nLab holomorphic symplectic manifold

Idea

A holomorphic symplectic manifold is a complex manifold equipped with a holomorphic symplectic form.

References

General

Survey:

• Arnaud Beauville, Holomorphic symplectic geometry, 2011 (pdf, pdf)

On Berezin quantization generalized to holomorphic symplectic manifolds:

• Joshua Lackman: Quantization of Holomorphic Symplectic Manifolds: Analytic Continuation of Path Integrals and Coherent States [arXiv:2501.05428]

Rozansky-Witten theory

That Rozansky-Witten invariants, originally conceived only for hyperkähler manifolds, are more generally defined for holomorphic symplectic manifolds is due to

• Mikhail Kapranov, Rozansky–Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71–113, MR2000h:57056, doi, alg-geom/9704009

See also

• Justin Roberts, Simon Willerton, On the Rozansky-Witten weight systems, Algebr. Geom. Topol. 10 (2010) 1455-1519 (arXiv:math/0602653)