nLab
polarized algebraic variety

Contents

Context

Complex geometry

Geometric quantization

Contents

Idea

A polarization on an algebraic variety is a choice of line bundle on it whose Chern class is analogous to the class represented by a Kähler form in complex analytic geometry.

In other words, the concept of polarized algebraic variety is the generalization of that of Kähler polarization from symplectic geometry/complex geometry to more general algebraic geometry. In fact it is the generalization of the concept of a holomorphic prequantum line bundle compatible with a Kähler polarization.

(Notice however the reversion of the logic: in symplectic geometry the symplectic form is given and then a complex structure is chosen to match it, whereas here in algebraic geometry the analog of the complex structure exists beforehand, given by the algebraic structure, and now a polarization conversely asks for the analog of a compatible symplectic form (and for its prequantization)).

Therefore a polarized variety together with a choice of Theta characteristic on it is the algebraic geometry version of a polarized phase space with metaplectic correction. See at geometric quantization for more on this.

Definition

A polarization of an algebraic variety XX over an algebraically closed field kk is a choice of ample element in its Néron-Severi group Pic X/Pic X 0Pic_X/Pic_X^0. In other words, a polarization is the choice of an equivalence class of an ample algebraic line bundle over XX (e.g. holomorphic line bundles if kk is the complex numbers) where two such are regarded as equivalent if they differ by tensoring with one whose underlying topological Chern class is trivial.

The polarization is called principal if it is of degree 1.

Examples

Jacobians and higher Jacobians

The Jacobian variety of an algebraic variety is principally polarized by the theta divisor.

See also this MO discussion.

More generally the higher intermediate Jacobians with their Weil complex structure are polarized. On the other hand, when equipped with the Griffith complex structure they carry a p-convex polarization which is symplectomorphic but not biholomorphic to the Weil polarization.

References

Polarization specifically of Jacobian varieties is discussed for instance in

  • Alexander Polishchuk, section 17 of Abelian varieties, Theta functions and the Fourier transform, Cambridge University Press (2003) (pdf)

Last revised on February 15, 2015 at 06:27:32. See the history of this page for a list of all contributions to it.