Prof. Dr. Daniel Greb - Publications


My preprints on arXiv. My published papers on MathSciNet (subscription required). My Google Scholar entry.



  • Moduli of K3 families over $\mathbb{P}^1$, cycle spaces of IHS period domains, and deformations of complex-hyperkähler metrics, arXiv:2311.13420
    (with Martin Schwald)

    In the spirit of the classical theory for K3 surfaces, we establish a moduli theory for families of marked K3 surfaces over smooth rational curves in the K3 period domain. In particular, we study the geometry of the space of such rational curves, which assumes the role of a period space for this moduli problem. Over an open subset containing all twistor cycles, we construct a family of such families that is a universal small deformation for every twistor family. Finally, small deformations of K3 twistor families are shown to induce complex-hyperkähler metrics on members of the families via the original complex version of Penrose's Non-linear Graviton construction. This answers a question posed independently by Fels--Huckleberry--Wolf and Looijenga. Generalizations to higher-dimensional irreducible holomorphic-symplectic manifolds are studied throughout.

  • Milnor-Wood inequality for klt varieties of general type and uniformization, arXiv:2308.05586
    (with Matteo Costantini)

    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 consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations.

  • Reductive quotients of klt singularities, arXiv:2111.02812
    (with Lukas Braun, Kevin Langlois, and Joaquín Moraga)

    We prove that the quotient of a klt type singularity by a reductive group is of klt type. In particular, given a klt variety $X$ edowed with the action of a reductive group $G$ and admitting a quasi-projective good quotient $X\rightarrow X/\!/G$, we can find a boundary $B$ on $X/\!/G$ so that the pair $(X/\!/G,B)$ is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler $G$-manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing $n$-dimensional K-polystable Fano manifolds of volume $v$ has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.

  • 1-rational singularities and quotients by reductive groups, arXiv:0901.3539

    This preprint will not be submitted; the results follow easily from those in my later paper ''Rational singularities and quotients by holomorphic group actions'' that appeared in Annali della Scuola Normale Superiore di Pisa; however, the preprint gives an independent and technically simpler proof in the algebraic case and avoids the technical difficulties encountered in the analytic setup.


CONFERENCE PROCEEDINGS (invited, without peer review)




Reviews written by myself for Mathematical Reviews can be found here (subscription required).