For elliptic surfaces with two multiple fibers of multiplicities 2 and m at least 3 and with chi equal to 1, every singular point of the moduli space M(c2) is canonical, and for compact M(c2) the Kodaira dimension is (dim M(c2)+1)/2.
Formality conjecture for K3 surfaces
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abstract
We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the DG algebra RHom(F,F) is formal for any sheaf F polystable with respect to an ample line bundle. Our main tool is the uniqueness of DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.
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The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces
For elliptic surfaces with two multiple fibers of multiplicities 2 and m at least 3 and with chi equal to 1, every singular point of the moduli space M(c2) is canonical, and for compact M(c2) the Kodaira dimension is (dim M(c2)+1)/2.