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A survey on symplectic singularities and symplectic resolutions. (English) Zbl 1116.14008

This is a survey paper concerning the specialized problem in algebraic geometry called symplectic resolution. If a normal variety \(W\) is a symplectic variety, i.e. its smooth part admits a holomorphic symplectic form \(\omega \) and pull-back of this form to any resolution \(\pi :Z\to W\) extends to a holomorphic 2-form \(\Omega \) on \(Z.\) If additionally the extended 2-form \(\Omega \) is a symplectic form then \(\pi \) is called a symplectic resolution. In the paper under review, the author describes basic properties of symplectic normal varieties and symplectic resolutions. Namikawa’s work, quotient singularities, nilpotent orbit closures, birational geometry, symplectic moduli spaces and the list of some conjectures on symplectic resolutions: on finiteness, on deformations, on cohomology, on derived equivalence and on birational geometry.

MSC:

14E05 Rational and birational maps
18E30 Derived categories, triangulated categories (MSC2010)

References:

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