nLab
p-convex polarization

Contents

Context

Geometric quantization

Complex geometry

Symplectic geometry

Contents

Idea

The concept of pp-convex polarization is the generalization of the concept of Kähler polarization to possibly indefinite Hermitean metrics.

For a Hermitean metric with precisely pp negative eigenvalues, one speaks of pp-convex polarization (e.g. Griffiths survey, p. 3).

Hence for (X,h)(X,h) a Hermitean manifold with Hermitean metric of signature (dim(X)p,p)(dim(X)-p,p), then a pp-convex polarization of it is a symplectic form ω\omega which in every local coordinate chart has the form

ω=i2 α,βh α,βdz αdz¯ β \omega = \tfrac{i}{2} \sum_{\alpha,\beta} h_{\alpha,\beta} d z^\alpha \wedge d\bar z^\beta

where (h α,β)(h_{\alpha,\beta}) are the components of the metric in the given chart.

Hence for p=0p = 0 then a pp-convex polarization reduces to a Kähler polarization or, algebraically, to a polarized variety.

Examples

The Griffiths intermediate Jacobians (see there for more) naturally carry a pp-convex polarization but not in general a Kähler polarization (Griffiths 68b).

References

  • Phillip Griffiths, Periods of integrals on algebraic manifolds. I Construction and properties of the modular varieties“, American Journal of Mathematics 90 (2): 568–626, (1968) doi:10.2307/2373545, ISSN 0002-9327, JSTOR 2373545, MR 0229641

  • Phillip GriffithsPeriods of integrals on algebraic manifolds. II Local study of the period mapping“, American Journal of Mathematics 90 (3): 805–865 (1968), doi:10.2307/2373485, ISSN 0002-9327, JSTOR 2373485, MR 0233825

  • Phillip Griffiths, On the periods of integrals on algebraic manifolds, survey (pdf)

Created on June 17, 2014 at 09:20:32. See the history of this page for a list of all contributions to it.