nLab p-convex polarization

Idea

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

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

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

$$\omega = \tfrac{i}{2} \sum_{\alpha,\beta} h_{\alpha,\beta} d z^\alpha \wedge d\bar z^\beta$$

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

Hence for $p = 0$ then a $p$-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 $p$-convex polarization but not in general a Kähler polarization (Griffiths 68b).

Related concepts

• real polarization

• polarized variety

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)