Prof. Dr. Daniel Greb - Publications

Databases

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



Publications


PREPRINTS


  • Arakelov inequalities and characterization of totally geodesic ball quotients in $\mathcal{A}_g$, arXiv:2609.37023
    (with Matteo Costantini and Carolina Tamborini)

    We establish an Arakelov inequality for variations of Hodge structures underlying families of principally polarized complex Abelian varieties. In the equality case, this leads to a numerical characterization of certain totally geodesic ball quotients inside the moduli space of Abelian varieties. Our result extends work of Möller-Viehweg-Zuo by removing the strong positivity conditions imposed in their statement. For families over compact base spaces, our proof involves showing that the period map associated with a family of Abelian varieties factors through certain MMP operations and then generalizing the results of Möller, Viehweg, and Zuo to an appropriate singular setting. In the quasiprojective surface case, we implement a new approach that does not pass through Miyaoka-Yau-type uniformisation theorems but uses an argument going back to an idea of Mok, symmetric space theory and semistability considerations instead.


  • Moduli of K3 families over $\mathbb{P}^1$ and complex-hyperkähler metrics induced by deformed twistor cycles, arXiv:2311.13420
    (with Martin Schwald)

    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 members of the families via Penrose's Non-linear Graviton construction. On the way to proving this result, we construct a Hausdorff fine moduli space for families of marked K3 surfaces over smooth rational curves in the K3 period domain. Over an open subset containing all twistor cycles we construct a family of such families, which is a universal small deformation for every twistor family. Whenever possible, we extend the results to higher-dimensional irreducible holomorphic-symplectic manifolds.


  • 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.


PEER-REVIEWED PUBLICATIONS



CONFERENCE PROCEEDINGS (invited, without peer review)


Habilitationsschrift


Dissertation




MathSciNet-Reviews


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