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Daniel Greb

Publications and source records attributed to Daniel Greb.

At least 37 records · Page 2Linked to original sources
arXiv2016

A Master Space for Moduli Spaces of Gieseker-Stable Sheaves

We consider a notion of stability for sheaves, which we call multi-Gieseker stability that depends on several ample polarisations $L_1, \dots, L_N$ and on an additional parameter $σ\in \mathbb{Q}_{\geq 0}^N\setminus\{0\}$. The set of semi stable sheaves admits a projective moduli space $\mathcal M_σ$. We prove that given a finite collection of parameters $σ$, there exists a sheaf- and representation-theoretically defined master space $Y$ such that each corresponding moduli space is obtained from $Y$ as a Geometric Invariant Theory (GIT) quotient. In particular, any two such spaces are related by a finite number of "Thaddeus-flips". As a corollary, we deduce that any two Gieseker-moduli space of sheaves (with respect to different polarisations $L_1$ and $L_2$) are related via a GIT-master space. This confirms an old expectation and generalises results from the surface case to arbitrary dimension.

math.AG↗
arXiv2016

Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties

Given a quasi-projective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite étale covers from the smooth locus $X_{\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y \rightarrow X$, ramified only over the singularities of $X$, such that the étale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every étale cover of $Y_{\mathrm{reg}}$ extends to an étale cover of $Y$. As first major application, we show that every flat holomorphic bundle defined on $Y_{\mathrm{reg}}$ extends to a flat bundle that is defined on all of $Y$. As a consequence, we generalise a classical result of Yau to the singular case: every variety with at worst terminal singularities and with vanishing first and second Chern class is a finite quotient of an Abelian variety. As a further application, we verify a conjecture of Nakayama and Zhang describing the structure of varieties that admit polarised endomorphisms.

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arXiv2015

Moduli of vector bundles on higher-dimensional base manifolds - Construction and Variation

We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson-Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker-Maruyama moduli spaces with respect to two different chosen polarisations are related via Thaddeus-flips through other "multi-Gieseker"-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding Donaldson-Uhlenbeck moduli space.

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arXiv2015

Variation of Gieseker moduli spaces via quiver GIT

We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker-stability. We prove, under a boundedness assumption, which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, that semistable sheaves have a projective coarse moduli space that depends on a natural stability parameter. We then give two applications of this machinery. First, we show that given a real ample class $ω\in N^1(X)_\mathbb{R}$ on a smooth projective threefold $X$ there exists a projective moduli space of sheaves that are Gieseker-semistable with respect to $ω$. Second, we prove that given any two ample line bundles on $X$ the corresponding Gieseker moduli spaces are related by Thaddeus-flips.

math.AG↗
arXiv2015

Semi-continuity of Stability for Sheaves and Variation of Gieseker Moduli Spaces

We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that generalises the classical notion of Gieseker-stability to allow for several polarisations. As such we are able to prove that on smooth threefolds certain moduli spaces of Gieseker-stable sheaves are related by a finite number of Thaddeus-flips (that is flips arising for Variation of Geometric Invariant Theory) whose intermediate spaces are themselves moduli spaces of sheaves.

math.AG↗
arXiv2015

Movable curves and semistable sheaves

This paper extends a number of known results on slope-semistable sheaves from the classical case to the setting where polarisations are given by movable curve classes. As applications, we obtain new flatness results for reflexive sheaves on singular varieties, as well as a characterisation of finite quotients of Abelian varieties via a Chern class condition.

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arXiv2014

Complex-analytic quotients of algebraic G-varieties

It is shown that any compact semistable quotient (in the sense of Heinzner and Snow) of a normal algebraic variety by a complex reductive Lie group $G$ is a good quotient. This reduces the investigation and classification of such complex-analytic quotients to the corresponding questions in the algebraic category. As a consequence of our main result, we show that every compact space in Nemirovski's class $\mathscr{Q}_G$ has a realisation as a good quotient, and that every complete algebraic variety in $\mathscr{Q}_G$ is unirational with finitely generated Cox ring and at worst rational singularities. In particular, every compact space in class $\mathscr{Q}_T$, where $T$ is an algebraic torus, is a toric variety.

math.CV↗
arXiv2014

Partial positivity: geometry and cohomology of q-ample line bundles

We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canonical pairs.

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arXiv2013

Etale covers of Kawamata log terminal spaces and their smooth loci

Working with a singular variety X, one is often interested in comparing the set of etale covers of X with that of its smooth locus X_reg. More precisely, one may ask: What are the obstructions to extend finite etale covers of X_reg to all of X? How do the etale fundamental groups of X and of its smooth locus compare? For projective varieties with Kawamata log terminal (klt) singularities we answer these questions in part. The main result of this paper asserts that there are no infinite towers of finite morphisms over a klt base variety where all morphisms are etale in codimension one, but branched over a small set. In a certain sense, this result can be seen as saying that the difference between the etale covers of X and those of X_reg is finite if X is klt. As an immediate application, we construct a finite covering Y of X, etale in codimension one, such that the etale fundamental groups of Y and Y_reg agree.

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arXiv2012

Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville

Let X be a compact hyperkähler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair (X, L), which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian fibration there exists a smooth good minimal model, i.e., a hyperkähler manifold birational to X on which the fibration is holomorphic.

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arXiv2012

Lagrangian fibrations on hyperkähler fourfolds

Let X be a projective hyperkähler manifold containing a Lagrangian subtorus L. We study intersections of deformations of L, defining a canonical almost holomorphic map called L-reduction, which is not birational if and only if X admits an almost holomorphic Lagrangian fibration with (strong) fibre L. In dimension four we prove that in the above situation there is always a holomorphic Lagrangian fibration with fibre L, thus answering a question of Beauville in this particular case.

math.AG↗
arXiv2012

Reflexive differential forms on singular spaces -- Geometry and Cohomology

Based on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials. First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is developed in an appendix to the paper. Moreover, we prove that Kodaira-Akizuki-Nakano vanishing for sheaves of reflexive differentials holds in certain extreme cases, and that it fails in general. Finally, topological and Hodge-theoretic properties of reflexive differentials are explored.

math.AG↗
arXiv2012

Singular spaces with trivial canonical class

The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-- and holomorphic-symplectic manifolds. The decomposition of the simply-connected part corresponds to a decomposition of the tangent bundle into a direct sum whose summands are integrable and stable with respect to any polarisation. Building on recent extension theorems for differential forms on singular spaces, we prove an analogous decomposition theorem for the tangent sheaf of projective varieties with canonical singularities and numerically trivial canonical class. In view of recent progress in minimal model theory, this result can be seen as a first step towards a structure theory of manifolds with Kodaira dimension zero. Based on our main result, we argue that the natural building blocks for any structure theory are two classes of canonical varieties, which generalise the notions of irreducible Calabi-Yau-- and irreducible holomorphic-symplectic manifolds, respectively.

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arXiv2011

Differential Forms on Log Canonical Spaces

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting. Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.

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arXiv2011

Torsion and cotorsion in the sheaf of Kähler differentials on some mild singularities

We give a criterion for the sheaf of Kähler differentials on a cone over a smooth projective variety to be torsion-free. Applying this to Veronese embeddings of projective space and using known results on differentials on quotient singularities we show that even for mild, e.g. Gorenstein terminal, singularities the sheaf of Kähler differentials will in general have torsion and cotorsion.

math.AG↗
arXiv2010

Invariant meromorphic functions on Stein spaces

In this paper we develop fundamental tools and methods to study meromorphic functions in an equivariant setup. As our main result we construct quotients of Rosenlicht-type for Stein spaces acted upon holomorphically by complex-reductive Lie groups and their algebraic subgroups. In particular, we show that in this setup invariant meromorphic functions separate orbits in general position. Applications to almost homogeneous spaces and principal orbit types are given. Furthermore, we use the main result to investigate the relation between holomorphic and meromorphic invariants for reductive group actions. As one important step in our proof we obtain a weak equivariant analogue of Narasimhan's embedding theorem for Stein spaces.

math.CV↗
arXiv2010

Rational singularities and quotients by holomorphic group actions

We prove that rational and 1-rational singularities of complex spaces are stable under taking quotients by holomorphic actions of reductive and compact Lie groups. This extends a result of Boutot to the analytic category and yields a refinement of his result in the algebraic category. As one of the main technical tools vanishing theorems for cohomology groups with support on fibres of resolutions are proven.

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