New Techniques in Geometric Invariant Theory 31st August - 4th September 2015 Abstracts Daniel Halpern-Leistner: Beyond Geometric Invariant Theory Many natural moduli problems in algebraic geometry are too big to admit quasi-projective good moduli spaces -- in technical language they are represented by locally finite type algebraic stacks which are not quasi-compact. Motivated by the classic example of vector bundles on a smooth curve, I will discuss a method of taming such a moduli problem by introducing a stratification of the stack which has a canonical modular interpretation (called a Θ-stratification). Rather than finding a GIT problem which approximates the moduli problem of interest, the goal is to transport as many of the results of GIT as possible to the more general setting of algebraic stacks, where they can be applied directly. I will discuss applications of such stratifications to studying the geometry of a stack, some classical and some more recent. In particular, for smooth stacks the stratification leads to a direct sum decomposition of the cohomology of the stack, and recently this has been categorified to statements about the derived category and K-theory (both algebraic and topological). Introducing a bit of derived algebraic geometry lets one treat the case of algebraic symplectic (or Hyperkaehler) quotients as well. Outline: Lecture 1 - Kempf-Ness stratification in GIT Background on Kempf's canonical one-parameter-subgroups Stratifications: Geometry of the stratification Morse theory perspective Main theorem of GIT from this perspective: existence of stratification whose piece have good moduli spaces Extending these stratifications to moduli problems: Moduli problems as algebraic stacks Quotient stacks as algebraic stacks Classification of vector bundles on a curve Lecture 2 - Categorical Kirwan surjectivity Kirwan's surjectivity theorem Equivariant derived categories of coherent sheaves Statement and proof of categorical Kirwan surjectivity Applications to flips and flops Extension to Landau-Ginzburg models Lecture 3 - Theta-reductive stacks and Theta-stratifications I Modular interpretation of the strata Mapping stack from $\mathbb{A}^1 / \mathbb{G}_m$ The linear case: Harder-Narasimhan filtrations Degeneration space associated to a point in a stack Generalization of fan for a toric variety Explicit computations -- for affine/reductive, a locally polyhedral subspaces of the spherical building Numerical invariants and formulation of the existence and uniqueness question Lecture 4 - Theta-reductive stacks and Theta-stratifications II Existence and uniqueness of optimal destabilizers for Theta-reductive stacks Uses Tannakian formalism and categorical Kirwan surjectivity Convexity implies uniqueness Boundedness implies existence Obtaining $\Theta$-stratifications of $\Theta$-reductive stacks Example: the moduli of objects in the heart of a t-structure Torsion theories, numerical invariants Lecture 5 - Applications Singular versions of Kirwan's surjectivity theorem extension of categorical Kirwan surjectivity using derived algebraic geometry recovering topological K-theory from derived categories non-abelian virtual localization theorems comparison with virtual localization theorems in cohomology Frances Kirwan: Non-reductive geometric invariant theory Mumford's geometric invariant theory (GIT) provides a method for constructing (projective completions of) quotient varieties for linear actions of complex reductive groups on affine and projective varieties, and has many applications (for example in the construction of moduli spaces in algebraic geometry). Mumford's GIT can be extended to actions of linear algebraic groups which are not necessarily reductive, but many of the nice properties belonging to reductive GIT no longer hold for non-reductive actions. The aim of these talks is firstly to describe Mumford's GIT for reductive group actions and extensions to actions of non-reductive groups, then to give some conditions under which the good properties satisfied by reductive GIT still hold for suitable non-reductive actions, and finally to discuss some examples and applications. Much of the material beyond the classical results will be based on joint work with Gergely Berczi, Andrew Dancer, Brent Doran and Tom Hawes. Gergely Berczi: Tautologial integrals on curvilinear Hilbert schemes The punctual Hilbert scheme of k points on a smooth projective variety X parametrises zero dimensional subschemes of X of length k supported at one point. A point of the punctual Hilbert scheme is called curvilinear if it sits in the germ of a smooth curve on X. The irreducible component of the punctual Hilbert scheme containing these points is called the curvilinear component. We give a description of the curvilinear Hilbert scheme as a projective completion of the non-reductive quotient of holomorphic map germs from the complex line into X by holomorphic polynomial reparametrisations. Using an algebraic model of this quotient and equivariant localisation we develop an iterated residue formula for tautological integrals over the curvilinear component. We discuss possible generalisations for other non-reductive moduli problems. Andrew Dancer: Symplectic and Hyperkahler implosion. We review implosion constructions in symplectic and hyperkahler geometry, emphasising the links with non-reductive geometric invariant theory. We also describe some recent work on aproaching hyperkahler implosion via moduli spaces of solutions to Nahm's equations. Brent Doran: Sources and Uses of Unipotent Actions in Algebraic Geometry and Arithmetic There are many directions to extend GIT (symplectic, analytic, stacky, non-reductive etc.) with its original aim - to construct and study moduli spaces as quotients - firmly in mind. We look at some sources of non-reductive group actions naturally arising in algebraic geometry (automorphisms of varieties, affine space bundles that are not algebraic vector bundles, etc.) and demonstrate their application, particularly via invariant theory and GIT, to novel areas beyond moduli problems - in particular, to well-known questions about the geometry of cycles and about counting rational points of bounded height in Diophantine geometry. Emilie Dufresne: Invariants and separating morphisms We study the invariants of an algebraic group action on an affine variety via separating morphisms, that is, dominant G-invariant morphism to another affine variety such that points which are separated by some invariant have distinct image. This is a more geometric take on the study of separating invariants, a new trend in invariant theory initiated by Derksen and Kemper. Our main result a refinement of Winkelmann's work on invariant rings and quotients of algebraic group actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient given by the possibly not finitely generated ring of invariants is "almost'' an algebraic variety, and that the quotient morphism has a number of nice properties. One of the main difficulties comes from the fact that the quotient morphism is not necessarily surjective. These general results are then refined for actions of the additive group, where we can say much more. The most complete results are obtained for representations. We also give a complete and detailed analysis of Roberts' famous example of a an action of the additive group on 7-dimensional affine space with a non-finitely generated ring of invariants. (Joint work with Hanspeter Kraft) Daniel Greb: Complex-analytic quotients of algebraic G-varieties In the 1980's Geometric Invariant Theory has been generalised to actions of complex-reductive groups on complex manifolds and complex spaces. When looking at an algebraic action of such a group G on a variety over the complex numbers, in principle we therefore have more options to take quotients in the category of complex spaces than just good quotients (in the category of algebraic spaces). I will present a number of equivariant GAGA-results stating that in the end every compact complex-analytic quotient is a good quotient. The talk is based on a paper with the same title. Jochen Heinloth: Using a stack theoretic GIT criterion to construct separated coarse moduli One of the applications of GIT is that it allows to single out parts of moduli problems that admit separated coarse moduli spaces. Similar to the methods used in Daniel Halpern-Leistner's talks one can also find stack theoretic criteria allowing to find separated substacks of moduli stacks, which sometimes allow one to avoid explicit GIT constructions. To illustrate the criterion we apply it to some generalizations of principal bundles, known as parahoric group schemes on curves. Kiumars Kaveh: Toric degenerations and symplectic geometry of projective varieties The purpose of this talk is to discuss some recent general results about symplectic geometry of smooth projective varieties using toric degenerations (motivated by commutative algebra). The main result is the following: Let X be a smooth n-dimensional complex projective variety embedded in a projective space and equipped with a Kahler structure induced from a Fubini-Study Kahler form. We show that X has an open subset U (in the usual topology) which is symplectomorphic to the algebraic torus $(\mathbb{C}^*)^n$ equipped with an integral toric Kahler form coming from a monomial embedding. We use this to obtain lower bounds on the Gromov width of X, in particular we show that the Gromov width of X is at least 1. Moreover, we show that X has symplectic packings by balls of capacity $1-\epsilon$. Yoshinori Namikawa: A characterization of nilpotent cones of complex semisimple Lie algebras A complex normal variety X is called a symplectic variety if it admits a holomorphic symplectic 2-form on the regular part and the 2-form extends to a holomorphic 2-form on a resolution Y of X. Compared with the compact case, there are a lot of examples of affine symplectic varieties. They are not only interesting objects in algebraic geometry, but also play important roles in geometric representation theory. The aim of this talk is to characterize the nilpotent variety of a complex semisimple Lie algebra among affine symplectic varieties. The main result is that if X is an affine singular symplectic variety embedded in an affine space as a complete intersection of homogeneous polynomials and the symplectic 2-form is homogeneous, then it coincides with the nilpotent cone N of a complex semisimple Lie algebra together with the Kostant-Kirillov 2-form. The proof of the main result uses the theory of Poisson deformation, holomorphic contact geometry, Mori theory and some elementary representation theory. Gerald Schwarz: Oka Principles and the Linearization Problem Let $X$ and $Y$ be Stein $G$-manifolds where $G$ is a reductive complex Lie group. We find sufficient conditions for $X$ and $Y$ to be $G$-biholomorphic. There is a quotient Stein space $Q_X$, and a morphism $\pi_X\colon X\to Q_X$ such that $(\pi_X)^*\mathcal{O}(Q_X)=\mathcal{O}(X)^G$. Similarly we have $p_Y\colon Y\to Q_Y$. Suppose that $\Phi\colon X\to Y$ is a $G$-biholomorphism. Then the induced mapping $\phi\colon Q_X\to Q_Y$ has the following property: for any $z\in Q_X$, $X_z:=\pi_X^{-1}(z)$ is $G$-isomorphic to $Y_{\phi(z)}$ (the fibers are actually affine $G$-varieties). We say that $\phi$ is admissible. Now given an admissible $\phi$, assume that we have a $G$-equivariant homeomorphism $\Phi\colon X\to Y$ lifting $\phi$. Our goal is to establish an Oka principle, saying that $\Phi$ has a deformation $\Phi_t$ through $G$-equivariant homeomorphisms with $\Phi_0=\Phi$ and $\Phi_1$ biholomorphic. We establish this in two main cases. One case is where $\Phi$ is a diffeomorphism that restricts to $G$-isomorphisms on the reduced fibers of $\pi_X$ and $\pi_Y$. The other case is where $\Phi$ restricts to $G$-isomorphisms on the fibers and $X$ satisfies an auxiliary condition, which usually holds. Finally, we give applications to the Holomorphic Linearization Problem. Let $G$ act holomorphically on $X=\mathbb{C}^n$. When is there a holomorphic change of coordinates such that the action of $G$ becomes linear? We prove that this is true, for $X$ satisfying the same auxiliary condition as before, if and only if the quotient $Q_X$ is admissibly biholomorphic to the quotient of a $G$-module.