Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSmoothness of non-reductive fixed point sets and cohomology of non-reductive GIT quotients
236
0
0.0
(
0
)
Authors
Eloise Hamilton
Ask ChatGPT about the research
No Arabic abstract
We establish a method for calculating the Poincare series of moduli spaces constructed as quotients of smooth varieties by suitable non-reductive group actions; examples of such moduli spaces include moduli spaces of unstable vector or Higgs bundles on a smooth projective curve, with a Harder-Narasimhan type of length two. To do so, we first prove a result concerning the smoothness of fixed point sets for suitable non-reductive group actions on smooth varieties. This enables us to prove that quotients of smooth varieties by such non-reductive group actions, which can be constructed using Non-Reductive GIT via a sequence of blow-ups, have at worst finite quotient singularities. We conclude the paper by providing explicit formulae for the Poincare series of these non-reductive GIT quotients.
rate research
Read More
Let $H$ be a complex linear algebraic group with internally graded unipotent radical acting on a complex projective variety $X$. Given an ample linearisation of the action and an associated Fubini-Study Kahler form which is invariant for a maximal compact subgroup $Q$ of $H$, we define a notion of moment map for the action of $H$, and under suitable conditions (that the linearisation is well-adapted and semistability coincides with stability) we describe the (non-reductive) GIT quotient $X/!/H$ introduced by Berczi, Doran, Hawes and Kirwan in terms of this moment map. Using this description we derive formulas for the Betti numbers of $X/!/H$ and express the rational cohomology ring of $X/!/H$ in terms of the rational cohomology ring of the GIT quotient $X/!/T^H$, where $T^H$ is a maximal torus in $H$. We relate intersection pairings on $X/!/H$ to intersection pairings on $X/!/T^H$, obtaining a residue formula for these pairings on $X/!/H$ analogous to the residue formula of Jeffrey-Kirwan. As an application, we announce a proof of the Green-Griffiths-Lang and Kobayashi conjectures for projective hypersurfaces with polynomial degree.
The wall-and-chamber structure of the dependence of the reductive GIT quotient on the choice of linearisation is well known. In this article, we first give a brief survey of recent results in non-reductive GIT, which apply when the unipotent radical is graded. We then examine the dependence of these non-reductive quotients on the linearisation and an additional parameter, the choice of one-parameter subgroup grading the unipotent radical, and arrive at a picture similar to the reductive one.
The Green--Griffiths--Lang and Kobayashi hyperbolicity conjectures for generic hypersurfaces of polynomial degree are proved using intersection theory for non-reductive geometric invariant theoretic quotients and recent work of Riedl and Yang.
We combine recently developed intersection theory for non-reductive geometric invariant theoretic quotients with equivariant localisation to prove a formula for Thom polynomials of Morin singularities. These formulas use only toric combinatorics of certain partition polyhedra, and our new approach circumvents the poorly understood Borel geometry of existing models.
We classify affine varieties with an action of a connected, reductive algebraic group such that the group is isomorphic to an open orbit in the variety. This is accomplished by associating a set of one-parameter subgroups of the group to the variety, characterizing such sets, and proving that sets of this type correspond to affine embeddings of the group. Applications of this classification to the existence of morphisms are then given.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
