Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence

Let G be a connected linear semisimple Lie group with Lie algebra , and let be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in and is the nilpotent -orbit in associated to Ω by the Kostant-Sekiguchi correspondence. We show that the corank of the Hamiltonian K-space Ω is twice the complexity of the variety .     

On the exact Hausdorff dimension of the set of Liouville numbers

Let denote the set of Liouville numbers. For a dimension function h, we write () for the h-dimensional Hausdorff measure of . In this paper we locate the exact cut-point at which the Hausdorff measure of drops from infinity to zero. Namely, if h is a dimension function that increases faster than any power function near 0, then ()=, and if h is a dimension function that increases slower than some power function near 0, then ()=0. This answers a question asked by R. D. Mauldin.Mathematics Subject Classification (2000): 28A80     

Tannaka duality on quotient stacks

Let = [X/G] be the quotient stack of a scheme X by an affine group scheme G over a field k. Assume that there is a line bundle on whose underlying line bundle on X is very ample. Let VB() be the category of vector bundles on .We show that is canonically isomorphic to the stack of fiber functors on VB(). This is an analogue of the Tannaka duality for affine groups. Partially supported by CNCSIS contract no. 33079/2004     

2-universal -lattices over real quadratic fields

Let be a real quadratic field with m a square-free positive rational integer, and be the ring of integers in F. An -lattice L on a totally positive definite quadratic space V over F is called r-universal if L represents all totally positive definite -lattices l with rank r over . We prove that there exists no 2-universal -lattice over F with rank less than 6, and there exists a 2-universal -lattice over F with rank 6 if and only if m=2, 5. Moreover there exists only one 2-universal -lattice with rank 6, up to isometry, over .     

Restriction of the cotangent bundle to elliptic curves and Hilbert-Kunz functions

We describe the possible restrictions of the cotangent bundle to an elliptic curve . We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R₊-primary ideal in the graded section ring .     

Conjugacy classes of affine automorphisms of and linear automorphisms of ℙ n in the Cremona groups

We describe the conjugacy classes of affine automorphisms in the group Aut(n,) (respectively Bir()) of automorphisms (respectively of birational maps) of . From this we deduce also the classification of conjugacy classes of automorphisms of ℙⁿ in the Cremona group Bir().     

The theory of energy for sub-Laplacians with an application to quasi-continuity

In this paper, we provide a suitable theory for the energy where μ is a Radon measure and Γ is the fundamental solution of a sub-Laplacian on a stratified group As a significant application, we prove the quasi-continuity of superharmonic functions related to . The proofs are elementary and mostly rely on the use of appropriate mean-value formulas and mean-integral operators relevant to the Potential Theory for .     

Invertible modules for commutative -algebras with residue fields

The aim of this note is to understand under which conditions invertible modules over a commutative -algebra in the sense of Elmendorf, Kriz, Mandell & May give rise to elements in the algebraic Picard group of invertible graded modules over the coefficient ring by taking homotopy groups. If a connective commutative -algebra R has coherent localizations for every maximal ideal , then for every invertible R-module U, U_*=π_*U is an invertible graded R_*-module. In some non-connective cases we can carry the result over under the additional assumption that the commutative -algebra has ‘residue fields’ for all maximal ideals if the global dimension of R_* is small or if R is 2-periodic with underlying Noetherian complete local regular ring R₀. We apply these results to finite abelian Galois extensions of Lubin-Tate spectra.     

On the rank of abelian varieties over function fields

Let be a smooth projective curve defined over a number field k, A/k() an abelian variety and (τ, B) the k()/k-trace of A. We estimate how the rank of A(k())/τB(k) varies when we take a finite geometrically abelian cover defined over k. This work was partially supported by CNPq research grant 304424/2003-0, Pronex 41.96.0830.00 and CNPq Edital Universal 470099/2003-8. I would like to thank Douglas Ulmer for comments on how to treat the case of arbitrary ramification, but the conductor prime to the ramification locus, in the case of elliptic fibrations. I would also like to thank Marc Hindry for comments on the inequality comparing the conductors of A and A'. Finally, I also thank the referee for his comments and criticisms.     

The Rees algebra of a positive normal affine semigroup ring

Let R be a positive normal affine semigroup ring of dimension d and let be the maximal homogeneous ideal of R. We show that the integral closure of is equal to for all n ∈ℕ with n ≥ d − 2. From this we derive that the Rees algebra R[t] is normal in case that d ≤ 3. If emb dim(R) = d + 1, we can give a necessary and sufficient condition for R[t] to be normal.     

Iwasawa invariants of galois deformations

Fix a residual ordinary representation :G_F→GL_n(k) of the absolute Galois group of a number field F. Generalizing work of Greenberg–Vatsal and Emerton–Pollack–Weston, we show that the Iwasawa invariants of Selmer groups of deformations of depends only on and the ramification of the deformation.     

Estimates for stable hypersurfaces of prescribed F-mean curvature

We consider immersed hypersurfaces :Mⁿ→ℝⁿ⁺¹ with prescribed anisotropic mean curvature . Such hypersurfaces can be characterized as critical points of parametric functionals of the type with an elliptic Lagrangian F depending on normal directions and a smooth vectorfield Q satisfying . We establish curvature estimates for stable hypersurfaces of dimension n≤5, provided F is C³-close to the area integrand.     

Norm Inequalities for Sums of Positive Operators. II

It is shown that if A and B are positive operators on a separable complex Hilbert space, then for every unitarily invariant norm. When specialized to the usual operator norm ||·|| and the Schatten p-norms ||·||_p, this inequality asserts that and These inequalities improve upon some earlier related inequalities. Other norm inequalities for sums of positive operators are also obtained.     

Courbes nodales et ramification sauvage virtuelle

Let S be the spectrum of a discrete valuation ring of residue characteristic p>0. Let Z be a scheme separated and of finite type over S, and denote by K_c(Z,) (l≠p) the Grothendieck group of -constructible sheaves on the etale site of Z. In this article, we introduce the subgroup of virtual sheaves with virtual wild ramification zero, and we prove that this subgroup is preserved by the formalism of six operations “à la Grothendieck”.     

Associated primes and cofiniteness of local cohomology modules

Let be an ideal of Noetherian ring R and let s be a non-negative integer. Let M be an R-module such that is finite R-module. If s is the first integer such that the local cohomology module is non -cofinite, then we show that is finite. In particular, the set of associated primes of is finite. Let be a local Noetherian ring and let M be a finite R-module. We study the last integer n such that the local cohomology module is not -cofinite and show that n just depends on the support of M.The research of the first author was supported in part by a grant from IPM (No. 83130114).The second author was supported by a grant from University of Tehran (No. 6103023/1/01).     

Fonctions plurisousharmoniques et courants positifs de type (1,1) sur une variété presque complexe

If (X, J) is an almost complex manifold, then a function u is said to be plurisubharmonic on X if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is conjectured that plurisubharmonicity is equivalent to the positivity of the (1,1)-current , (the (1,1)-current need not be closed here!). The conjecture is trivial if u is of class The result is elementary in the complex integrable case because the operator can be written as an operator with constant coefficients in complex coordinates. Hence the positivity of the current is preserved by regularising with usual convolution kernels. This is not possible in the almost complex non integrable case and the proof of the result requires a much more intrinsic study. In this chapter we prove the necessity of the positivity of the (1,1)-current . We prove also the sufficiency of the positivity in the particular case of an upper semi-continuous function f which is continuous in the complement of the singular locus f⁻¹(−∞).

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Résume Une fonction semi-continue supérieurement u sur une variété presque complexe (X, J) est dite plurisousharmonique si la restriction à toute courbe pseudo-holomorphe locale est sous-harmonique. Comme dans le cas analytique complexe, nous conjecturons que la notion de plurisousharmonicité pour une fonction u est équivalente à la positivité du (1,1)-courant , (lequel n'est pas forcément fermé dans le cas non intégrable). La conjecture est triviale dans le cas d'une fonction u de classe Le résultat en question est élémentaire dans le cas complexe intégrable car l'opérateur s'écrit comme un opérateur à coefficients constants dans des coordonnées complexes. On peut donc facilement conserver la positivité du courant en régularisant avec des noyaux usuels. Dans le cas presque complexe non intégrable ceci ce n'est pas possible et la preuve du résultat exige un étude beaucoup plus intrinsèque. Nous montrons la nécessité de la positivité du (1,1)-courant en utilisant la théorie locale des courbes J-holomorphes. Nous montrons aussi la suffisance de la positivité dans le cas particulier d'une fonction f semi-continue supérieurement et continue en dehors du lieu singulier f⁻¹(−∞).

     

Isoperimetric inequalities and random walks on quotients of graphs and buildings

Let be a Euclidean or hyperbolic building and let GAut be a locally compact unimodular group, which acts strongly transitively on . We use graphs , quasi-isometric to , to study asymptotic properties of quotients , where is a discrete subgroup of G. If G has Kazhdans property (T) we show that such quotients satisfy strong isoperimetric inequalities. This yields new examples of graphs with positive Cheeger constant. Such graphs cannot be bi-Lipschitz embedded into Hilbert space. Moreover, simple random walks on such quotients are shown to be recurrent if and only if is a uniform lattice in G.Mathematics Subject Classification (1991): 11E95, 22E40, 22E50, 51E24, 60G50in final form: 10 October 2003     

Exceptional points of an endomorphism of the projective plane

Let f an endomorphism of of degree >1. We show how to obtain a bound (depending only on n) on the number of codimension-two subspaces in which are completely invariant for f (where L is completely invariant for f means that f^–1(L)=L set-theoretically).     

Diophantine inequalities involving several power sums

Let denote the ring of power sums, i.e. complex functions of the form for some and _iA, where is a multiplicative semigroup. Moreover, let We consider Diophantine inequalities of the form where >1 is a quantity depending on the dominant roots of the power sums appearing as coefficients in F(n,y), and show that all its solutions have y parametrized by some power sums from a finite set. This is a continuation of the work of Corvaja and Zannier [4–6] and of the authors [10, 18] on such problems.Mathematics Subject Classification (2000):11D45,11D61Revised version: 6 May 2004