Kahler

Tags
- #physics #geomtop/differential-geometry #AG/complex
Refs:
- https://people.maths.ox.ac.uk/brantner/hodge.pdf #resources/notes
Links:
- supersymmetry
- almost complex structure
- Calabi-Yau
- Kahler problem
- Dolbeaut cohomology
- Kobayashi-Hitchin correspondence
- Higgs bundle
- nonabelian Hodge theorem
- Hodge index theorem
- The package:

Kahler

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Relation to Hodge theory: attachments/Pasted%20image%2020220424181951.png

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These can all fail for singular varieties, leading to intersection cohomology.

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Examples

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attachments/Pasted%20image%2020220422233320.png # hyperkahler

For orbifolds, a hyperkahler resolution ensures that Gromov-Witten invariants vanish.

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Examples

The only compact hyperkahler fourfold is the K3 surface.

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Lefschetz decomposition and Hard Lefschetz

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See hard Lefschetz

Links to this page

weak and hard Lefschetz theorems

For symplectic manifolds \((X, \omega)\) of real dimension \(2n\): define a map \(L: H_\mathrm{dR}(X) \to \Sigma^{2}H_{\mathrm{dR}}(X)\) by \([\alpha] \mapsto [\omega \wedge \alpha]\). Then the iterates \(L^i\) restrict to \(L^i: H^{n-i}_\mathrm{dR}(X) \to H_\mathrm{dR}^{n+i}(X)\), and Hard Lefschetz states that this is an isomorphism for compact Kahlers.
unresolved links output

Kobayashi-Hitchin correspondence in Unsorted/Kahler

fourfold in Unsorted/Kahler

nonabelian Hodge theorem in Unsorted/Kahler
p-adic Hodge theory

kahler
kahler identities

Tags: ? Refs: ? Links: kahler
character variety

See Kahler manifolds and hard Lefschetz.
Talk Surface bundles, mapping class groups, moduli spaces, and cohomology

Note: the moduli space is a Kähler manifold, so you get the Weil-Peterson 2-form which is the Kähler class in \(H^2\).
Mirror Symmetry

Calabi-Yau, kahler,
Hodge theory

See kahler manifold
Hodge diamonds

kahler
Hodge decomposition

For kahler manifolds:
Hitchin moduli space

hyperkahler
CY metric

In terms of a kahler metric.

#physics #geomtop/differential-geometry #AG/complex #resources/notes