I am a PhD student in the mathematics department at Radboud University Nijmegen. I am part of the project 'Arithmetic and Geometry Beyond Shimura Varieties', led by Ben Moonen and Lenny Taelman. Currently I am working on a project about derived equivalences of generalized Kummer varieties, under the supervision of Lenny Taelman and Lie Fu.
Before this I was a master student in the complex geometry workgroup at the Mathematical Institute of University Bonn, supervised by Daniel Huybrechts, where I wrote my master's thesis about 'the Tate conjecture and finiteness results for K3 surfaces via cubic fourfolds'. I obtained my bachelor's degree at the university of Bonn in 2016 with a thesis on 'p-adic integration and birational Calabi-Yau varieties'.
My mathematical interests are:
Hyperkähler varieties, in particular K3 surfaces and generalized Kummer varieties
Derived categories of algebraic varieties and derived equivalences between them
Comparison of derived equivalence, birationality, L-equivalnce, etc.
Kernels of categorical resolutions of nodal singularities. Joint with Warren Cattani, Franco Giovenzana, Shengxuan Liu, Luigi Martinelli, Laura Pertusi, and Jieao Song. Preprint, 2022.
In this paper we study derived categories of nodal singularities. We show that for all nodal singularities there is a categorical resolution whose kernel is generated by a 2 or 3-spherical object, depending on the dimension. We apply this result to the case of nodal cubic fourfolds, where we describe the kernel generator of the categorical resolution as an object in the bounded derived category of the associated degree six K3 surface.
Derived equivalences of generalized Kummer varieties. Preprint, 2022.
In this article we study derived (auto)equivalences of generalized Kummer varieties Kumn(A). We provide an answer to a question raised by Namikawa by showing that the generalized Kummer varieties Kumn(A) and Kumn(A∨) are derived equivalent as long as n is even and the abelian surface A admits a polarization whose exponent is coprime to n+1. Furthermore we obtain exact sequences involving groups of autoequivalences in the style of Orlov's short exact sequence for autoequivalences of abelian varieties.
Derived equivalences of generalized Kummer varieties. PhD thesis, in preparation 2022.
Finiteness results and the Tate conjecture for K3 surfaces via cubic fourfolds. Master's thesis, supervised by Daniel Huybrechts, 2018.
We explore the arithmetic of Hassett’s association, which assigns K3 surfaces to special cubic fourfolds. We demonstrate that this association, which a priori is transcendental, descends to number fields.
The motivation for this is the Tate Conjecture for K3 surfaces over finite fields, and its relation to finiteness statements. More precisely, by a Theorem of Lieblich–Maulik–Snowden, the Tate Conjecture for K3 surfaces over finite fields is true if and only if for each fixed finite field there exist only finitely many K3 surfaces over this field. Now, the idea is that we embed the moduli spaces of polarized K3 surfaces into a fixed moduli space (in our case the moduli space of cubic fourfolds) which has only finitely many points over a finite field.
We work towards a proof of the Tate Conjecture in special cases using this approach via a finiteness statement. For this we continue Charles study of the proof of Lieblich–Maulik–Snowden’s theorem and discuss which (weaker) finiteness statements are sufficient to deduce the Tate Conjecture for a given K3 surface.
p-adic integration and birational Calabi-Yau varieties. Bachelor's thesis, supervised by Daniel Huybrechts, 2016.
This bachelor thesis is an exposition of Batyrev's theorem that two birationally equivalent Calabi-Yau varieties have equal Betti numbers.
The proof is based on methods of p-adic analysis, especially p-adic integration on the analytification of smooth varieties over a p-adic field. By a theorem of Weil one can relate the volume of these p-adic analytic manifolds with the number of points in a reduction to a finite field of the variety under consideration. This reduces the comparison of local zeta functions (and in particular Betti numbers, by the Weil conjectures) to the comparison of volumes of certain p-adic analytic manifolds. Batyrev’s theorem follows by showning that the p-adic volumes of two birationally equivalent Calabi–Yau varieties coincide.
In order to realize the indicated strategy formally, we work out a few technicalities. In particular, the method of “spreading out” is explained, which lets us move from varieteis over the complex numbers to varieties over a p-adic field.
Talks, Seminars, Conferences
Derived equivalences of generalized Kummer varieties. Fano varieties and Hyper-Kähler varieties. Strasbourg, January 2023 (Upcoming)
Derived equivalences of generalized Kummer varieties. AMS-SMF-EMS Joint International Meeting: Special Session on Derived Categories and Rationality. Grenoble, July 2022
Selected seminar talks:
Göttsche's formula II: Macdonald’s formula, and refined intersection theory. PhD seminar Amsterdam-Nijmegen on Hilbert schemes of surfaces. Amsterdam SS20.
Basic Hodge theory and special cubic fourfolds. PhD Colloquium. Nijmegen SS19
The Tate conjecture and finiteness results for K3 surfaces via cubic fourfolds (3 talks). Master Seminar. Bonn WS17/18-SS18.
Néron models of abelian varieties. SFB-Transregio-45-Seminar zur Algebraischen Geometrie (Degenerations of algebraic varieties and motivic integration). Bonn SS18
Counterexamples to the integral Hodge conjecture. Graduate Seminar on Advanced Geometry (Topics in Complex Geometry and Hodge Theory). Bonn WS17/18
Symmetric powers and Grothendieck-Deligne norm map. Graduate Seminar on Algebraic Geometry (Jacobians of Curves). Bonn SS17
Review of sheaf cohomology and holomorphic vector bundles. Graduate Seminar on Algebraic Geometry (Complex Geometry). Bonn SS17