Semistability and finite maps

Let ${f : Y \longrightarrow M}Let f : Y ? M{f : Y \longrightarrow M} be a surjective holomorphic map between compact connected K?hler manifolds such that each fiber of f is a finite subset of Y. Let ω be a K?hler form on M. Using a criterion of Demailly and Paun (Ann. Math. 159 (2004), 1247–1274) it follows that the form f*ω represents a K?hler class. Using this we prove that for any semistable sheaf E ? M{E\, \longrightarrow\,M} , the pullback f*E is also semistable. Furthermore, f*E is shown to be polystable provided E is reflexive and polystable. These results remain valid for principal bundles on M and also for Higgs G-sheaves.     

Smooth and strongly smooth points in symmetric spaces of measurable operators

We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators E(M,t){E(\mathcal{M},\tau)} associated to a semifinite von Neumann algebra (M, t){(\mathcal{M}, \tau)}. We prove that x is a smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function f ? S_E^×{f\in S_{E^{\times}}} supporting μ(x), or s(x ^*) = 1. Under the assumption that the trace τ on M{\mathcal{M}} is σ-finite, we show that x is strongly smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and E(M,t){E(\mathcal{M},\tau)}.     

On the analyticity of CR mappings between nonminimal hypersurfaces

Let be a connected real-analytic hypersurface containing a connected complex hypersurface , and let be a smooth CR mapping sending M into another real-analytic hypersurface . In this paper, we prove that if f does not collapse E to a point and does not collapse M into the image of E, and if the Levi form of M vanishes to first order along E, then f is real-analytic in a neighborhood of E. In general, the corresponding statement is false if the Levi form of M vanishes to second order or higher, in view of an example due to the author. We also show analogous results in higher dimensions provided that the target M' satisfies a certain nondegeneracy condition. The main ingredient in the proof, which seems to be of independent interest, is the prolongation of the system defining a CR mapping sending M into M' to a Pfaffian system on M with singularities along E. The nature of the singularity is described by the order of vanishing of the Levi form along E. Received: 12 February 2001 / Published online: 18 January 2002     

A note on localized directional weak lower semi-continuity

In minimization problems for functionals f : M → R, M ? E a subset of some infinite dimensional Banach space E, we typically have to rely on weak (sequential) lower semi-continuity of f on the whole space E even if M is a proper subset of E. The main reason for this lack of 'localized' weak lower semi-continuity seems to be that it is not known how to get and/or to characterize weak sequential lower semi-continuity on a subset M without knowing it on the whole space. As a first step to overcome this difficulty we propose the concept of 'localized directional weak sequential lower semi-continuity' and offer a way to implement it, namely in terms of conditions on the Gateaux derivative f′ of f (weak K-monotonicity). This allows to formulate a criterium and new sufficient conditions for the existence of a minimizer.

We conclude with a discussion of applications to the variational approach to the solution of (systems of) nonlinear partial differential equations where we focus on the case of integral functionals of vector fields for which the integrand is not assumed to be quasi-convex.     

Problemi variazionali conformi

Letf: (M,g)→(N,g′) be a differentiable map between the riemannian manifoldsM andN, M being compact.K. Uhlenbeck pointed out a functionalE ^m(f), related to the energy density off, that depends only on the conformal structure ofM. In this paper we prove thatE ^m(f) is stationary with respect to deformations of the riemannian metric ofM if and only iff is weakly conformal; in this casef provides a local minimum ofE ^m.     

Infima of universal energy functionals on homotopy classes

We consider the infima (f) on homotopy classes of energy functionals E defined on smooth maps f: Mⁿ → V^k between compact connected Riemannian manifolds. If M contains a sub‐manifold L of codimension greater than the degree of E then (f) is determined by the homotopy class of the restriction of f to M \ L. Conversely if the infimum on a homotopy class of a functional of at least conformal degree vanishes then the map is trivial in homology of high degrees. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximation on Subsets and Holomorphy in Neighborhood

Let M be an arbitrary subset of the complex plane C, and let a function f; be defined on M. The connection between the holomorphy of f; and the degree of the best polynomial approximation of f; on compact subsets of M having a unique limit point is described.     

Hochschild (Co)Homology of Hopf Crossed Products*

For a general crossed product E = A#_f H, of an algebra A by a Hopf algebra H, we obtain complexes smaller than the canonical ones, giving the Hochschild homology and cohomology of E. These complexes are equipped with natural filtrations. The spectral sequences associated to them coincide with the ones obtained using a natural generalization of the direct method introduced in Trans. Amer. Math. Soc. 74 (1953) 110–134. We also get that if the 2-cocycle f takes its values in a separable subalgebra of A, then the Hochschild (co)homology of E with coefficients in M is the (co)homology of H with coefficients in a (co)chain complex.     

The automorphism group of a Banach principal bundle with {1}-structure

A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach–Lie group acting smoothly on P provided the Lie algebra of infinitesimal automorphisms consists of complete vector fields. As a consequence we show that the automorphism group of a connected geodesically complete affine Banach manifold M can be turned into a Banach–Lie group acting smoothly on M.     

Tighter Bounds on the Solution of a Divide-and-Conquer Maximin Recurrence

LetM(n) be defined by the recurrencewherefis an arbitrary nondecreasing function andM(1) is given. The recurrenceM(n) is a divide-and-conquer maximin recurrence, which occurs in a variety of problems in the analysis of algorithms. In this paper, a new upper bound onM(n) is first derived. The derived bound is smaller than the one proposed previously by Li and Reingold. It is at most two times the exact solution ofM(n). Using the bound, we further show thatM(n) ≤ 2E(n), whereE(n) is defined by the recurrenceE(n) = E(⌊n/2⌋) + E(⌈n/2⌉) + f(⌊n/2⌋). From this result, we can conclude that a divide-and-conquer algorithm whose time complexity is expressed asM(n) is as efficient as a divide-and-conquer algorithm whose time complexity is expressed asE(n).     

On Extending Peano Derivatives

Let E be a closed set with inf E = a and sup E = b, and k be a positive integer. Let f : E Rbe such that the k-th Peano derivative of f relative to E, f _(k) (x, E), exists. It is proved under certain condition on the function f, that an extension F : [a, b] Rof f exists such that the ordinary derivative of F of order k, F ^<k> (x) exists on [a, b] and is continuous on [a, b], and f _<> (x, E) = F ^<i> (x) on E, for i = 1, 2, &, k.     

Coreflectivity of E-Monads and Algebraic Hulls

Let Xbe a category with a given(E,M)-factorization structure for morphisms, M Mono X. In general, an arbitrary endofunctor T of X fails badly to preserve the E-class. If T carries a monad structure, then T(E) E implies that the corresponding category of Eilenberg–Moore-algebras admits (E,M)-factorizations and vice versa. In order to get T as close as possible to this nice algebraic behaviour, a couniversal modification T T with (E) E is constructed in two different ways using mild and natural assumptions on E and M, respectively. T inherits its monad structure from T. In case of T = U F, F U, the Eilenberg–Moore-category of T contains a universal (E,M-algebraic hull (completion) of U [2, 3]. There are further applications to varietal hulls [4] and to function spaces.     

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Let X be a locally compact topological space and (X, E, X_ω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X_ω ⊆ X, such that all internal subsets of X_ω are relatively compact in the induced topology and X is homeomorphic to the quotient X_ω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function X ? ^*\Bbb CX \rightarrow {}{^{\ast}{\Bbb C}} . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient M_w(X)/M₀(X){\cal M}_{\omega}(X)/{\cal M}_0(X) , for certain external subspaces M₀(X), M_w(X){\cal M}_0(X), {\cal M}_{\omega}(X) of the hyperfinite dimensional Banach space ^*\Bbb C^X{}{^{\ast}{\Bbb C}}^X , with the norm ‖f‖₁ = ∑_{x ∈ X} |f(x)|. If additionally X = G is a hyperfinite group, X_ω = G_ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G₀ of G_ω, and G is isomorphic to the locally compact group G_ω/G₀, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and M_w(G)/M₀(G){\cal M}_{\omega}(G)/{\cal M}_0(G) are isometrically isomorphic as Banach algebras.     

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.     

Uniqueness theorems for Korenblum type spaces

For a scale of spaces X of functions analytic in the unit disc, including the Korenblum space, and for some natural families ɛ of uniqueness subsets for X, we describe minorants for (X, ɛ), that is, non-decreasing functions M: (0, 1) → (0, ∞) such that f ∈ X, E ∈ ɛ, and log |f(z)| ≤ −M(|z|) on E imply f = 0. We give an application of this result to approximation by simple fractions with restrictions on the coefficients. The first author was partially supported by the ANR project DYNOP. The second author was partially supported by the Research Council of Norway, grant 160192/V30.     

An inner product relation on Saito-Kurokawa lifts

Let f be a newform of weight 2k−2 and level M with M an odd square-free integer. Via the Saito-Kurokawa correspondence there is associated to f a Siegel newform F _f of weight k and level M. In this paper we provide a formula relating the Petersson products {F _f , F _f } and {f, f}. We use this result to give a new proof of a special case of a well-known result of Shimura on the algebraicity of a special value of a Rankin convolution L-function. 2000 Mathematics Subject ClassificationPrimary—11F32; Secondary—11F46, 11F67     

The Problem of Prescribed Critical Functions

Let (M,g) be a compact Riemannian manifold on dimension n ≥ 4 not conformally diffeomorphic to the sphere Sⁿ. We prove that a smooth function f on M is a critical function for a metric g conformal to g if and only if there exists x ∈ M such that f(x) > 0.Mathematics Subject Classifications (2000): 53C21, 46E35, 26D10.     

Spherical Maximal Operators on Radial Functions

Let A_tf(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function M_E f(X) = sup_t∈_E |Atf(x)| where E is a fixed set in IR⁺ and f is a radial function ∈ L^p(IR^d). Let P_d = d/(d?1) (the critical exponent for Stein's maximal function). For the cases (i) p < p_d, d ? 2, and (ii) p = p_d, d ? 3, and for p ? q ? ∞ we prove necessary and sufficient conditions on E for M_E to map radial functions in L^p to the Lorentz space L^P,q.     

A general approach to index theorems for holomorphic maps and foliations

Let M be a smooth complex manifold, and S(⊂ M) be a compact irreducible subvariety with dim _C S > 0. Let be given either a holomorphic map f : M → M with f _|S  = id _S , f ≠ id _M , or a holomorphic foliation on M: we describe an approach that can be applied to both map and foliation in order to obtain index theorems. Partially supported by GNSAGA, Centro de Giorgi, M.U.R.S.T.