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# Dmitry Kaledin

## Symplectic singularities in algebraic geometry

Let $Y$ be an affine algebraic variety, and let $X$ be a
smooth resolution of singularities of the variety $Y$. Singularities
of $Y$ are said to be symplectic if $X/Y$ carries a closed
holomorphic 2-form which is non-degenerate outside of the
exceptional set. It was discovered recently by A. Beauville,
Y. Namikawa, J. Wierzba and others that symplectic singularities
possess a rich internal structure -- it is possible to claim and
prove much stronger results than one would expect from the viewpoint
of usual algebraic geometry. We will give an overview of these
recent advances and outline directions of further research. We will
pay particular attention to the especially interesting case when the
symplectic form on $X$ is non-degenerate everywhere, and even
stronger, to the case when the variety $Y$ is the quotient $Y=V/G$
of a symplectic vector space $V$ by an action of a finite group $G$.