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The dual abelian variety, the Poincare bundle and polarizations

Statement

Assume AC and DC as inherited from projectivity and the supplied cohomology machinery. Let $A$ be an abelian variety over a field $k$ , of dimension $g$ . Then:

(a) [existence and duality] the degree-zero part of the rigidified relative Picard functor of $A / k$ (The rigidified relative Picard functor and the dual abelian variety) is representable by an abelian variety $\hat{A}$ , the dual abelian variety, of dimension $g$ , with universal Poincare sheaf $P$ on $A \times_{k} \hat{A}$ ; the canonical homomorphism $A \to \hat{\hat{A}}$ is an isomorphism;

(b) [functoriality] $A \mapsto \hat{A}$ is a contravariant functor on abelian varieties over $k$ , and for every isogeny $f : A \to B$ the dual $f^{\lor} : \hat{B} \to \hat{A}$ is an isogeny with kernel the Cartier dual of $\ker f$ and degree $\deg f^{\lor} = \deg f$ ;

(c) [Mumford maps] for every invertible sheaf $L$ on $A$ the Mumford homomorphism $φ_{L} : A \to \hat{A}$ exists; if $L$ is ample then $φ_{L}$ is a symmetric isogeny with finite kernel $K (L)$ ; every symmetric homomorphism $A \to \hat{A}$ is $φ_{L}$ for some invertible sheaf $L$ after base change to a separably closed field;

(d) [polarizations] an ample $L$ makes $φ_{L}$ a polarization, every abelian variety admits a polarization, the degree of a polarization is a perfect square, and $A$ is projective.

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ of dimension $g$ over a field $k$ , and the rigidified relative Picard functor of The rigidified relative Picard functor and the dual abelian variety.

[F1]

The algebraically trivial rigidified Picard subfunctor is represented by an abelian variety $A^{\lor}$ of dimension $g$ with a normalized Poincare bundle on $A \times_{k} A^{\lor}$ , and the formation and universal property are compatible with field extension (Finite-field descent of the dual and the Poincare bundle).

[F2]

Duality is contravariantly functorial on homomorphisms: for composable homomorphisms $f : A \to B$ and $g : B \to C$ , $(g \circ f)^{\lor} = f^{\lor} \circ g^{\lor}$ ; it is additive for parallel homomorphisms $f, g : A \to B$ , so $(f + g)^{\lor} = f^{\lor} + g^{\lor}$ . If $f$ is an isogeny, then $f^{\lor}$ is an isogeny with kernel $(\ker f)^{D}$ and degree $\deg f^{\lor} = \deg f$ . The canonical biduality morphism $κ_{A} : A \to A^{\lor \lor}$ is an isomorphism and is natural in $A$ (Dual isogenies, Cartier-dual kernels and canonical biduality, Normal subgroup quotients of finite-type group schemes exist as fppf scheme quotients).

[F3]

The theorem of the square makes $φ_{L}$ a homomorphism into the degree-zero part for every invertible sheaf $L$ (The theorem of the square and the Mumford homomorphism into the Picard group); ample bundles give symmetric isogenies and every symmetric homomorphism is a Mumford map over a separably closed field, with finite separable realization in general (Polarizations and ampleness under Picard twists, Symmetric homomorphisms are Mumford maps).

[F4]

Every abelian variety is projective and therefore carries an ample invertible sheaf; the degree of any polarization equals $χ (A, L)^{2}$ for an ample $L$ realizing it (Every abelian variety over a field is projective, The square degree of a Mumford map, Polarizations and the Mumford isogeny attached to an ample line bundle).

Proof

technique · direct: assemble the commissioned clauses from the constructed dual, the dual-isogeny calculus, the symmetric-map realization and the square-degree computation

1.1F1F2givenconstruct

Clause (a) is [F1]: the degree-zero rigidified Picard subfunctor is represented by an abelian variety $\hat{A} = A^{\lor}$ of dimension $\dim A = g$ with universal normalized Poincare sheaf $P$ on $A \times_{k} A^{\lor}$ , and the formation is compatible with field extension. The canonical morphism $κ_{A} : A \to A^{\lor \lor}$ is an isomorphism by [F2], which is the biduality statement of (a).

1.2F2givenalgebra

Clause (b) is [F2]: pullback of rigidified bundles defines the dual homomorphism for every homomorphism. Composition is contravariant for composable homomorphisms $f : A \to B$ , $g : B \to C$ , and additivity $(f + g)^{\lor} = f^{\lor} + g^{\lor}$ is for parallel homomorphisms $f, g : A \to B$ . When $f$ is an isogeny, $f^{\lor}$ is an isogeny with kernel $(\ker f)^{D}$ and degree $\deg f$ . Thus $A \mapsto \hat{A}$ is a contravariant functor, and the duality identities used here have the required domains.

1.3F3givenconstruct

Clause (c): for an invertible sheaf $L$ the Mumford map $φ_{L}$ is a homomorphism into $\hat{A}$ by [F3]; if $L$ is ample, [F3] gives that $φ_{L}$ is a symmetric isogeny with finite kernel $K (L)$ . Conversely, if $λ : A \to \hat{A}$ is symmetric, then over a separably closed extension field [F3] realizes $λ$ as $φ_{L}$ for an invertible $L$ ; over an arbitrary field the realization exists after a finite separable extension in general, as stated.

2.1F3F4givenalgebra∎

Clause (d): if $L$ is ample, [F3] shows that $φ_{L}$ is a symmetric isogeny with $(id, φ_{L})^{*} P$ ample, so it is a polarization by definition; every $A$ is projective by [F4] and hence carries an ample $L$ , giving a polarization. For any polarization $λ$ realized by an ample $L$ over an algebraic closure, [F4] gives $\deg λ = χ (A_{\overset{ˉ}{k}}, L)^{2}$ , a perfect square, and the degree is unchanged by field extension. Projectivity of $A$ is [F4].

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Sources

B. Edixhoven, G. van der Geer, B. Moonen, Abelian Varieties (2012), Chapters 6, 8, 9 and 11 (dual abelian variety, Poincare bundle, Mumford maps and polarizations) (standard reference, not scraped)
S. Bosch, W. Lutkebohmert, M. Raynaud, Neron Models (1990), 8.1 (rigidified relative Picard functor and dual abelian variety) (standard reference, not scraped)