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Infinite Grassmannians and moduli spaces of Gbundles
 Math. Annalen
, 1994
"... These are notes for my eight lectures given at the C.I.M.E. session on “Vector bundles on curves. New directions ” held at Cetraro (Italy) in June 1995. The work presented here was done in collaboration with M.S. Narasimhan and A. Ramanathan and appeared in [KNR]. These notes differ from [KNR] in th ..."
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Cited by 70 (4 self)
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These are notes for my eight lectures given at the C.I.M.E. session on “Vector bundles on curves. New directions ” held at Cetraro (Italy) in June 1995. The work presented here was done in collaboration with M.S. Narasimhan and A. Ramanathan and appeared in [KNR]. These notes differ from [KNR] in that we have
Equidistribution of small points, rational dynamics, and potential theory
 Ann. Inst. Fourier (Grenoble
, 2006
"... Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points ..."
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Cited by 46 (7 self)
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Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points in P 1 (k) whose ϕcanonical heights tend to zero, then the zn’s and their Galois conjugates are equidistributed with respect to µϕ,v. In the archimedean case, µϕ,v coincides with the wellknown canonical measure associated to ϕ. This theorem generalizes a result of BakerHsia [BH] when ϕ(z) is a polynomial. The proof uses a polynomial lift F (x, y) = (F1(x, y), F2(x, y)) of ϕ to construct a twovariable ArakelovGreen’s function gϕ,v(x, y) for each v. The measure µϕ,v is obtained by taking the Berkovich space Laplacian of gϕ,v(x, y), using a theory developed in [RB]. The other ingredients in the proof are (i) a potentialtheoretic energy minimization principle which says that � � gϕ,v(x, y) dν(x)dν(y) is uniquely minimized over all probability measures ν on P 1 Berk,v when ν = µϕ,v, and (ii) a formula for homogeneous transfinite diameter of the vadic filled Julia set KF,v ⊂ C 2 v in terms of the resultant Res(F) of F1 and F2. The resultant formula, which generalizes a formula of DeMarco [DeM], is proved using results
From the GinzburgLandau Model to Vortex Lattice Problems
, 2012
"... We introduce a “Coulombian renormalized energy” W which is a logarithmic type of interaction between points in the plane, computed by a “renormalization.” We prove various of its properties, such as the existence of minimizers, and show in particular, using results from number theory, that among lat ..."
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Cited by 26 (13 self)
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We introduce a “Coulombian renormalized energy” W which is a logarithmic type of interaction between points in the plane, computed by a “renormalization.” We prove various of its properties, such as the existence of minimizers, and show in particular, using results from number theory, that among lattice configurations the triangular lattice is the unique minimizer. Its minimization in general remains open. Our motivation is the study of minimizers of the twodimensional GinzburgLandau energy with applied magnetic field, between the first and second critical fields Hc1 and Hc2. In that regime, minimizing configurations exhibit densely packed triangular vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, W as a Γlimit of the GinzburgLandau energy. More precisely we show that the vortices of minimizers of GinzburgLandau, blownup at a suitable scale, converge to minimizers of W, thus providing a first rigorous hint at the Abrikosov lattice. This is a next order effect compared to the meanfield type results we previously established. The derivation of W uses energy methods: the framework of Γconvergence, and an abstract scheme for obtaining lower bounds for “2scale energies ” via the ergodic theorem that we introduce.
Free Bosons and TauFunctions for Compact Riemann Surfaces and Closed Smooth Jordan Curves I. Current Correlation Functions
, 2001
"... We study families of quantum eld theories of free bosons on a compact Riemann surface of genus g. For the case g > 0 these theories are parameterized by holomorphic line bundles of degree g 1, and for the case g = 0  by smooth closed Jordan curves on the complex plane. In both cases we dene a n ..."
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Cited by 17 (0 self)
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We study families of quantum eld theories of free bosons on a compact Riemann surface of genus g. For the case g > 0 these theories are parameterized by holomorphic line bundles of degree g 1, and for the case g = 0  by smooth closed Jordan curves on the complex plane. In both cases we dene a notion of function as a partition function of the theory and evaluate it explicitly. For the case g > 0 the function is an analytic torsion [3], and for the case g = 0  the regularized energy of a certain natural pseudomeasure on the interior domain of a closed curve. For these cases we rigorously prove the Ward identities for the current correlation functions and determine them explicitly. For the case g > 0 these functions coincide with those obtained in [21, 36] using bosonization. For the case g = 0 the function we have dened coincides with the function introduced in [29, 44, 24] as a dispersionless limit of the Sato's function for the twodimensional Toda hierarchy. As a corollary of the Ward identities, we obtain recent results [44, 24] on relations between conformal maps of exterior domains and functions. For this case we also dene a Hermitian metric on the space of all contours of given area. As another corollary of the Ward identities we prove that the introduced metric is Kahler and the logarithm of the function is its Kahler potential. Contents
Hermitian vector bundles and extension groups on arithmetic schemes II. THE ARITHMETIC ATIYAH EXTENSION
, 2008
"... In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element — the arithmetic Atiyah class — in a suitable arithmetic extension group. Namely, if E is a ..."
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Cited by 16 (2 self)
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In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element — the arithmetic Atiyah class — in a suitable arithmetic extension group. Namely, if E is a hermitian vector bundle on an arithmetic scheme X, its arithmetic Atiyah class bat X/Z(E) lies in the group d Ext 1 X(E, E ⊗ Ω1), and is an obstruction to the algebraicity over X of the X/Z unitary connection on the vector bundle EC over the complex manifold X(C) that is compatible with its holomorphic structure. In the first sections of this article, we develop the basic properties of the arithmetic Atiyah class which can be used to define characteristic classes in arithmetic Hodge cohomology. Then we study the vanishing of the first Chern class ĉH 1 (L) of a hermitian line bundle L in the arithmetic Hodge cohomology group d Ext 1
Models of Curves and Finite Covers
 COMPOSITIO MATHEMATICA
, 1999
"... Let K be a discrete valuation field with ring of integers OK.Letf: X → Y beafinite morphism of curves over K. In this article, we study some possible relationships between the models over OK of X and of Y. Three such relationships are listed below. Consider a Galois cover f: X → Y of degree prime t ..."
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Cited by 16 (2 self)
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Let K be a discrete valuation field with ring of integers OK.Letf: X → Y beafinite morphism of curves over K. In this article, we study some possible relationships between the models over OK of X and of Y. Three such relationships are listed below. Consider a Galois cover f: X → Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semistable reduction over K, thenX achieves semistable reduction over some explicit tame extension of K(B). When K is strictly henselian, we determine the minimal extension L/K with the property that XL has semistable reduction. Let f: X → Y be a finite morphism, with g(Y) � 2. We show that if X has a stable model X over OK, then Y has a stable model Y over OK, and the morphism f extends to a morphism X → Y. Finally, given any finite morphism f: X → Y, is it possible to choose suitable regular models X and Y of X and Y over OK such that f extends to a finite morphism X → Y? As was shown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situations, with f a cyclic cover of any order � 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3.
A lower bound for average values of dynamical Green’s functions
, 2006
"... Abstract. We provide a Mahler/Elkiesstyle lower bound for the average values of dynamical Green’s functions on P 1 over an arbitrary valued field, and give some dynamical and arithmetic applications. 1. ..."
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Cited by 13 (1 self)
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Abstract. We provide a Mahler/Elkiesstyle lower bound for the average values of dynamical Green’s functions on P 1 over an arbitrary valued field, and give some dynamical and arithmetic applications. 1.
Arakelov Invariants of Riemann Surfaces
 DOCUMENTA MATH.
, 2005
"... We derive closed formulas for the ArakelovGreen function and the Faltings deltainvariant of a compact Riemann surface. ..."
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Cited by 12 (7 self)
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We derive closed formulas for the ArakelovGreen function and the Faltings deltainvariant of a compact Riemann surface.
Bounds on Faltings’s Delta Function Through Covers
, 2006
"... Let X be a compact Riemann surface of genus gX ≥ 1. In [7], G. Faltings introduced a new invariant δFal(X) associated to X. In this paper we give explicit bounds for δFal(X) in terms of fundamental differential geometric invariants arising from X, when gX> 1. As an application, we are able to g ..."
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Cited by 11 (1 self)
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Let X be a compact Riemann surface of genus gX ≥ 1. In [7], G. Faltings introduced a new invariant δFal(X) associated to X. In this paper we give explicit bounds for δFal(X) in terms of fundamental differential geometric invariants arising from X, when gX> 1. As an application, we are able to give bounds for Faltings’s delta function for the family of modular curves X0(N) in terms of the genus only. In combination with work of A. Abbes, P. Michel and E. Ullmo this leads to an asymptotic formula for the Faltings height of the Jacobian J0(N) associated to X0(N).