The translation-invariant vector fields

Abstract

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

• a) a family of tangent vectors D(x) ∈ T_{X, x}, all x ∈ X such that in local charts

• 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) ∈ T_{x, 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.