Let G be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic p, assumed to be larger than the Coxeter number. The "support variety" of a G-module M is a certain closed subvariety of the nilpotent cone of G, defined in terms of cohomology for the first Frobenius kernel G 1 . In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for G = SLn in earlier work of the second author.In this paper, we show that for any G, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when p is sufficiently large. We also prove variants of these statements involving "relative support varieties."1.2. The Humphreys conjecture. We fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ G, denote by X the character lattice of T , and let X + ⊂ X be the subset of dominant weights (for the choice of positive roots such that B is the P.A.

We construct a co-t-structure on the derived category of coherent sheaves on the nilpotent cone, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the "exotic parity objects" (considered in [AHR1]), along the (classical) Springer resolution, give indecomposable objects inside the coheart of the co-t-structure on N . We also demonstrate how the various parabolic co-t-structures can be related by introducing an analogue to the usual translation functors.Additional applications include a proof of the scheme-theoretic formulation of Humphreys conjecture on support varieties of tilting modules in type A for p > h, as well as a verification of the conjecture in arbitrary type, (for p > h), over a large class of orbits, which includes all Richardson orbits. We also obtain new information regarding the structure of certain p-canonical cells and thick-tensor-ideals of tilting modules.

In this paper, we carry out several computations involving graded (or Gm-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the Gm-action on certain normalized (or "canonical") simple objects, confirming an old prediction of Ostrik. In the second part of the paper, we explicitly describe all simple perverse coherent sheaves for G = P GL 3 , in every characteristic other than 2 or 3. Applications include an explicit description of the cohomology of tilting modules for the corresponding quantum group, as well as a proof that PCoh Gm (N ) never admits a positive grading when the characteristic of the field is greater than 3. P.A. was supported by NSF Grant No. DMS-1500890. Preliminaries on nilpotent orbits2.1. Notation and conventions. Let k be an algebraically closed field. For a graded k-vector space V = m∈Z V m , we define V n to be the graded k-vector space given by (V n ) m = V m+n . It is sometimes convenient to think of V as a

Let G be a reductive algebraic group scheme defined over Fp and let G1 denote the Frobenius kernel of G. To each finite-dimensional G-module M , one can define the support variety VG 1 pM q, which can be regarded as a G-stable closed subvariety of the nilpotent cone.

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