共查询到20条相似文献,搜索用时 31 毫秒
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Ognjen Milatovic 《Differential Geometry and its Applications》2004,21(3):361-377
We consider a family of Schrödinger-type differential expressions L(κ)=D2+V+κV(1), where κ∈C, and D is the Dirac operator associated with a Clifford bundle (E,∇E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(κ)=*(∇F)∇F+V+κV(1), where κ∈C, and ∇F is a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V(1) are self-adjoint locally integrable sections of EndF. We give sufficient conditions for L(κ) and I(κ) to have a realization in L2(E) and L2(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Kato's inequality for Bochner Laplacian operator and Weitzenböck formula. 相似文献
3.
Yaroslav Kurylev 《Advances in Mathematics》2009,221(1):170-216
We consider a Dirac-type operator DP on a vector bundle V over a compact Riemannian manifold (M,g) with a non-empty boundary. The operator DP is specified by a boundary condition P(u|∂M)=0 where P is a projector which may be a non-local, i.e., a pseudodifferential operator. We assume the existence of a chirality operator which decomposes L2(M,V) into two orthogonal subspaces X+⊕X−. Under certain conditions, the operator DP restricted to X+ and X− defines a pair of Fredholm operators which maps X+→X− and X−→X+ correspondingly, giving rise to a superstructure on V. In this paper we consider the questions of determining the index of DP and the reconstruction of and DP from the boundary data on ∂M. The data used is either the Cauchy data, i.e., the restrictions to ∂M×R+ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e., the set of the eigenvalues and the boundary values of the eigenfunctions of DP. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in M×C4, M⊂R3. 相似文献
4.
A. Lerario 《Discrete and Computational Geometry》2012,48(4):1025-1047
We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space ?P n (e.g. X is the intersection of two real quadrics). We give explicit formulas for its Betti numbers and for those of its double cover in the sphere S n ; we also give similar formulas for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)??2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular?X. In the nondegenerate case we also prove the bound on each specific Betti number b k (X)??2(k+2). 相似文献
5.
A complete Riemannian manifold X with negative curvature satisfying −b2?KX?−a2<0 for some constants a,b, is naturally mapped in the space of probability measures on the ideal boundary ∂X by assigning the Poisson kernels. We show that this map is embedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to constant provided X is a rank one symmetric space of non-compact type. Furthermore, we give a geometric meaning of the embedding. 相似文献
6.
Lucia Marino 《Rendiconti del Circolo Matematico di Palermo》2003,52(2):263-280
IfX is a set of distinct points in ℙ2 with given graded Betti numbers, we produce a new set of pointsY with the same graded Betti numbers asX which admits all possible conductor degrees according to the graded Betti numbers. Moreover, for such schemes we can compute
the conductor degree for each point.
We conclude by generalizing the construction of these schemes, obtaining again the same results. 相似文献
7.
Yu. A. Kordyukov 《Acta Appl Math》1991,23(3):223-260
This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in L
pspaces. We prove the coincidence of minimal and maximal extensions of an operator of a considered type with a positive principal symbol, the existence of holomorphic semigroup, generated by it, and the estimates of L
p-norms of the operators of this semigroup. Some spectral properties of such operators in L
pspaces are also studied. 相似文献
8.
Weiping Zhang 《Topology》2005,44(6):1093-1131
We generalize a theorem of Bismut-Zhang, which extends the Cheeger-Müller theorem on Ray-Singer torsion and Reidemeister torsion, to the case of infinite Galois covering spaces. Our result is stated in the framework of extended cohomology, and generalizes in this case a recent result of Braverman-Carey-Farber-Mathai. It does not use the determinant class condition and thus also (potentially) generalizes several results on L2-torsions due to Burghelea, Friedlander, Kappeler and McDonald. We combine the framework developed by Braverman-Carey-Farber-Mathai on the determinant of extended cohomology with the heat kernel method developed in the original paper of Bismut-Zhang to prove our result. 相似文献
9.
Euisung Park 《Journal of Pure and Applied Algebra》2010,214(2):101-111
Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P1 be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f∗OP1(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L. 相似文献
10.
Yoshiaki Fukuma 《Journal of Pure and Applied Algebra》2011,215(2):168-184
Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this paper, we provide a lower bound for h0(m(KX+L)) under the assumption that κ(KX+L)≥0. In particular, we get the following: (1) if 0≤κ(KX+L)≤2, then h0(KX+L)>0 holds. (2) If κ(KX+L)=3, then h0(2(KX+L))≥3 holds. Moreover we get a classification of (X,L) with κ(KX+L)=3 and h0(2(KX+L))=3 or 4. 相似文献
11.
Yoshiaki Fukuma 《Journal of Pure and Applied Algebra》2007,209(1):99-117
Let (X,L) be a polarized manifold of dimension n. In this paper, for any integer i with 0≤i≤n we introduce the notion of the ith sectional invariant of (X,L). We define the ith sectional Euler number ei(X,L), the ith sectional Betti number bi(X,L), and the ith sectional Hodge number of type (j,i−j) of (X,L) and we will study some properties of these. 相似文献
12.
In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the BCAP, then L∞/X has the BCAP. We also show that X* has the λ-BCAP with conjugate operators if and only if the pair (X, Y) has the λ-BCAP for each finite codimensional subspace Y∈X. Let M be a closed subspace of X such that M⊥ is complemented in X*. If X has the (bounded) approximation property of order p, then M has the (bounded) approximation property of order p. 相似文献
13.
Finnur Lárusson 《Journal of Geometric Analysis》1995,5(2):281-291
LetX be a projective manifold of dimension n ≥ 2 andY →X an infinite covering space. EmbedX into projective space by sections of a sufficiently ample line bundle. We prove that any holomorphic function of sufficiently slow growth on the preimage of a transverse intersection ofX by a linear subspace of codimension <n extends toY. The proof uses a Hausdorff duality theorem for L2 cohomology. We also show that every projective manifold has a finite branched covering whose universal covering space is Stein. 相似文献
14.
Philippe Eyssidieux 《Mathematische Annalen》2000,317(3):527-566
In order to study the group of holomorphic sections of the pull-back to the universal covering space of an holomorphic vector bundle on a compact complex
manifold, it would be convenient to have a cohomological formalism, generalizing Atiyah's index theorem. In [Eys99], such a formalism is proposed in a restricted context. To each coherent analytic sheaf on a n-dimensionnal smooth projective variety and each Galois infinite unramified covering , whose Galois group is denoted by , cohomology groups denoted by are attached, such that:
1. The underly a cohomological functor on the abelian category of coherent analytic sheaves on X.
2. If is locally free, is the group of holomorphic sections of the pull-back to of the holomorphic vector bundle underlying .
3. belongs to a category of -modules on which a dimension function with real values is defined.
4. Atiyah's index theorem holds [Ati76]:
The present work constructs such a formalism in the natural context of complex analytic spaces. Here is a sketch of the main
ideas of this construction, which is a Cartan-Serre version of [Ati76]. A major ingredient will be the construction [Farb96]
of an abelian category containing every closed -submodule of the left regular representation. In topology, this device enables one to use standard sheaf theoretic methods
to study Betti numbers [Ati76] and Novikov-Shubin invariants [NovShu87]. It will play a similar r?le here. We first construct a -cohomology theory () for coherent analytic sheaves on a complex space endowed with a proper action of a group such that conditions 1-2 are fulfilled. The -cohomology on the Galois covering of a coherent analytic sheaf onX is the ordinary cohomology of a sheaf on X obtained by an adequate completion of the tensor product of by the locally constant sheaf on X associated to the left regular representation of the discrete group in the space of functions on . Then, we introduce an homological algebra device, montelian modules, which can be used to calculate the derived category
of and are a good model of the Čech complex calculating -cohomology. Using this we prove that , if X is compact. This is stronger than condition 3, since this also yields Novikov-Shubin type invariants. To explain the title
of the article, Betti numbers and Novikov-Shubin invariants of are the Von Neumann invariants of the coherent analytic sheaf . We also make the connection with Atiyah's -index theorem [Ati76] thanks to a Leray-Serre spectral sequence. From this, condition 4 is easily deduced.
Received: 30 October 1998 / Published online: 8 May 2000 相似文献
Received: 30 October 1998 / Published online: 8 May 2000 相似文献
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Pierre Albin 《Advances in Mathematics》2007,213(1):1-52
We study the Gauss-Bonnet theorem as a renormalized index theorem for edge metrics. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence and the asymptotically cylindrical metrics of the Atiyah-Patodi-Singer index theorem. We use renormalization to make sense of the curvature integral and the dimensions of the L2-cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod xm, we show that the finite time supertrace of the heat kernel on conformally compact manifolds renormalizes independently of the choice of special boundary defining function. 相似文献
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Nikolaos Mandouvalos 《Bulletin des Sciences Mathématiques》2009,133(2):134-144
We assume that the discrete part of the spectrum of the Laplacian on a non-compact locally symmetric space is non-empty and we prove that the Riesz transform is bounded on Lp for all p in an interval around 2. 相似文献
19.
We answer in the affirmative the following question raised by H. H. Corson in 1961: “Is it possible to cover every Banach space X by bounded convex sets with non-empty interior in such a way that no point of X belongs to infinitely many of them?”Actually, we show the way to produce in every Banach space X a bounded convex tiling of order 2, i.e., a covering of X by bounded convex closed sets with non-empty interior (tiles) such that the interiors are pairwise disjoint and no point of X belongs to more than two tiles. 相似文献
20.
Let K be a finite simplicial complex.
We are interested in the asymptotic behavior of the Betti
numbers of a sequence of finite sheeted covers of $K$, when
normalized by the index of the covers. W. Lück, has proved
that for regular coverings, these sequences of numbers converge
to the $l^2$ Betti numbers of the associated (in
general infinite) limit regular cover of K.
In this article we investigate the non regular case. We show that the
sequences of normalized Betti numbers still converge. But this time
the good limit object is no longer the associated limit cover of
K, but a lamination by
simplicial complexes. We prove that the limits of sequences of
normalized Betti numbers are equal to the $l^2$
Betti numbers of this lamination.
Even if the associated limit cover of
K is contractible, its $l^2$
Betti numbers are in general different from those of the lamination.
We construct such examples. We also give a dynamical condition for
these numbers to be equal. It turns out that this condition is
equivalent to a former criterion due to M. Farber. We hope that our
results clarify its meaning and show to which extent it is optimal.
In a second part of this paper we study non free measure-preserving
ergodic actions of a countable group $\Gamma$ on a standard Borel
probability space. Extending group-theoretic similar results of the
second author, we obtain relations between the $l^{2}$ Betti numbers
of $\Gamma$ and those of the generic stabilizers. For example, if
$b_1^{(2)} (\Gamma ) \neq 0$, then either almost each stabilizer is
finite or almost each stabilizer has an infinite first $l^2$ Betti number.
Asymptotique des nombres de Betti, invariants $l^2$ et laminations相似文献