Derived equivalence and non-vanishing loci

The paper proposes and motivates a conjecture on the invariance of cohomological support loci under derived equivalence. It contains a proof in the case of surfaces, and explains further developments and consequences.Comment: 9 pages; to appear in the Clay volume in honor of Joe Harris' 60th birthda

Generalized support varieties for finite group schemes

We construct two families of refinements of the (projectivized) support variety of a finite dimensional module M for a finite group scheme G. For an arbitrary finite group scheme, we associate a family of non maximal rank varieties Γj (G)M, 1 ≤ j ≤ p −1, to a kG-module M. For G infinitesimal, we construct a finer family of locally closed subvarieties V a (G)M of the variety of one parameter subgroups of G for any partition a of dimM. For an arbitrary finite group scheme G, a kG-module M of constant rank, and a cohomology class ζ in H1 (G, M) we introduce the zero locus Z(ζ) ⊂ Π(G). We show that Z(ζ) is a closed subvariety, and relate it to the non-maximal rank varieties. We also extend the construction of Z(ζ) to an arbitrary extension class ζ ∈ Extn G (M,N) whenever M and N are kG-modules of constant Jordan type