nLab symplectic singularity

Idea

A normal variety is called a symplectic variety? if its smooth part admits a holomorphic symplectic form, whose pull-back to any resolution extends to a holomorphic 2-form. If this 2-form is itself symplectic, the given resolution is called a symplectic resolution. More generally, an algebraic variety is said to have symplectic singularities if every point in the variety has a neighborhood which carries a symplectic structure.

Symplectic resolutions are analogous to hyper Kähler manifolds?.

Related concepts

• symplectic duality

References

The term was introduced in

• Arnaud Beauville, Symplectic singularities (1999) (arXiv:math/9903070)

Useful surveys include:

• Dmitry Kaledin, Geometry and topology of symplectic resolutions, Algebraic Geometry: 2005 Summer Research Institute, edited by Dan Abramovich (arXiv:math.AG/0608143)
• Baohua Fu, A survey on symplectic singularities and symplectic resolution, Annales Mathématiques Blaise Pascal 13 (2006) (web)

from which some of the above text is taken.