A rigidity theorem for the pair ${\cal q}{\Bbb C} P^n$ (complex hyperquadric, complex projective space)

Given a compact Kähler manifold M of real dimension 2n, let P be either a compact complex hypersurface of M or a compact totally real submanifold of dimension n. Let q\cal q (resp. \Bbb R Pⁿ{\Bbb R} P^n) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space \Bbb C Pⁿ{\Bbb C} P^n of constant holomorphic sectional curvature 4l \lambda . We prove that if the Ricci and some (n-1)-Ricci curvatures of M (and, when P is complex, the mean absolute curvature of P) are bounded from below by some special constants and volume (P) / volume (M) ￡\leq volume (q\cal q)/ volume (\Bbb C Pⁿ)({\Bbb C} P^n) (resp. ￡\leq volume (\Bbb R Pⁿ)({\Bbb R} P^n) / volume (\Bbb C Pⁿ)({\Bbb C} P^n)), then there is a holomorphic isometry between M and \Bbb C Pⁿ{\Bbb C} P^n taking P isometrically onto q\cal q (resp. \Bbb R Pⁿ{\Bbb R} P^n). We also classify the Kähler manifolds with boundary which are tubes of radius r around totally real and totally geodesic submanifolds of half dimension, have the holomorphic sectional and some (n-1)-Ricci curvatures bounded from below by those of the tube \Bbb R Pⁿ_r{\Bbb R} P^n_r of radius r around \Bbb R Pⁿ{\Bbb R} P^n in \Bbb C Pⁿ{\Bbb C} P^n and have the first Dirichlet eigenvalue not lower than that of \Bbb R Pⁿ_r{\Bbb R} P^n_r.     

Some blow-up results for a generalized Ginzburg-Landau equation

We investigate the blow-up of the solution to a complex Ginzburg-Landau like equation in u coupled with a Poisson equation in f\phi defined on the whole space \Bbb Rⁿ, n = 1{\Bbb R}^n, n = 1 or 2.     

Analytic Fourier integral operators, Monge-Ampère equation and holomorphic factorization

We will show that the factorization condition for the Fourier integral operators I_r ^m (X,Y;L )I_\rho ^\mu (X,Y;\it\Lambda ) leads to a parametrized parabolic Monge-Ampère equation. For an analytic operator, the fibration by the kernels of the Hessian of phase function is shown to be analytic in a number of cases, by considering a more general continuation problem for the level sets of a holomorphic mapping. The results are applied to obtain L^p-continuity for translation invariant operators in \Bbb Rⁿ{\Bbb R}^n with n ￡ 4n\leq 4 and for arbitrary \Bbb Rⁿ{\Bbb R}^n with dp_X×Y|_L ￡ n+2d\pi _{X\times Y}|_\Lambda \leq n+2.     

On the singularities of residual intertwining operators

Let G be a reductive algebraic group defined over \Bbb Q {\Bbb Q} . Let P, P' be parabolic subgroups of G, defined over \Bbb Q {\Bbb Q} , and let _boxclose_boxclose, a_P') t \in W({\frak a}_{P}, {\frak a}_{P'}) . In this paper we study the intertwining operator M_P￠|P(t,l), l ? \frak a^*_{P,\Bbb C} M_{P' \vert P}(t,\lambda),\,\lambda \in {\frak a}^*_{P,{\Bbb C}} , acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of M_P￠|P(t,l) M_{P' \vert P}(t,\lambda) is a meromorphic function of order ￡ n + 1 \le n + 1 , where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles.     

Integral Lie powers of a lattice for the cyclic group of order 2

Let L_n denote the n-th homogeneous component of the free Lie ring L(W) on a given \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice W. This paper gives explicit formulae for the multiplicities of the three indecomposable \Bbb ZC₂{{\Bbb Z}}C_{2}-lattices in a Krull-Schmidt decomposition of L_n. In the case where W is a free \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice, L_n is shown to have no non-zero direct summand on which C₂ acts trivially - this extends a result of R. M. Bryant for the special case where W is the regular \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice. As an application, the structure of the higher dimensional modules associated to a non-cyclic free presentation of C₂ is determined.     

Solvability of systems of partial differential equations for functions defined on nonconvex sets

We consider systems of partial differential equations with constant coefficients of the form ( R(D_x, D_y)f = 0, P(D_x)f = g), f,g ? C^￥(W),\big ( R(D_x, D_y)f = 0, P(D_x)f = {g}\big ), f,g \in {C}^{\infty}(\Omega),, where R (and P) are operators in (n + 1) variables (and in n variables, respectively), g satisfies the compatibility condition R(D_x, D_y)g = 0  and  W ì \Bbb Rⁿ⁺¹R(D_x, D_y){g} = 0 \ {\rm and} \ \Omega \subset {\Bbb R}^{n+1} is open. Let R be elliptic. We show that the solvability of such systems for certain nonconvex sets W\Omega implies that any localization at ￥\infty of the principle part P_m of P is hyperbolic. In contrast to this result such systems can always be solved on convex open sets W\Omega by the fundamental principle of Ehrenpreis-Palamodov.     

A Seidel-Walsh theorem with linear differential operators

Assume that {S_n}₁^￥ \{S_n\}_1^\infty is a sequence of automorphisms of the open unit disk \Bbb D{\Bbb D} and that {T_n}₁^￥\{T_n\}_1^\infty is a sequence of linear differential operators with constant coefficients, both of them satisfying suitable conditions. We prove that for certain spaces X of holomorphic functions in the open unit disk, the set of functions f ? Xf \in X such that {(T_n f) °S_n:  n ? \Bbb N}\{(T_n\,f) \circ S_n: \, n \in {\Bbb N}\} is dense in H(\Bbb D)H({\Bbb D}) is residual in X. This extends the Seidel-Walsh theorem together with some subsequent results.     

Straightening and bounded cohomology of hyperbolic groups

It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology Hⁿ_b(G,\Bbb R) H^n_b(G,{\Bbb R}) to Hⁿ(G,\Bbb R) H^n(G,{\Bbb R}) induced by inclusion is surjective for n 3 2 n \ge 2 . We introduce a homological analogue of straightening simplices, which works for any hyperbolic group. This implies that the map Hⁿ_b(G,V) ? Hⁿ(G,V) H^n_b(G,V) \to H^n(G,V) is surjective for n 3 2 n \ge 2 when V is any bounded \Bbb QG {\Bbb Q}G -module and when V is any finitely generated abelian group.     

On the infinitesimal affine rigidity of ellipsoids in the n-space

Due to R. Schneider 1967 an ellipsoid E in the affine space \Bbb Aⁿ\Bbb A^n is affinely rigid, i.e. every other ovaloid F in \Bbb Aⁿ\Bbb A^n with the same affine Blaschke metric as for E equals E up to an equiaffine motion of E. Due to M. Kozlowski 1985 resp. W. Blaschke 1922 for n = 3 ellipsoids are moreover S-rigid resp. infinitesimally S-rigid in the sense of equal resp. infinitesimally equal affine scalar curvature S (unknown until now for n >3). - In this article it is proved that ellipsoids in \Bbb Aⁿ\Bbb A^n are also infinitesimally S-rigid for any n.     

Arithmetic progressions in sumsets

We prove several results concerning arithmetic progressions in sets of integers. Suppose, for example, that a \alpha and b \beta are positive reals, that N is a large prime and that C,D í \Bbb Z/N\Bbb Z C,D \subseteq {\Bbb Z}/N{\Bbb Z} have sizes gN \gamma N and dN \delta N respectively. Then the sumset C + D contains an AP of length at least e^{c ?{log} N} e^{c \sqrt{\rm log} N} , where c > 0 depends only on g \gamma and d \delta . In deriving these results we introduce the concept of hereditary non-uniformity (HNU) for subsets of \Bbb Z/N\Bbb Z {\Bbb Z}/N{\Bbb Z} , and prove a structural result for sets with this property.     

Idempotents in a complex group algebra, projective modules, and the von Neumann algebra

The complex group algebra \Bbb CG{\Bbb C}G of a countable group G can be imbedded in the von Neumann algebra NG of G. If G is torsion-free, and if P is a finitely generated projective module over \Bbb CG{\Bbb C}G it is proved that the central-valued trace of NG?_{\Bbb CG}PNG\otimes _{{\Bbb C}G}P, i.e. of an idempotent \Bbb CG{\Bbb C}G-matrix A defining P is equal to the canonical trace k(P)\kappa (P) times identity I. It follows that k(P)\kappa (P) characterizes the isomorphism type of NG?_{\Bbb CG}PNG\otimes _{{\Bbb C}G}P.¶If k(P)\kappa (P) is an integer, e.g., if the weak Bass conjecture holds for G then NG?_{\Bbb C G}PNG\otimes _{{\Bbb C} G}P is free. It is also shown that for certain classes of groups geometric arguments can be used to prove the Bass conjecture.     

Lagrangian barriers and symplectic embeddings

We prove that every symplectic Kähler manifold (M;W) (M;\Omega) with integral [W] [\Omega] decomposes into a disjoint union (M,W) = (E,w₀) \coprod D (M,\Omega) = (E,\omega_0) \coprod \Delta , where (E,w₀) (E,\omega_0) is a disc bundle endowed with a standard symplectic form w₀ \omega_0 and D \Delta is an isotropic CW-complex. We perform explicit computations of this decomposition on several examples.¶As an application we establish the following symplectic intersection phenomenon: There exist symplectically irremovable intersections between contractible domains and Lagrangian submanifolds. For example, we prove that every symplectic embedding j:B²ⁿ(l) ? \Bbb CPⁿ \varphi:B^{2n}(\lambda) \to {\Bbb C}P^n of a ball of radius l² 3 1/2 \lambda^2 \ge 1/2 must intersect the standard Lagrangian real projective space \Bbb RPⁿ ì \Bbb CPⁿ {\Bbb R}P^n \subset {\Bbb C}P^n .     

Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space

We establish a close link between the amenability property of a unitary representation p \pi of a group G (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system (\Bbb S_p, G) ({\Bbb S}_{\pi}, G) , where \Bbb S_H {\Bbb S}_{\cal H} is the unit sphere the Hilbert space of representation. We prove that p \pi is amenable if and only if either p \pi contains a finite-dimensional subrepresentation or the maximal uniform compactification of (\Bbb S_p ({\Bbb S}_{\pi} has a G-fixed point. Equivalently, the latter means that the G-space (\Bbb S_p, G) ({\Bbb S}_{\pi}, G) has the concentration property: every finite cover of the sphere \Bbb S_p {\Bbb S}_{\pi} contains a set A such that for every e > 0 \epsilon > 0 the e \epsilon -neighbourhoods of the translations of A by finitely many elements of G always intersect. As a corollary, amenability of p \pi is equivalent to the existence of a G-invariant mean on the uniformly continuous bounded functions on \Bbb S_p {\Bbb S}_{\pi} . As another corollary, a locally compact group G is amenable if and only if for every strongly continuous unitary representation of G in an infinite-dimensional Hilbert space H {\cal H} the system (\Bbb S_H, G) ({\Bbb S}_{\cal H}, G) has the property of concentration.     

Bounded solutions of the Golab—Schinzel equation

Summary. Let \Bbb K {\Bbb K} be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over \Bbb K {\Bbb K} n be a positive integer, and f : X ? \Bbb K f : X \to {\Bbb K} be a solution of the functional equation¶¶f(x + f(x)ⁿ y) = f(x) f(y) f(x + f(x)^n y) = f(x) f(y) .¶We prove that, if there is a real positive a such that the set { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} contains a subset of second category and with the Baire property, then f is continuous or { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} for every x ? X x \in X . As a consequence of this we obtain the following fact: Every Baire measurable solution f : X ? \Bbb K f : X \to {\Bbb K} of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set A ì X A \subset X with f(X \A) = { 0 }) f(X \backslash A) = \{ 0 \}) .     

Base point free pencils of small degree on certain smooth curves

We give sufficient conditions on numbers d and m such that a linear system of degree m on the normalization C of a plane curve [`(C)]\overline {C} of degree d which is in a certain sense not too singular is in the natural way induced by either a pencil of lines or a pencil of conics in the plane. Those results generalize results on nodal and cuspidal plane curves and seem to complement the recent results of [2]. We present a new approach via the geometry of curves in \Bbb P₁×\Bbb P₂{\Bbb P}_1\times {\Bbb P}_2.     

The differential Hilbert series of a local algebra

The differential Hilbert series of a commutative local algebra R/R₀ which is essentially of finite type is the generating function of the numerical function which associates with each n ? \Bbb N n\in \Bbb N the minimal number of generators of the algebra Pⁿ_R/R0P^n_{R/R_0} of principal parts of order n, considered as an R-module. It can be expressed as a rational function over the integers. We wish to compute this rational function in terms of other invariants of the local algebra or at least give estimates of it. We obtain formulas which generalize wellknown facts about the minimal number of generators of the module of Kähler differentials.     

Integration of the lifting formulas and the cyclic homology of the algebras of differential operators

We integrate the Lifting cocycles Y_2n+1, Y_2n+3, Y_2n+5,? ([Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,([\rm Sh1,2]) on the Lie algebra Dif_n of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem [FT1]:¶¶ H^·_Lie(\frak g\frak l^fin_￥(Dif_n);\Bbb C) = ù^·(Y_2n+1, Y_2n+3, Y_2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) .     

Idempotent multipliers for LP (\Bbb R)(\Bbb R)

The algebra B_p(\Bbb R){\cal B}_p({\Bbb R}), p ? (1,￥)\{2}p\in (1,\infty )\setminus \{2\}, consisting of all measurable sets in \Bbb R{\Bbb R} whose characteristic function is a Fourier p-multiplier, forms an algebra of sets containing many interesting and non-trivial elements (e.g. all intervals and their finite unions, certain periodic sets, arbitrary countable unions of dyadic intervals, etc.). However, B_p(\Bbb R){\cal B}_p({\Bbb R}) fails to be a s\sigma -algebra. It has been shown by V. Lebedev and A. Olevskii [4] that if E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}), then E must coincide a.e. with an open set, a remarkable topological constraint on E. In this note we show if $2 < p < \infty $2 < p < \infty , then there exists E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}) which is not in B_q(\Bbb R){\cal B}_q({\Bbb R}) for any q > pq>p.     

Characterization of field homomorphisms and derivations by functional equations

Summary. Let K and [`(K)] \overline K be fields containing \Bbb Q {\Bbb Q} . We characterize pairs of additive functions f,g: K ?[`(K)] f,g: K \to \overline K satisfying a functional equation¶¶ g(x^ln) = f(x^l)ⁿ     \textrespectively        g(x^ln) = Ax^ln + x^ln-lf(x^l) g(x^{ln}) = f(x^l)^n \quad \text{respectively} \qquad g(x^{ln}) = Ax^{ln} + x^{ln-l}f(x^l) ,¶where n ? \Bbb Z \{0,1} n \in {\Bbb Z} \setminus \{0,1\} , l ? \Bbb N l\in {\Bbb N} and A ? K A \in K .     

The Łojasiewicz exponent of an analytic function at an isolated zero

Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rⁿ 0 \in {\Bbb R}^n such that f^-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|^a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|^b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|^q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|^(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.