Let $X$ be a smooth connected projective $K3$ surface over the complex numbers and let $Spl(r; c_1, c_2)$ be the moduli space of simple sheaves on $X$ of fixed rank $r$ and Chern classes $c_1$ and $c_2$. In 1984, Mukai proved that $Spl(r; c_1, c_2)$ is a smooth algebraic space of dimension $2rc_2-(r-1)c_1^2-2r^2+2$ with a natural symplectic sstricture, i.e., it has a non-degenerate closed holomorphic 2-form. In my talk, I will present a useful method to construct isotropic and Lagrangian subspaces of $Spl(r; c_1, c_2)$. This is joint work with Barbara Fantechi.

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Website: http://math.ipm.ac.ir/agnt/