geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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
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.
The Griffiths intermediate Jacobians (see there for more) naturally carry a $p$-convex polarization but not in general a Kähler polarization (Griffiths 68b).
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.