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) ∈ TX, 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
When X is an abelian variety, then translations on X define isomorphisms
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
For translation-invariant vector fields, the commutativity of X implies that bracket is zero (see Abelian Varieties, D. Mumford, Oxford Univ. Press, p. 100.
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© 2007 Birkhäuser Boston
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Mumford, D. (2007). The translation-invariant vector fields. In: Tata Lectures on Theta II. Modern Birkhäuser Classics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4578-6_4
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DOI: https://doi.org/10.1007/978-0-8176-4578-6_4
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