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The translation-invariant vector fields

  • David Mumford
Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

Let X be a variety. Then a vector field D on X is given equivalently by:
  1. a)

    a family of tangent vectors D(x) ∈ TX, x, all x ∈ X such that in local charts Open image in new window

     
  2. b)

    a derivation D: \( \mathcal{O}_X \to \mathcal{O}_X \).

     
In fact, given D(x), \( f \in \Gamma \left( {U,\mathcal{O}_X } \right) \), define Df by
$$ Df\left( x \right) = D\left( x \right)\left( f \right). $$
When X is an abelian variety, then translations on X define isomorphisms
$$ T_{X,O} \xrightarrow{ \sim }T_{X,x} $$
for all x ∈ X (O = identity), so we may speak of translation-invariant vector fields. It is easy to see that for all D(O) ∈ Tx, o′ there is a unique translation-invariant vector field with this value at O. In general, the vector fields on X form a Lie algebra under commutators: 4.c EQ
$$ \left[ {D_1 ,D_2 } \right]\left( f \right) = D_1 D_2 f - D_2 D_1 f. $$
For translation-invariant vector fields, the commutativity of X implies that bracket is zero (see Abelian Varieties, D. Mumford, Oxford Univ. Press, p. 100.

Keywords

Vector Field Branch Point Tangent Vector Abelian Variety Principal Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • David Mumford
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

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