Daniel Greb
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View article: Miyaoka–Yau inequalities and the topological characterization of certain klt varieties
Miyaoka–Yau inequalities and the topological characterization of certain klt varieties Open
Ball quotients, hyperelliptic varieties, and projective spaces are characterized by their Chern classes, as the varieties where the Miyaoka–Yau inequality becomes an equality. Ball quotients, Abelian varieties, and projective spaces are al…
View article: Milnor–Wood inequality for klt varieties of general type and uniformization
Milnor–Wood inequality for klt varieties of general type and uniformization Open
We generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to …
View article: Moduli of K3 families over $\mathbb{P}^1$ and complex-hyperkähler metrics induced by deformed twistor cycles
Moduli of K3 families over $\mathbb{P}^1$ and complex-hyperkähler metrics induced by deformed twistor cycles Open
We answer a question posed independently by Fels-Huckleberry-Wolf and Looijenga concerning the geometric meaning of small deformations of twistor cycles in the K3 period domain. These are shown to induce complex-hyperkähler metrics on memb…
View article: Miyaoka-Yau inequalities and the topological characterization of certain klt varieties
Miyaoka-Yau inequalities and the topological characterization of certain klt varieties Open
Ball quotients, hyperelliptic varieties, and projective spaces are characterized by their Chern classes, as the varieties where the Miyaoka-Yau inequality becomes an equality. Ball quotients, Abelian varieties, and projective spaces are al…
View article: Milnor-Wood inequality for klt varieties of general type and uniformization
Milnor-Wood inequality for klt varieties of general type and uniformization Open
We generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to …
View article: Momentum maps and the Kähler property for base spaces of reductive principal bundles
Momentum maps and the Kähler property for base spaces of reductive principal bundles Open
We investigate the complex geometry of total spaces of reductive principal bundles over compact base spaces and establish a close relation between the Kähler property of the base, momentum maps for the action of a maximal compact subgroup …
View article: Momentum maps and the Kähler property for base spaces of reductive principal bundles
Momentum maps and the Kähler property for base spaces of reductive principal bundles Open
We investigate the complex geometry of total spaces of reductive principal bundles over compact base spaces and establish a close relation between the Kähler property of the base, momentum maps for the action of a maximal compact subgroup …
View article: Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor
Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor Open
We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite…
View article: Reductive quotients of klt singularities
Reductive quotients of klt singularities Open
We prove that the quotient of a klt type singularity by a reductive group is of klt type. In particular, given a klt variety $X$ endowed with the action of a reductive group $G$ and admitting a quasi-projective good quotient $X\rightarrow …
View article: Projectively flat klt varieties
Projectively flat klt varieties Open
In the context of uniformisation problems, we study projective varieties with klt singularities whose cotangent sheaf admits a projectively flat structure over the smooth locus. Generalising work of Jahnke-Radloff, we show that torus quoti…
View article: Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor
Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor Open
We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite…
View article: Singular spaces with trivial canonical class
Singular spaces with trivial canonical class Open
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 C…
View article: Uniformisation of higher-dimensional minimal varieties
Uniformisation of higher-dimensional minimal varieties Open
After a historical discussion of classical uniformisation results for Riemann\nsurfaces, of problems appearing in higher dimensions, and of uniformisation\nresults for projective manifolds with trivial or ample canonical bundle, we\nintrod…
View article: Moduli of vector bundles on higher-dimensional base manifolds — Construction and variation
Moduli of vector bundles on higher-dimensional base manifolds — Construction and variation Open
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 ho…
View article: Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties
Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties Open
Given a quasiprojective 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 …