Holomorphic symplectic form - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold. A holomorphic symplectic form on $X$ is a holomorphic section

\sigma\in H^0\left(X,\bigwedge\nolimits^2T^{*1,0}X\right)

of the second exterior power of the holomorphic cotangent bundle such that $d\sigma=0$ and $\sigma$ is pointwise nondegenerate. Nondegeneracy means that the bundle map $T^{1,0}X\to T^{*1,0}X$ , $v\mapsto\sigma(v,\mathord{-})$ , is an isomorphism. Thus closedness is a differential condition and nondegeneracy is a fiberwise linear-algebra condition; both are part of the definition adopted here.

Immediate consequences

The complex dimension of $X$ must be even, say $2m$ , and

\sigma^m=\underbrace{\sigma\wedge\cdots\wedge\sigma}_{m\ \mathrm{factors}}

is a nowhere-vanishing holomorphic section of the canonical bundle. Hence a holomorphic symplectic form canonically trivializes that bundle. The inverse bivector $\sigma^{-1}$ is holomorphic, and the equation $d\sigma=0$ is equivalent to its Poisson bracket satisfying the Jacobi identity.

Writing $\sigma=\alpha+i\beta$ as real forms on the underlying smooth manifold produces two closed, nondegenerate real $2$ -forms. Therefore both $\alpha$ and $\beta$ are symplectic forms, although neither alone records the full holomorphic structure.

Canonical example

If $Y$ is a complex manifold, the total space of its holomorphic cotangent bundle has a tautological holomorphic $1$ -form $\theta$ . In local coordinates $(z^i,\xi_i)$ ,

\theta=\sum_i\xi_i\,dz^i, \qquad \sigma=d\theta=\sum_i d\xi_i\wedge dz^i.

This $\sigma$ is closed and nondegenerate, so it is a holomorphic symplectic form.

A decisive near-miss

On $\mathbb C^4$ , the holomorphic form

\tau=dz^1\wedge dz^2+e^{z^1}dz^3\wedge dz^4

is nondegenerate everywhere, but

d\tau=e^{z^1}dz^1\wedge dz^3\wedge dz^4\neq0.

It is therefore a nondegenerate holomorphic $2$ -form but not a holomorphic symplectic form under the closedness convention.

References

Daniel Huybrechts, “Compact Hyperkähler Manifolds: Basic Results,” Inventiones Mathematicae 135 (1999), 63–113. DOI record. Relevant: §1, holomorphic symplectic forms and irreducible symplectic manifolds.
Arnaud Beauville, “Holomorphic Symplectic Geometry: A Problem List,” in Complex and Differential Geometry, Springer Proceedings in Mathematics 8, 2011. Author-hosted paper. Relevant: §1.1, basic definitions and canonical forms.