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In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field.
For an abelian variety A {\displaystyle A} defined over a field F {\displaystyle F} with ring of integers R {\displaystyle R} , consider the Néron model of A {\displaystyle A} , which is a 'best possible' model of A {\displaystyle A} defined over R {\displaystyle R}. This model may be represented as a scheme over S p e c {\displaystyle \mathrm {Spec} } for which the generic fibre constructed by means of the morphism S p e c → S p e c {\displaystyle \mathrm {Spec} \to \mathrm {Spec} } gives back A {\displaystyle A}. The Néron model is a smooth group scheme, so we can consider A 0 {\displaystyle A^{0}} , the connected component of the Néron model which contains the identity for the group law. This is an open subgroup scheme of the Néron model. For a residue field k {\displaystyle k} , A k 0 {\displaystyle A_{k}^{0}} is a group variety over k {\displaystyle k} , hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus, so that A k 0 {\displaystyle A_{k}^{0}} is a semiabelian variety, then A {\displaystyle A} has semistable reduction at the prime corresponding to k {\displaystyle k}. If F {\displaystyle F} is a global field, then A {\displaystyle A} is semistable if it has good or semistable reduction at all primes.
The semistable reduction theorem of Alexander Grothendieck states that an abelian variety acquires semistable reduction over a finite extension of F {\displaystyle F}.