arXiv subjects

Gerald W. Schwarz

Publications and source records attributed to Gerald W. Schwarz.

At least 19 recordsLinked to original sources

arXiv2026

Parametric equivariant Oka principle

Let $G$ be a reductive complex Lie group and $K$ be a maximal compact subgroup of $G$. Let $X$ be a reduced Stein $G$-space and $Y$ be a $G$-elliptic manifold. We prove the following parametric equivariant Oka principle. The inclusion of the space of holomorphic $G$-maps $X\to Y$ into the space of continuous $K$-maps $X\to Y$ is a weak homotopy equivalence with respect to the compact-open topology. The proof is divided into a homotopy-theoretic part, which is handled by an abstract theorem of Studer, and an analytic part, for which we prove equivariant versions of the homotopy approximation theorem and the nonlinear splitting lemma that are key tools in Oka theory. The principle can be strengthened so as to allow interpolation on a $G$-invariant subvariety of $X$ and approximation on a $K$-invariant holomorphically convex compact subset of $X$.

math.CV↗

arXiv2023

Gromov's Oka principle for equivariant maps

We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that they satisfy all the expected basic properties, and present examples. Our main theorem is an equivariant Oka principle saying that if a finite group $G$ acts on a Stein manifold $X$ and another manifold $Y$ in such a way that $Y$ is $G$-Oka, then every $G$-equivariant continuous map $X\to Y$ can be deformed, through such maps, to a $G$-equivariant holomorphic map. Approximation on a $G$-invariant holomorphically convex compact subset of $X$ and jet interpolation along a $G$-invariant subvariety of $X$ can be built into the theorem. We conjecture that the theorem holds for actions of arbitrary reductive complex Lie groups and prove partial results to this effect.

math.CV↗

arXiv2023

Isomorphisms of Symplectic Torus Quotients

We call a reductive complex group $G$ quasi-toral if $G^0$ is a torus. Let $G$ be quasi-toral and let $V$ be a faithful $1$-modular $G$-module. Let $N$ (the shell) be the zero fiber of the canonical moment mapping $μ\colon V\oplus V^*\to\mathfrak{g}^*$. Then $N$ is a complete intersection variety with rational singularities. Let $M$ denote the categorical quotient $N/\!\!/ G$. We show that $M$ determines $V\oplus V^*$ and $G$, up to isomorphism, if $\operatorname{codim}_N N_\mathrm{sing}\geq 4$. If $\operatorname{codim}_NN_\mathrm{sing}=3$, the lowest possible, then there is a process to produce an algebraic (hence quasi-toral) subgroup $G'\subset G$ and a faithful $1$-modular $G'$-submodule $V'\subset V$ with shell $N'$ such that $\operatorname{codim}_{N'}(N')_\mathrm{sing}\geq 4$. Moreover, there is a $G'$-equivariant morphism $N'\to N$ inducing an isomorphism $N'/\!\!/ G'\xrightarrow{\sim} N/\!\!/ G$. Thus, up to isomorphism, $M$ determines $V'\oplus (V')^*$ and $G'$, hence also $N'$. We establish similar results for real shells and real symplectic quotients associated to unitary modules for compact Lie groups.

math.SG↗

arXiv2023

When does the zero fiber of the moment map have rational singularities?

Let $G$ be a complex reductive group and $V$ a $G$-module. There is a natural moment mapping $μ\colon V\oplus V^*\to\mathfrak{g}^*$ and we denote $μ^{-1}(0)$ (the shell) by $N_V$. We use invariant theory and results of Mustaţă [Mus01] to find criteria for $N_V$ to have rational singularities and for the categorical quotient $N_V /\!\!/ G$ to have symplectic singularities, the latter results improving upon [HSS20]. It turns out that for ``most'' $G$-modules $V$, the shell $N_V$ has rational singularities. For the case of direct sums of classical representations of the classical groups, $N_V$ has rational singularities and $N_V /\!\!/ G$ has symplectic singularities if $N_V$ is a reduced and irreducible complete intersection. Another important special case is $V=p\,\mathfrak{g}$ (the direct sum of $p$ copies of the Lie algebra of $G$) where $p\geq 2$. We show that $N_V$ has rational singularities and that $N_V /\!\!/ G$ has symplectic singularities, improving upon results of [Bud19], [AA16], [Kap19] and [GH20]. Let $π=π_1(Σ)$ where $Σ$ is a closed Riemann surface of genus $p\geq 2$. Let $G$ be semisimple and let $\operatorname{Hom}(π,G)$ and $\mathscr X\!(π,G)$ be the corresponding representation variety and character variety. We show that $\operatorname{Hom}(π,G)$ is a complete intersection with rational singularities and that $\mathscr X\!(π,G)$ has symplectic singularities. If $p>2$ or $G$ contains no simple factor of rank $1$, then the singularities of $\operatorname{Hom}(π,G)$ and $\mathscr X\!(π,G)$ are in codimension at least four and $\operatorname{Hom}(π,G)$ is locally factorial. If, in addition, $G$ is simply connected, then $\mathscr X\!(π,G)$ is locally factorial.

math.AG↗

arXiv2021

Equivariant Oka theory: Survey of recent progress

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group $G$ acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle $E$ of homogeneous spaces for a group bundle $\mathscr G$, all over a reduced Stein space $X$ with compatible actions of a reductive complex group on $E$, $\mathscr G$, and $X$. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov's Oka principle based on a new notion of a $G$-manifold being $G$-Oka.

math.CV↗

arXiv2021

A characterization of linearizability for holomorphic $\mathbb{C}^*$-actions

Let $G$ be a reductive complex Lie group acting holomorphically on $X=\mathbb{C}^n$. The (holomorphic) Linearization Problem asks if there is a holomorphic change of coordinates on $\mathbb{C}^n$ such that the $G$-action becomes linear. Equivalently, is there a $G$-equivariant biholomorphism $Φ\colon X\to V$ where $V$ is a $G$-module? There is an intrinsic stratification of the categorical quotient $X /\!/G$, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of $G$. Suppose that there is a $Φ$ as above. Then $Φ$ induces a biholomorphism $ϕ\colon X/\!/G\to V/\!/G$ which is stratified, i.e., the stratum of $ X/\!/G$ with a given label is sent isomorphically to the stratum of $V/\!/G$ with the same label. The counterexamples to the Linearization Problem construct an action of $G$ such that $X/\!/G$ is not stratified biholomorphic to any $V/\!/G$. Our main theorem shows that, for a reductive group $G$ with $G^0=\mathbb{C}^*$, the existence of a stratified biholomorphism of $X/\!/G$ to some $V/\!/G$ is not only necessary but also sufficient for linearization. In fact, we do not have to assume that $X$ is biholomorphic to $\mathbb{C}^n$, only that $X$ is a Stein manifold.

math.CV↗

arXiv2019

Symplectic quotients have symplectic singularities

Let $K$ be a compact Lie group with complexification $G$, and let $V$ be a unitary $K$-module. We consider the real symplectic quotient $M_0$ at level $0$ of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of $M_0$. We show that if $(V, G)$ is $3$-large, a condition that holds generically, then the complex symplectic quotient has symplectic singularities and is graded Gorenstein. This in particular implies that the real symplectic quotient is graded Gorenstein. In the case that $K$ is a torus or $\operatorname{SU}_2$, we show that these results hold without the hypothesis that $(V,G)$ is $3$-large.

math.SG↗

arXiv2017

Homotopy principles for equivariant isomorphisms

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. When is there an equivariant biholomorphism of $X$ and $Y$? A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are biholomorphic and that the biholomorphism $ϕ$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $ϕ$. We demonstrate two homotopy principles in this situation. The first result says that if there is a $G$-diffeomorphism $Φ\colon X\to Y$, inducing $ϕ$, which is $G$-biholomorphic on the reduced fibres of the quotient mappings, then $Φ$ is homotopic, through $G$-diffeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. The second result roughly says that if we have a $G$-homeomorphism $Φ\colon X\to Y$ which induces a continuous family of $G$-equivariant biholomorphisms of the fibres $p_X^{-1}(q)$ and $p_Y^{-1}(ϕ(q))$ for $q\in Q_X$ and if $X$ satisfies an auxiliary property (which holds for most $X$), then $Φ$ is homotopic, through $G$-homeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. Our results improve upon earlier work of the authors and use new ideas and techniques.

math.CV↗

arXiv2017

An Oka principle for Stein G-manifolds

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. Assume that we have a biholomorphism $Q:= Q_X\to Q_Y$ and an open cover $\{U_i\}$ of $Q$ and $G$-biholomorphisms $Φ_i\colon p_X^{-1}(U_i)\to p_Y^{-1}(U_i)$ inducing the identity on $U_i$. There is a sheaf of groups $\mathcal A$ on $Q$ such that the isomorphism classes of all possible $Y$ is the cohomology set $H^1(Q,\mathcal A)$. The main question we address is to what extent $H^1(Q,\mathcal A)$ contains only topological information. For example, if $G$ acts freely on $X$ and $Y$, then $X$ and $Y$ are principal $G$-bundles over $Q$, and Grauert's Oka Principle says that the set of isomorphism classes of holomorphic principal $G$-bundles over $Q$ is canonically the same as the set of isomorphism classes of topological principal $G$-bundles over $Q$. We investigate to what extent we have an Oka principle for $H^1(Q,\mathcal A)$.

math.CV↗

arXiv2016

An equivariant parametric Oka principle for bundles of homogeneous spaces

We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle $E$ with a structure group bundle $\mathscr G$ on a reduced Stein space $X$, such that the fibre of $E$ is a homogeneous space of the fibre of $\mathscr G$, with the complexification $K^\mathbb C$ of a compact real Lie group $K$ acting on $X$, $\mathscr G$, and $E$. Our main result is that the inclusion of the space of $K^\mathbb C$-equivariant holomorphic sections of $E$ over $X$ into the space of $K$-equivariant continuous sections is a weak homotopy equivalence. The result has a wide scope; we describe several diverse special cases. We use the result to strengthen Heinzner and Kutzschebauch's classification of equivariant principal bundles, and to strengthen an Oka principle for equivariant isomorphisms proved by us in a previous paper.

math.CV↗

arXiv2016

Sufficient Conditions for Holomorphic Linearisation

Let $G$ be a reductive complex Lie group acting holomorphically on $X={\mathbb C}^n$. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ${\mathbb C}^n$ such that the $G$-action becomes linear. Equivalently, is there a $G$-equivariant biholomorphism $Φ\colon X\to V$ where $V$ is a $G$-module? There is an intrinsic stratification of the categorical quotient $Q_X$, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of $G$. Suppose that there is a $Φ$ as above. Then $Φ$ induces a biholomorphism $ϕ\colon Q_X\to Q_V$ which is stratified, i.e., the stratum of $Q_X$ with a given label is sent isomorphically to the stratum of $Q_V$ with the same label. The counterexamples to the Linearisation Problem construct an action of $G$ such that $Q_X$ is not stratified biholomorphic to any $Q_V$. Our main theorem shows that, for most $X$, a stratified biholomorphism of $Q_X$ to some $Q_V$ is sufficient for linearisation. In fact, we do not have to assume that $X$ is biholomorphic to ${\mathbb C}^n$, only that $X$ is a Stein manifold.

math.CV↗

arXiv2015

Jet schemes and invariant theory

Let $G$ be a complex reductive group and $V$ a $G$-module. Then the $m$th jet scheme $G_m$ acts on the $m$th jet scheme $V_m$ for all $m\geq 0$. We are interested in the invariant ring $\mathcal{O}(V_m)^{G_m}$ and whether the map $p_m^*\colon\mathcal{O}((V//G)_m) \rightarrow \mathcal{O}(V_m)^{G_m}$ induced by the categorical quotient map $p\colon V\rightarrow V//G$ is an isomorphism, surjective, or neither. Using Luna's slice theorem, we give criteria for $p_m^*$ to be an isomorphism for all $m$, and we prove this when $G=SL_n$, $GL_n$, $SO_n$, or $Sp_{2n}$ and $V$ is a sum of copies of the standard representation and its dual, such that $V//G$ is smooth or a complete intersection. We classify all representations of $\mathbb{C}^*$ for which $p^*_{\infty}$ is surjective or an isomorphism. Finally, we give examples where $p^*_m$ is surjective for $m=\infty$ but not for finite $m$, and where it is surjective but not injective.

math.AG↗

arXiv2015

Arc spaces and the vertex algebra commutant problem

Given a vertex algebra $\mathcal{V}$ and a subalgebra $\mathcal{A}\subset \mathcal{V}$, the commutant $\text{Com}(\mathcal{A},\mathcal{V})$ is the subalgebra of $\mathcal{V}$ which commutes with all elements of $\mathcal{A}$. This construction is analogous to the ordinary commutant in the theory of associative algebras, and is important in physics in the construction of coset conformal field theories. When $\mathcal{A}$ is an affine vertex algebra, $\text{Com}(\mathcal{A},\mathcal{V})$ is closely related to rings of invariant functions on arc spaces. We find strong finite generating sets for a family of examples where $\mathcal{A}$ is affine and $\mathcal{V}$ is a $βγ$-system, $bc$-system, or $bcβγ$-system.

math.RT↗

arXiv2014

When is a symplectic quotient an orbifold?

Let $K$ be a compact Lie group of positive dimension. We show that for most unitary $K$-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When $K$ is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian $K$-manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary $\operatorname{SU}_2$-modules yield symplectic quotients that are $\mathbb{Z}$-graded regularly symplectomorphic to a linear symplectic orbifold. We similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold.

math.SG↗

arXiv2013

Lifting automorphisms of quotients of adjoint representations

Let $\mathfrak g_i$ be a simple complex Lie algebra, $1\leq i \leq d$, and let $G=G_1\times...\times G_d$ be the corresponding adjoint group. Consider the $G$-module $V=\oplus r_i\mathfrak g_i$ where $r_i\geq 1$ for all $i$. We say that $V$ is \emph{large} if all $r_i\geq 2$ and $r_i\geq 3$ if $G_i$ has rank 1. In [Schwarz12] we showed that when $V$ is large any algebraic automorphism $ψ$ of the quotient $Z:= V//G$ lifts to an algebraic mapping $Ψ\colon V\to V$ which sends the fiber over $z$ to the fiber over $ψ(z)$, $z\in Z$. (Most cases were already handled in [Kuttler11]). We also showed that one can choose a biholomorphic lift $Ψ$ such that $Ψ(gv)=σ(g)Ψ(v)$, $g\in G$, $v\in V$, where $σ$ is an automorphism of $G$. This leaves open the following questions: Can one lift holomorphic automorphisms of $Z$? Which automorphisms lift if $V$ is not large? We answer the first question in the affirmative and also answer the second question. Part of the proof involves establishing the following result for $V$ large. Any algebraic differential operator of order $k$ on $Z$ lifts to a $G$-invariant algebraic differential operator of order $k$ on $V$. We also consider the analogues of the questions above for actions of compact Lie groups.

math.RT↗

arXiv2013

An Oka principle for equivariant isomorphisms

Let $G$ be a reductive complex Lie group acting holomorphically on normal Stein spaces $X$ and $Y$, which are locally $G$-biholomorphic over a common categorical quotient $Q$. When is there a global $G$-biholomorphism $X\to Y$? If the actions of $G$ on $X$ and $Y$ are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that $X$ and $Y$ are $G$-biholomorphic if $X$ is $K$-contractible, where $K$ is a maximal compact subgroup of $G$, or if $X$ and $Y$ are smooth and there is a $G$-diffeomorphism $ψ:X\to Y$ over $Q$, which is holomorphic when restricted to each fibre of the quotient map $X\to Q$. We prove a similar theorem when $ψ$ is only a $G$-homeomorphism, but with an assumption about its action on $G$-finite functions. When $G$ is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of $G$-biholomorphisms from $X$ to $Y$ over $Q$. This sheaf can be badly singular, even for a low-dimensional representation of $\mathrm{SL}_2(\C)$. Our work is in part motivated by the linearisation problem for actions on $\C^n$. It follows from one of our main results that a holomorphic $G$-action on $\C^n$, which is locally $G$-biholomorphic over a common quotient to a generic linear action, is linearisable.

math.CV↗

arXiv2013

Quotients, automorphisms and differential operators

Let $V$ be a $G$-module where $G$ is a complex reductive group. Let $Z:=\quot VG$ denote the categorical quotient and let $π\colon V\to Z$ be the morphism dual to the inclusion $Ø(V)^G\subsetØ(V)$. Let $ϕ\colon Z\to Z$ be an algebraic automorphism. Then one can ask if there is an algebraic map $Φ\colon V\to V$ which lifts $ϕ$, i.e., $π(Φ(v))=ϕ(π(v))$ for all $v\in V$. In \cite{Kuttler} the case is treated where $V=r\lieg$ is a multiple of the adjoint representation of $G$. It is shown that, for $r$ sufficiently large (often $r\geq 2$ will do), any $ϕ$ has a lift. We consider the case of general representations (satisfying some mild assumptions). It turns out that it is natural to consider holomorphic lifting of holomorphic automorphisms of $Z$, and we show that if a holomorphic $ϕ$ and its inverse lift holomorphically, then $ϕ$ has a lift $Φ$ which is an automorphism such that $Φ(gv)=σ(g)Φ(v)$, $v\in V$, $g\in G$ where $σ$ is an automorphism of $G$. We reduce the lifting problem to the group of automorphisms of $Z$ which preserve the natural grading of $Ø(Z)\simeqØ(V)^G$. Lifting does not always hold, but we show that it always does for representations of tori in which case algebraic automorphisms lift to algebraic automorphisms. We extend Kuttler's methods to show lifting in case $V$ contains a copy of $\lieg$.

math.GR↗

arXiv2013

The Koszul complex of a moment map

Let $K\to U(V)$ be a unitary representation of the compact Lie group $K$. Then there is a canonical moment mapping $ρ\colon V\to\mathfrak k^*$. We have the Koszul complex ${\mathcal K}(ρ,\mathcal C^\infty(V))$ of the component functions $ρ_1,...,ρ_k$ of $ρ$. Let $G=K_{\mathbb C}$, the complexification of $K$. We show that the Koszul complex is a resolution of the smooth functions on $ρ^{-1}(0)$ if and only if $G\to\GL(V)$ is 1-large, a concept introduced in earlier work of the second author. Now let $M$ be a symplectic manifold with a Hamiltonian action of $K$. Let $ρ$ be a moment mapping and consider the Koszul complex given by the component functions of $ρ$. We show that the Koszul complex is a resolution of the smooth functions on $Z=ρ^{-1}(0)$ if and only if the complexification of each symplectic slice representation at a point of $Z$ is 1-large.

math.SG↗