STRATIFICATIONS ASSOCIATED TO REDUCTIVE GROUP ACTIONS ON AFFINE SPACES

For a complex reductive group G acting linearly on a complex affine space V with respect to a character ρ, we show two stratifications of V associated to this action (and a choice of invariant inner product on the Lie algebra of the maximal compact subgroup of G) coincide. The first is Hesselink's stratification by adapted 1-parameter subgroups and the second is the Morse theoretic stratification associated to the norm square of the moment map. We also give a proof of a version of the Kempf-Ness theorem, which states that the geometric invariant theory quotient is homeomorphic to the symplectic reduction (both taken with respect to ρ). Finally, for the space of representations of a quiver of fixed dimension, we show that the Morse theoretic stratification and Hesselink's stratification coincide with the stratification by Harder-Narasimhan type

Complete moduli of cubic threefolds and their intermediate Jacobians

The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the moduli space of principally polarized abelian fivefolds. A much better "wonderful" compactification of the space of cubic threefolds was constructed by the first and fourth authors --- it has a modular interpretation, and divisorial normal crossing boundary. We prove that the intermediate Jacobian map extends to a morphism from the wonderful compactification to the second Voronoi toroidal compactification of the moduli of principally polarized abelian fivefolds --- the first and fourth author previously showed that it extends to the Satake compactification. Since the second Voronoi compactification has a modular interpretation, our extended intermediate Jacobian map encodes all of the geometric information about the degenerations of intermediate Jacobians, and allows for the study of the geometry of cubic threefolds via degeneration techniques. As one application we give a complete classification of all degenerations of intermediate Jacobians of cubic threefolds of torus rank 1 and 2.Comment: 56 pages; v2: multiple updates and clarification in response to detailed referee's comment