scholarly journals ASYMPTOTIC TORSION AND TOEPLITZ OPERATORS

2015 ◽  
Vol 16 (2) ◽  
pp. 223-349 ◽  
Author(s):  
Jean-Michel Bismut ◽  
Xiaonan Ma ◽  
Weiping Zhang
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Differential Form ◽  
Toeplitz Operators ◽  
Vector Bundles ◽  
Direct Image ◽  
Analytic Torsion ◽  
Leading Term ◽  
Positive Line Bundle ◽  
Positive Line

We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles $F_{p}$. For $p\in \mathbf{N}$, the flat vector bundle $F_{p}$ is the direct image of $L^{p}$, where $L$ is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kähler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.

scholarly journals The asymptotics of the analytic torsion on CR manifolds with S1 action

2019 ◽  
Vol 21 (04) ◽  
pp. 1750094 ◽  
Author(s):  
Chin-Yu Hsiao ◽  
Rung-Tzung Huang
Keyword(s):  
Line Bundle ◽  
Asymptotic Formula ◽  
Analytic Torsion ◽  
Cr Manifold ◽  
Cr Manifolds ◽  
Dimension Formula ◽  
Positive Line Bundle ◽  
Positive Line

Let [Formula: see text] be a compact connected strongly pseudoconvex CR manifold of dimension [Formula: see text], [Formula: see text] with a transversal CR [Formula: see text]-action on [Formula: see text]. We introduce the Fourier components of the Ray–Singer analytic torsion on [Formula: see text] with respect to the [Formula: see text]-action. We establish an asymptotic formula for the Fourier components of the analytic torsion with respect to the [Formula: see text]-action. This generalizes the asymptotic formula of Bismut and Vasserot on the holomorphic Ray–Singer torsion associated with high powers of a positive line bundle to strongly pseudoconvex CR manifolds with a transversal CR [Formula: see text]-action.

scholarly journals (k, s)-POSITIVITY AND VANISHING THEOREMS FOR COMPACT KÄHLER MANIFOLDS

2011 ◽  
Vol 22 (04) ◽  
pp. 545-576 ◽  
Author(s):  
QILIN YANG
Keyword(s):  
Line Bundle ◽  
Vector Bundles ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Vanishing Theorems ◽  
Positive Vector ◽  
Holomorphic Vector Bundles ◽  
Positive Line Bundle ◽  
Algebraic Manifolds ◽  
Positive Line

We study the (k, s)-positivity for holomorphic vector bundles on compact complex manifolds. (0, s)-positivity is exactly the Demailly s-positivity and a (k, 1)-positive line bundle is just a k-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for (k, s)-positive vector bundles are proved and the vanishing theorems for k-ample vector bundles on projective algebraic manifolds are generalized to k-positive vector bundles on compact Kähler manifolds.

scholarly journals SCALING ASYMPTOTICS FOR QUANTIZED HAMILTONIAN FLOWS

2012 ◽  
Vol 23 (10) ◽  
pp. 1250102 ◽  
Author(s):  
ROBERTO PAOLETTI
Keyword(s):  
Line Bundle ◽  
Symplectic Manifold ◽  
Natural Generalization ◽  
Local Scaling ◽  
Hamiltonian Flows ◽  
Szegő Kernel ◽  
Szego Kernel ◽  
Positive Line Bundle ◽  
Positive Line ◽  
Compact Symplectic Manifold

In recent years, the near diagonal asymptotics of the equivariant components of the Szegö kernel of a positive line bundle on a compact symplectic manifold have been studied extensively by many authors. As a natural generalization of this theme, here we consider the local scaling asymptotics of the Toeplitz quantization of a Hamiltonian symplectomorphism, and specifically how they concentrate on the graph of the underlying classical map.

scholarly journals Käahler identity on Levi flat manifolds and application to the embedding

2000 ◽  
Vol 158 ◽  
pp. 87-93 ◽  
Author(s):  
Takeo Ohsawa ◽  
Nessim Sibony
Keyword(s):  
Line Bundle ◽  
Finite Order ◽  
Flat Manifold ◽  
Positive Line Bundle ◽  
Flat Manifolds ◽  
Positive Line

AbstractIt is shown that any compact Levi flat manifold admitting a positive line bundle is embeddable into ℙn by a CR mapping with an arbitrarily high, though finite, order of regularity.

THE INDEX OF TOEPLITZ OPERATORS ON FREE TRANSFORMATION GROUP C*-ALGEBRAS

2002 ◽  
Vol 34 (1) ◽  
pp. 84-90 ◽  
Author(s):  
EFTON PARK
Keyword(s):  
Vector Bundle ◽  
Differential Operator ◽  
Differential Form ◽  
Compact Manifold ◽  
Toeplitz Operators ◽  
Discrete Group ◽  
Transformation Group ◽  
Hermitian Vector Bundle ◽  
First Order ◽  
C Algebras

Let Γ be a discrete group acting on a compact manifold X, let V be a Γ-equivalent Hermitian vector bundle over X, and let D be a first-order elliptic self-adjoint Γ-equivalent differential operator acting on sections of V. This data is used to define Toeplitz operators with symbols in the transformation group C*-algebra C(X)[rtimes ]Γ, and it is shown that if the symbol of such a Toeplitz operator is invertible, then the operator is Fredholm. In the case where Γ is finite and acts freely on X, a geometric-topological formula for the index is stated that involves an explicitly constructed differential form associated to the symbol.

scholarly journals Local index theory and the Riemann–Roch–Grothendieck theorem for complex flat vector bundles

2018 ◽  
Vol 12 (04) ◽  
pp. 941-987
Author(s):  
Man-Ho Ho
Keyword(s):  
Vector Bundle ◽  
Dirac Operator ◽  
Differential Form ◽  
Vector Bundles ◽  
Variational Formula ◽  
Index Theory ◽  
Index Theorem ◽  
Local Index ◽  
Chern Simons ◽  
Local Family

The purpose of this paper is to give a proof of the real part of the Riemann–Roch–Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac operator, a variational formula of the Bismut–Cheeger eta form without the kernel bundle assumption in the even dimensional fiber case, and some properties of the Cheeger–Chern–Simons class of complex flat vector bundle.

Sums of stably trivial vector bundles

1980 ◽  
Vol 87 (1) ◽  
pp. 97-107 ◽  
Author(s):  
Jacques Allard
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Vector Bundles ◽  
Real Vector

We say that a real vector bundle ξ over a finite C.W. complex X is stably trivial of type (n, k) or, simply, of type (n, k) if ξ ⊕ kε ≅ nε, where ε denotes a trivial line bundle. The following theorem is an immediate corollary (see (12)) of a theorem of T. Y. Lam ((9), theorem 2).

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