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38
Stability conditions on a noncompact CalabiYau threefold
"... Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the ..."
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Cited by 27 (1 self)
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Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the quantum cohomology of P 2. We give some evidence from mirror symmetry for this conjecture. 1.
The fiftytwo icosahedral solutions to Painlevé VI
 J. Reine Angew. Math
"... Abstract. The solutions of the (nonlinear) Painlevé VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto’s affine F4 Weyl group action and many properties of the solutions will be given. There are 52 classes, the first ten of which cor ..."
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Cited by 24 (5 self)
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Abstract. The solutions of the (nonlinear) Painlevé VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto’s affine F4 Weyl group action and many properties of the solutions will be given. There are 52 classes, the first ten of which correspond directly to the ten icosahedral entries on Schwarz’s list of algebraic solutions of the hypergeometric equation. The next nine solutions are simple deformations of known PVI solutions (and have less than five branches) and five of the larger solutions are already known, due to work of Dubrovin, Mazzocco and Kitaev. Of the remaining 28 solutions we will find 12 explicitly (including all those of genus zero) and will show that the others may be found, in principle, by the method of [3], using Jimbo’s formula (the largest solution would be a degree 72 polynomial representing a certain genus seven plane curve). One example that will be constructed is an explicit algebraic solution with 12 branches that is “generic ” in the sense that its parameters lie on none of the affine F4 hyperplanes. This is the only known explicit generic solution to Painlevé VI.
Dessins d’enfants, their deformations and algebraic the sixth Painlevé and Gauss hypergeometric functions
"... We consider an application of Grothendieck’s dessins d’enfants to the theory of the sixth Painlevé and Gauss hypergeometric functions: two classical special functions of the isomonodromy type. It is shown that, higher order transformations and the Schwarz table for the Gauss hypergeometric function ..."
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Cited by 23 (1 self)
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We consider an application of Grothendieck’s dessins d’enfants to the theory of the sixth Painlevé and Gauss hypergeometric functions: two classical special functions of the isomonodromy type. It is shown that, higher order transformations and the Schwarz table for the Gauss hypergeometric function are closely related with some particular Belyi functions. Moreover, we introduce a notion of deformation of the dessins d’enfants and show that one dimensional deformations are a useful tool for construction of algebraic the sixth Painlevé functions.
MODULI OF STABLE PARABOLIC CONNECTIONS, RIEMANNHILBERT CORRESPONDENCE AND GEOMETRY OF PAINLEVÉ EQUATION OF TYPE VI, Part I
, 2003
"... In this paper, we will give a complete geometric background for the geometry of Painlevé V I and Garnier equations. By geometric invariant theory, we will construct a smooth coarse moduli space M α n (t, λ, L) of stable parabolic connection on P1 with logarithmic poles at D(t) = t1 + · · · + tn a ..."
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Cited by 21 (8 self)
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In this paper, we will give a complete geometric background for the geometry of Painlevé V I and Garnier equations. By geometric invariant theory, we will construct a smooth coarse moduli space M α n (t, λ, L) of stable parabolic connection on P1 with logarithmic poles at D(t) = t1 + · · · + tn as well as its natural compactification. Moreover the moduli space R(Pn,t)a of Jordan equivalence classes of SL2(C)representations of the fundamental group π1(P 1 \D(t), ∗) are defined as the categorical quotient. We define the RiemannHilbert correspondence RH: M α n (t, λ, L) − → R(Pn,t)a and prove that RH is a bimeromorphic proper surjective analytic map. Painlevé and Garnier equations can be derived from the isomonodromic flows and Painlevé property of these equations are easily derived from the properties of RH. We also prove that the smooth parts of both moduli spaces have natural symplectic structures and RH is a symplectic resolution of singularities of R(Pn,t)a, from which one can give geometric backgrounds for other interesting phenomena, like Hamiltonian structures, Bäcklund transformations, special solutions of these equations.
The discrete and continuous Painlevé VI hierarchy and the Garnier systems
 Glasgow Math. J. 43A
"... We present a general scheme to derive higherorder members of the Painlevé VI (PVI) hierarchy of ODE’s as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations ..."
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Cited by 21 (6 self)
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We present a general scheme to derive higherorder members of the Painlevé VI (PVI) hierarchy of ODE’s as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations on a multidimensional lattice. The connection with the isomonodromic Garnier systems is discussed.
Some explicit solutions to the Riemann–Hilbert problem
"... Abstract. Explicit solutions to the Riemann–Hilbert problem will be found realising some irreducible nonrigid local systems. The relation to isomonodromy and the sixth Painlevé equation will be described. 1. ..."
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Cited by 12 (2 self)
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Abstract. Explicit solutions to the Riemann–Hilbert problem will be found realising some irreducible nonrigid local systems. The relation to isomonodromy and the sixth Painlevé equation will be described. 1.
Higher genus icosahedral Painlevé curves, epreprint http://xyz.lanl.gov, math.DG/0506407
 Sydney University
, 1996
"... Abstract. We will write down the higher genus algebraic curves supporting icosahedral solutions of the sixth Painlevé equation, including the largest (genus seven) curve. 1. ..."
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Cited by 7 (1 self)
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Abstract. We will write down the higher genus algebraic curves supporting icosahedral solutions of the sixth Painlevé equation, including the largest (genus seven) curve. 1.
Holomorphic dynamics, Painlevé VI equation and character varieties, Preprint arXiv: 0711.1579
, 2007
"... 1.1. Character variety 3 1.2. Automorphisms and modular groups 4 1.3. Projective structures 6 ..."
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Cited by 7 (0 self)
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1.1. Character variety 3 1.2. Automorphisms and modular groups 4 1.3. Projective structures 6