The motivic fundamental group of P1∖{0,1,∞} and the theorem of Siegel

September 2005

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110 Reads

In this paper, we establish a link between the structure theory of the
pro-unipotent motivic fundamental group of the projective line minus three
points and Diophantine geometry. In particular, we give a p-adic proof of
Siegel's theorem.

Exact Lagrangian submanifolds in simply-connected cotangent bundles,” math.SG/0701783

January 2007

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114 Reads

We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler, using a different approach. Comment: 28 pages, 3 figures. Version 2 -- derivation and discussion of the spectral sequence considerably expanded. Other minor changes

Another way to enumerate rational curves with torus actions, Invent math 142

May 1999

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27 Reads

A new proof of the mirror conjecture for Fano and Calabi-Yau complete intersections in P^n is given, using only the circle action on the graph space. The proof applies to projective bundles as well, with applications to "linear" relative Calabi-Yau's and to Schubert calculus. Comment: 30 pages, LaTeX. Section 4 has been eliminated and the proofs of Lemmas 4.4 and 5.1 have been improved. The introduction has also been rewritten to better indicate the new ideas in this paper and to emphasize that it contains a proof of the mirror conjecture which is simpler than previous proofs and completely independent of them

The Existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles

November 2000

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89 Reads

Hilbert-Arnold (HA) problem, motivated by Hilbert 16-th problem, is to prove that for a generic k-parameter family of smooth vector fields {\dot x=v(x,\eps)}_{\eps\in B^k} on the 2-dimensional sphere S^2 has uniformly bounded number of limit cycles (isolated periodic solutions), denoted by LC(\eps), over the parameter \eps, i.e. max_{\eps \in B^k} LC(\eps) <= K < \infty for some K. The HA problem can be reduced to so-called Local Hilbert-Arnold (LHA) problem. Suppose that a generic k-parameter family {\dot x=v(x,\eps)}_{\eps \in B^k}, x\in S^2 for some parameter \eps^*\in B^k has a polycycle (separatrix polygon) gamma consisting of equilibrium points as vertices and connecting separatrices as sides. LHA problem is to estimate B(k)--- the maximal number of limit cycles that can be born in a neighbourhood of gamma for a field \dot x=v(x,\eps), where \eps is close to \eps^*.

Moeckel, R.: Finiteness of relative equilibria of the four-body problem. Invent. Math. 163, 289-312

February 2006

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96 Reads

We show that the number of relative equilibria of the Newtonian four-body problem is finite, up to symmetry. In fact, we show
that this number is always between 32 and 8472. The proof is based on symbolic and exact integer computations which are carried
out by computer.

The infinitesimal 16th Hilbert problem in the quadratic case

March 2001

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38 Reads

Let H(x,y) be a real cubic polynomial with four distinct critical values (in a complex domain) and let X
H
=H
y
\(\)-H
x
\(\) be the corresponding Hamiltonian vector field. We show that there is a neighborhood ? of X
H
in the space of all quadratic plane vector fields, such that any X∈? has at most two limit cycles.

A counterexample to a 1961 “theorem” in homological algebra

May 2002

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188 Reads

In 1961, Jan-Erik Roos published a “theorem”, which says that in an [AB4*] abelian category, lim1 vanishes on Mittag–Leffler sequences. See Propositions 1 and 5 in [4]. This is a “theorem” that many people since have known
and used. In this article, we outline a counterexample. We construct some strange abelian categories, which are perhaps of
some independent interest.¶These abelian categories come up naturally in the study of triangulated categories. A much fuller
discussion may be found in [3]. Here we provide a brief, self contained, non–technical account. The idea is to make the counterexample
easy to read for all the people who have used the result in their work.¶In the appendix, Deligne gives another way to look
at the counterexample.

Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle

January 2008

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92 Reads

We prove that stable ergodicity is C
r
open and dense among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, for all r∈[2,∞].
The proof follows the Pugh–Shub program [29]: among conservative partially hyperbolic diffeomorphisms with one-dimensional
center bundle, accessibility is C
r
open and dense, and essential accessibility implies ergodicity.

Universality in the 2D Ising model and conformal invariance of fermionic
observables

October 2009

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88 Reads

It is widely believed that the celebrated 2D Ising model at criticality has a
universal and conformally invariant scaling limit, which is used in deriving
many of its properties. However, no mathematical proof of universality and
conformal invariance has ever been given, and even physics arguments support (a
priori weaker) M\"obius invariance. We introduce discrete holomorphic fermions
for the 2D Ising model at criticality on a large family of planar graphs. We
show that on bounded domains with appropriate boundary conditions, those have
universal and conformally invariant scaling limits, thus proving the
universality and conformal invariance conjectures.

Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

April 2006

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131 Reads

We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity. Comment: 7 pages

The structure of 2D semi-simple field theories

December 2007

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69 Reads

I classify all cohomological 2D field theories based on a semi-simple complex
Frobenius algebra A. They are controlled by a linear combination of
kappa-classes and by an extension datum to the Deligne-Mumford boundary. Their
effect on the Gromov-Witten potential is described by Givental's Fock space
formulae. This leads to the reconstruction of Gromov-Witten invariants from the
quantum cup-product at a single semi-simple point and from the first Chern
class, confirming Givental's higher-genus reconstruction conjecture. The proof
uses the Mumford conjecture proved by Madsen and Weiss.

Strong rigidity of II$_1$ factors arising from malleable actions of w-rigid groups, II

August 2004

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23 Reads

We prove that any isomorphism $\theta:M_0\simeq M$ of group measure space II$_1$ factors, $M_0=L^\infty(X_0, \mu_0) \rtimes_{\sigma_0} G_0$, $M=L^\infty(X, \mu) \rtimes_{\sigma} G$, with $G_0$ containing infinite normal subgroups with the relative property (T) of Kazhdan-Margulis (i.e. $G_0$ {\it w-rigid}) and $G$ an ICC group acting by Bernoulli shifts $\sigma$, essentially comes from an isomorphism of probability spaces which conjugates the actions. Moreover, any isomorphism $\theta$ of $M_0$ onto a ``corner'' $pMp$ of $M$, for $p\in M$ an idempotent, forces $p=1$. In particular, all group measure space factors associated with Bernoulli shift actions of w-rigid ICC groups have trivial fundamental group and all isomorphisms between such factors come from isomorphisms of the corresponding groups. This settles a ``group measure space version'' of Connes rigidity conjecture, shown in fact to hold true in a greater generality than just for ICC property (T) groups. We apply these results to ergodic theory, establishing new strong rigidity and superrigidity results for orbit equivalence relations.

On the tautological ring of $\M _g$

January 1995

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61 Reads

We prove among other things that any product of tautological classes of
$\M_g$ of degree $d$ (in the Chow ring of $\M _g$ with rational coefficients)
vanishes for $d>g-2$ and is proportional to the class of the hyperelliptic
locus in degree $g-2$.

A complete characterization of coefficients of a.e. convergent orthogonal series and majorizing measures

February 2010

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39 Reads

We characterize sequences of numbers (a
n
) such that ∑n≥1a
n
Φn
converges a.e. for any orthonormal system (Φn
) in any L
2-space. In our criteria we use the set A={∑m≥n||a
m
|2;n≥1}, majorizing measures on A, and new descriptions of the complexity of A.

The Abel–Jacobi map for higher Chow groups, II

November 2007

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35 Reads

We explicitly describe cycle-class maps c_H from motivic cohomology to absolute Hodge cohomology, for smooth quasi-projective and (some) proper singular varieties, and compute special cases of the latter. For smooth projective varieties, we also study Hodge-theoretically defined "higher Abel-Jacobi maps" on the kernel of c_H; this leads to new results on nontrivial indecomposable higher Chow cycles in the regulator kernel. Comment: 68 pages, 3 figures

Geometry of KDV. IV: Abel sums, Jacobi variety, and theta function in the scattering case

December 1990

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14 Reads

The study of the general projective curve X, defined by the vanishing of an irreducible polynomial p(x, y), originated in the discovery of addition formulas for integrals ∫ ∞ v r(x, y) dx of rational functions on such a curve

Connected components of the moduli space of Abelian differentials with prescribed singularities

January 2002

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88 Reads

Consider the moduli space of pairs (C,w) where C is a smooth compact complex curve of a given genus and w is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces. Comment: 42 pages, 12 figures, LaTeX

Stability of degenerate abelian varieties

May 1999

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13 Reads

We compactify canonically the moduli scheme of abelian schemes over Z[+N,1/N] by introducing the noncommutative level structures. Any degenerate abelian scheme on the boundary of the compactification is a singular one among our models - projectively stable quasi-abelian schemes. We prove that a degenerate abelian scheme has Kempf-stable Hilbert points if and only if it is a projectively stable quasi-abelian scheme.

Vector Bundles on Abelian Surfaces

September 1971

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55 Reads

Very much about line bundles on abelian varieties and vector bundles on elliptic curves are known by now. Various complications arise, however, for vector bundles on abelian varieties, already when the dimension and the rank are both two. We indicate some of them here. More precisely, we show that there are simple vector bundles other than those obtained as the direct image of a line bundle under an isogeny. We also show that the triviality of the Chern classes of an indecomposable vector bundle on an abelian variety does not necessarily imply the existence of an algebraic (or holomorphic) connection. For simplicity we restrict our attention to abelian varieties over the field C of complex numbers, or more generally, complex tori. In Section 1 we review known results for convenience. We first state the method of constructing simple vector bundles by taking the direct image of a line bundle under an isogeny, discussed in Oda [13]. We next give a characterization of them, the main part of which is due to Morikawa [9], and a characterization in terms of their Chern classes when the dimension is two. Then we recall the notion of algebraic or holomorphic connection and state a theorem about it by Matsushima [8] and Morimoto [10]. In Section 2 we review and slightly improve a theorem of Schwarzenberger [14] about non-simple indecomposable vector bundles of rank two on a complete non-singular surface. We then state a construction of various vector bundles of rank two on a complete non-singular surface which is essentially due to Schwarzenberger (loc. cir.). In Section 3 we first review the cohomology of line bundles on abelian varieties recently obtained by Mumford [11] and MumfordKempf [12]. We then apply it to the construction method of Section 2 and construct various simple or non-simple but inde~zomposable vector bundles of rank two on abelian surfaces. They give examples mentioned at the beginning. F. Takemoto generalized the notion of stability of vector bundles on a curve to that of H-stability (H being an ample line bundle) on higher dimensional varieties. For the relation of our results here to this notion, we refer the reader to his forthcoming paper [15].

Formal subgroups of abelian varieties

January 2001

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24 Reads

In this paper, we generalize the result of [12] in the following sense. Let A be an abelian variety over a number field k, let ? be the Néron model of A over the ring of integers O
k
of k. Completing ? along its zero section defines a formal group \(\widehat{\mathcal{A}}\) over O
k
. We prove that any formal subgroup of the generic fiber of \(\widehat{\mathcal{A}}\) whose closure in \(\widehat{\mathcal{A}}\) is smooth over an open subset of Spec O
k
arises in fact from an abelian subvariety of A. The proof is of a transcendental nature and uses the Arakelovian formalism introduced by Bost [3].

The Toda lattice, Dynkin diagrams, singularities and Abelian varieties

January 1991

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9 Reads

The periodic l+1 particle Toda lattices coresponding to extended Dynkin diagrams have l+1 polynomial constants of the motion, namely as many as there are dots in the Dynkin diagram. For most values of the constants, their intersection defines as l-dimensional affine invariant manifold which completes into a complex algebraic torus (Abelian variety) by glueing on a divisor entirely specified by the extended Dynkin diagram: Therefore the global geometry of the complex invariant tori, such as polarization, dimension of certain linear systems, divisor equivalences, etc...., is entirely given by the extended Dynkin diagram. In particular, an explicit geometric description for the case of three-particle Toda lattices is given which describes the divisors, their singularities and how the various divisors intersect

Local André-Oort conjecture for the universal abelian variety

October 2006

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25 Reads

We prove a p-adic version of the André-Oort conjecture for subvarieties of the universal abelian varieties. Let g and n be integers with n≥3 and p a prime number not dividing n. Let R be a finite extension of \(W[\mathbb{F}_{p}^{alg}]\), the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli space \(\mathcal{A}=\mathcal{A}_{g,1,n}\) of g-dimensional principally polarized abelian varieties with full level n-structure as well as the universal abelian variety \(\pi:\mathcal{X}\to\mathcal{A}\) over \(\mathcal{A}\) may be defined over R. We call a point \(\xi\in\mathcal{X}(R)\)
R-special if \(\mathcal{X}_{\pi(\xi)}\) is a canonical lift and ξ is a torsion point of its fibre. Employing the model theory of difference fields and work of Moonen on special subvarieties of \(\mathcal{A}\), we show that an irreducible subvariety of \(\mathcal{X}_{R}\) containing a dense set of R-special points must be a special subvariety in the sense of mixed Shimura varieties.