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On vector bundles of finite order

Abstract

We define Finsler metrics of finite order on a holomorphic vector bundle by imposing estimates on the holomorphic bisectional curvature. We generalize the vanishing theorem of Griffiths and Cornalba regarding Hermitian bundles of finite order to the Finsler context. We develop a value distribution theory for holomorphic maps from the projectivization of a vector bundle to projective space. We show that the projectivization of a Finsler bundle of finite order can be immersed into a projective space of sufficiently large dimension via a map of finite order.

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Abbreviations

X :

special affine variety of dimension n

\(\mathcal{O}_{\rm f.o.}\) :

sheaf of holomorphic functions on X of finite order, cf. (2.6)

\(\tau, \rho=e^{\tau /2}\) :

strictly plurisubharmonic exhaustive functions on X, cf. (1.3)

\(\tau', \rho'=e^{\tau' /2}\) :

plurisubharmonic exhaustive functions on X

d c :

the twisted derivative \(\frac{1}{4\pi \i}(\partial - \bar{\partial})\)

dd c :

the complex Hessian (also known as Levi form) \(\frac{1}{2\pi \i}\partial \bar{\partial}\)

\(\varphi = dd^c \tau\) :

Kähler form on X

Φ:

the volume-form φn of φ

\(\psi = dd^c \tau'\) :

semipositive-definite form on X satisfying ψn = 0

\({\mathfrak{p}}\) :

generic projection from X to \({\mathbb{C}}^n\)

E :

holomorphic vector bundle of rank r + 1 on X

E*:

the dual of E

\({\mathcal{O}}_{\text{f.o.}}(E^*)\) :

sheaf of holomorphic sections of finite order of E*, cf. (5.11)

\({\mathbb{P}}(E)\) :

projectivization of E

E o :

complement of the zero-section in E

\(\mathcal{V}\) :

the vertical tangent bundle inside TE o

\(\mathcal {H}\) :

the horizontal tangent bundle inside TE o

p :

projection from E onto the base X

π:

projection from \({\mathbb{P}}(E)\) onto the base X

\({\mathcal{L}}\) :

the hyperplane line bundle \({\mathcal{O}}_{{\mathbb{P}}(E)}(1)\)

\({\tilde \sigma}\) :

section of \(\mathcal{L}\) corresponding to section σ of E*, cf. (4.7)

g :

metric of finite order on det(E)

h :

Hermitian or Finsler metric on E

\(\tilde{h}\) :

Hermitian metric on \({\mathcal{L}}\) induced by h

\((G_{i{\rm \bar{j}}})\) :

Hermitian metric on \(\mathcal {V}\) induced by h

\({\tilde l}\) :

the Hermitian metric \(\tilde {h} \cdot \text{det}(G_{i\bar{\jmath}})^{-1} \cdot \pi^* g\) on \({\mathcal{L}}\)

\(\tilde \varphi\) :

Kähler form on \({\mathbb{P}}(E)\) , cf. (5.7)

\(\tilde \Phi\) :

the volume-form \(\tilde \varphi^{n+r}\) of \(\tilde \varphi\)

\(\tilde{\varphi}^{\mathcal{V}} ,\tilde{\varphi}^{\mathcal{H}}\) :

the vertical, resp. the horizontal part of \(\tilde \varphi\) , cf. paragraph after (3.23)

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Maican, M. On vector bundles of finite order. manuscripta math. 124, 97–137 (2007). https://doi.org/10.1007/s00229-007-0106-2

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Keywords

  • Vector Bundle
  • Line Bundle
  • Global Section
  • Holomorphic Section
  • Holomorphic Vector Bundle