We study the class of the classifying stack of a finite group in a Grothendieck group of algebraic stacks introduced previously. We show that this class is trivial in a number of examples most notably for all symmetric groups. We also give some examples where it is not trivial. The latter uses counterexamples of Saltman and Swan to the problem of Noether.
This survey paper explains how one can attach geometric invariants to semialgebraic sets defined over non-archimedean fields, using the theory of motivic integration of Hrushovski and Kazhdan. It also discusses tropical methods to compute these invariants in concrete cases, as well as an application to refined curve counting, developed in collaboration with Sam Payne and Franziska Schroeter.
In this paper we consider plane quartics with to involutions. We compute the Dixmier invariants, the bitangents and the Matrix representation problem of these curves, showing that they have symbolic solutions for the last two questions.
Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to construct moduli spaces, and explain two new developments extending this theory to non-reductive groups and to stacks, which enable the construction of new moduli spaces.
Given a normal surface singularity (X,0), its link, M is a closed differentiable three dimensional manifold which carries much analytic information. It is an interesting question to ask whether, under suitable analytic and topological conditions, the geometric genus (or other analytic invariants) can be recovered from the link. The Casson invariant conjecture predicts that p_g can be identified using the Casson invariant in the case when (X,0) is a complete intersection and M has trivial first homology with integral coefficients. The Seiberg-Witten invariant conjecture predicts that the geomet…
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 study the Borel moment map $μ_B:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*$, given by $(r,s,i,j)\mapsto [r,s]+ij$, and describe our algorithm to construct the geometric invariant theory (GIT) quotients $μ_B^{-1}(0)/\!\!/_{\det}B$ and $μ_B^{-1}(0)/\!\!/_{\det^{-1}}B$, and the affine quotient $μ_B^{-1}(0)/\!\!/B$. We also provide an insight of the singular locus of $2^n$ irreducible components of $μ_B$. Finally, analogous to the Hilbert--Chow morphism, we discuss that the GIT quotient for the Borel setting is a resolution of singularities.
Perverse schobers are categorifications of perverse sheaves. We construct a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation X of a reductive group G, extending the local system of triangulated categories established by them. The triangulated categories appearing in our perverse schober are subcategories of the derived category of the quotient stack X/G.
In this paper, we explicitly prove that statistical manifolds, related to exponential families and with flat structure connection have a Frobenius manifold structure. This latter object, at the interplay of beautiful interactions between topology and quantum field theory, raises natural questions, concerning the existence of Gromov--Witten invariants for those statistical manifolds. We prove that an analog of Gromov--Witten invariants for those statistical manifolds (GWS) exists. Similarly to its original version, these new invariants have a geometric interpretation concerning intersection poi…
In [arXiv:2008.04625] the authors constructed a classifying space for polystable holomorphic vector bundles on a compact Kähler manifold using analytic GIT theory. The aim of this article is to show that this classifying space taken in the weakly normal category is a coarse moduli space in the sense of complex geometry when the topology is fixed as induced by the space of Hermite-Einstein connections modulo the group of unitary gauge transformations.
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to distinguish between two different ones. The aim of this note is to generalize the classical concept of curvature and torsion to so-called higher algebraic curvatures and to show that each analytic branch of a complex curve is already uniquely defined by them.
This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed.
For linear actions of real reductive Lie groups we prove the Kempf-Ness Theorem about closed orbits and the Kirwan-Ness Stratification Theorem of the null cone. Since our completely self-contained proof focuses strongly on geometric and analytic methods, essentially avoiding any deep algebraic result, it applies also to non-rational linear actions.
Given a compact Kähler manifold, Geometric Invariant Theory is applied to construct analytic GIT-quotients that are local models for a classifying space of (poly)stable holomorphic vector bundles containing the coarse moduli space of stable bundles as an open subspace. For local models invariant generalized Weil-Petersson forms exist on the parameter spaces, which are restrictions of symplectic forms on smooth ambient spaces. If the underlying Kähler manifold is of Hodge type, then the Weil-Petersson form on the moduli space of stable vector bundles is known to be the Chern form of a certain d…
