given by fiber integration
of the cup product
Over a Riemann surface , the intersection pairing on has a quadratic refinement by the function that sends a Theta characteristic to the mod 2-dimension of its space of sections. See Theta characteristic – Over Riemann surfaces.
For the case that the cohomology in question is ordinary differential cohomology,
the cup product is the Beilinson-Deligne cup product;
The differentially refined intersection pairing is non-trivial and interesting also on manifolds of dimension less than , where the integral intersection pairing vanishes: it provides a secondary characteristic class, a secondary intersection pairing.
|manifold dimension||invariant||quadratic form||quadratic refinement|
|signature genus||intersection pairing||integral Wu structure|
|line bundle||square root||choice corresponds to|
|canonical bundle||Theta characteristic||over Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure|
|density bundle||half-density bundle|
|canonical bundle of Lagrangian submanifold||metalinear structure||metaplectic correction|
|determinant line bundle||Pfaffian line bundle|
|quadratic secondary intersection pairing||partition function of self-dual higher gauge theory||integral Wu structure|