scholarly journals The asymptotics of the Ray-Singer analytic torsion of the symmetric powers of a positive vector bundle

1990 ◽  
Vol 40 (4) ◽  
pp. 835-848 ◽  
Author(s):  
Jean-Michel Bismut ◽  
E. Vasserot
Keyword(s):  
Vector Bundle ◽  
Analytic Torsion ◽  
Symmetric Powers ◽  
Positive Vector ◽  
Positive Vector Bundle

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 A remark on characterizations of Griffiths positivity through asymptotic conditions

2021 ◽  
pp. 2150087
Author(s):  
Genki Hosono ◽  
Takahiro Inayama
Keyword(s):  
Vector Bundle ◽  
Holomorphic Vector Bundle ◽  
Extension Theorems ◽  
Symmetric Powers

In this paper, we study characterizations of Griffiths semi-positivity through [Formula: see text]-estimates of the [Formula: see text]-equation and [Formula: see text]-extension theorems for symmetric powers of a holomorphic vector bundle. We also investigate several versions of the converse of the Demailly–Skoda theorem.

scholarly journals Analytic torsion of space forms of certain compact symmetric spaces

1977 ◽  
Vol 67 ◽  
pp. 65-88
Author(s):  
Hajime Urakawa
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Fundamental Group ◽  
Vector Space ◽  
Symmetric Spaces ◽  
Analytic Torsion ◽  
Dimensional Representation ◽  
Space Forms ◽  
Finite Dimensional Representation ◽  
Finite Dimensional

Let M be a compact, oriented Riemannian manifold of dimension dr and let Γ be the fundamental group of M. For a finite dimensional representation ρ of Γ on a vector space F, Ray and Singer [10] have defined the analytic torsion T(M, ρ) as follows: We denote by E the vector bundle over M with typical fibre F defined by the representation ρ.

Euler classes of vector bundles over manifolds

2021 ◽  
Vol 71 (1) ◽  
pp. 199-210
Author(s):  
Aniruddha C. Naolekar
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Euler Class ◽  
Homology Sphere ◽  
Rational Homology ◽  
Rational Homology Sphere ◽  
Orientable Manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

scholarly journals WEAKLY UNIFORM RANK TWO VECTOR BUNDLES ON MULTIPROJECTIVE SPACES

2011 ◽  
Vol 84 (2) ◽  
pp. 255-260
Author(s):  
EDOARDO BALLICO ◽  
FRANCESCO MALASPINA
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Splitting Type ◽  
Multiprojective Spaces

AbstractHere we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover, we show that every rank r>2 weakly uniform vector bundle with splitting type a1,1=⋯=ar,s=0 is trivial and every rank r>2 uniform vector bundle with splitting type a1>⋯>ar splits.

scholarly journals Le complémentaire des puissances -ièmes dans un corps de nombres est un ensemble diophantien

2015 ◽  
Vol 151 (10) ◽  
pp. 1965-1980 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Jan Van Geel
Keyword(s):  
Prime Number ◽  
Rational Points ◽  
Parameter Family ◽  
Symmetric Power ◽  
Symmetric Powers

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

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