# 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

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

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