determine the absolute
The Moduli of Weierstrass Fibrations Over IP 1
Let k be an algebraically closed field of characteristic 4 = 2, 3. Let X p , Y be a flat proper map of reduced irreducible k-schemes such that every geometric fibre is either (a) an elliptic curve,
Lectures On The Mordell-Weil Theorem
Contents: Heights - Nomalized heights - The Mordell-Weil theorem - Mordell's conjecture - Local calculation of normalized heights - Siegel's method - Baker's method - Hilbert's irreducibility theorem
Monodromy of certain Painlevé–VI transcendents and reflection groups
Abstract.We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β=γ=0, δ=1/2 and 2α=(2μ-1)2 with arbitrary μ, 2μ≠∈ℤ. We introduce
On Isomonodromic Deformations of Fuchsian Systems
Isomonodromic deformations of Fuchsian systems are considered. A description of all possible forms of such isomonodromic deformations is presented.
...
1
2
3
4
5
...