Research papers

• Curves on powers of hyperelliptic Jacobians (with Stefan Schreieder)
  arXiv preprint, 2024. [+]
  For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. As an application, we show that, for a very general principally polarized abelian variety of dimension at least four, or the intermediate Jacobian of a very general cubic threefold, no power is isogenous to a product of Jacobians of curves. This confirms some cases of the Coleman-Oort conjecture. We further deduce from our results some progress on the question whether the integral Hodge conjecture fails for such abelian varieties.

• Hyperbolic geometry and real moduli of five points on the line
  arXiv preprint, 2023. [+]
  We show that each connected component of the moduli space of smooth real binary quintics is isomorphic to an arithmetic quotient of an open subset of the real hyperbolic plane. Moreover, our main result says that the induced metric on this moduli space extends to a complete real hyperbolic orbifold structure on the space of stable real binary quintics. This turns the moduli space of stable real binary quintics into a real two-dimensional ball quotient, whose fundamental domain is given by the hyperbolic triangle of angles π/3,π/5 and π/10, and whose fundamental group is non-arithmetic.

• Non-arithmetic uniformization of metric spaces attached to unitary Shimura varieties
  arXiv preprint, 2023. [+]
  We develop a new method of constructing non-arithmetic lattices in the projective orthogonal group PO(n,1) for every integer n larger than one. The technique is to consider anti-holomorphic involutions on a complex arithmetic ball quotient, glue their fixed loci along geodesic subspaces, and show that the resulting metric space carries canonically the structure of a complete real hyperbolic orbifold.

• On the integral Hodge conjecture for real abelian threefolds
  Journal für die reine und angewandte Mathematik, 2023. [+]
  We prove the real integral Hodge conjecture for several classes of real abelian threefolds. For instance, we prove the property for real abelian threefolds A whose real locus A(ℝ) is connected, and for real abelian threefolds A which are a product A=B×E of an abelian surface B and an elliptic curve E with connected real locus E(ℝ). Moreover, we show that every real abelian threefold satisfies the real integral Hodge conjecture modulo torsion, and reduce the general case to the Jacobian case.

• Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties (with Thorsten Beckmann)
  
  Compositio Mathematica 159(6)6: 1188-1213, 2023. [+]
  We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, the Jacobian of a smooth projective curve over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.

• Real moduli spaces and density of non-simple real abelian varieties.
  
  The Quarterly Journal of Mathematics, 73 no. 3: 969-989, 2022. [+]
  For fixed k < g and a family of polarized abelian varieties of dimension g over ℝ⁠, we give a criterion for the density in the parameter space of those abelian varieties over ℝ containing a k-dimensional abelian subvariety over ℝ⁠. As application, we prove density of such a set in the moduli space of polarized real abelian varieties of dimension g and density of real algebraic curves mapping non-trivially to real k-dimensional abelian varieties in the moduli space of real algebraic curves as well as in the space of real plane curves. This extends to the real setting results by Colombo and Pirola [10]. We then consider the real locus of an algebraic stack over ℝ⁠, attaching a topological space to it. For a real moduli stack, this defines a real moduli space. We show that for Mg and Ag⁠, the real moduli spaces that arise in this way coincide with the moduli spaces of Gross–Harris and Seppälä–Silhol.

PhD thesis

• Moduli spaces and algebraic cycles in real algebraic geometry
  
  2022 [+]
  This thesis intends to make a contribution to the theories of algebraic cycles and moduli spaces over the real numbers. In the study of the subvarieties of a projective algebraic variety, smooth over the field of real numbers, the cycle class map between the Chow ring and the equivariant cohomology ring plays an important role. The image of the cycle class map remains difficult to describe in general; we study this group in detail in the case of real abelian varieties. To do so, we construct integral Fourier transforms on Chow rings of abelian varieties over any field. They allow us to prove the integral Hodge conjecture for one-cycles on complex Jacobian varieties, and the real integral Hodge conjecture modulo torsion for real abelian threefolds. For the theory of real algebraic cycles, and for several other purposes in real algebraic geometry, it is useful to have moduli spaces of real varieties to our disposal. Insight in the topology of a real moduli space provides insight in the geometry of a real variety that defines a point in it, and the other way around. In the moduli space of real abelian varieties, as well as in the Torelli locus contained in it, we prove density of the set of moduli points attached to abelian varieties containing an abelian subvariety of fixed dimension. Moreover, we provide the moduli space of stable real binary quintics with a hyperbolic orbifold structure, compatible with the period map on the locus of smooth quintics. This identifies the moduli space of stable real binary quintics with a non-arithmetic ball quotient.

Selected Conference Talks

Past conference talks:

• Talk at the conference Curves, Abelian Varieties and Related Topics (CAVARET), Barcelona (Spain), June 2024
• Research School in Real Algebraic Geometry, Trieste (Italy), November 2023
• Birational Geometry and Regulous Functions, Le Croisic, France
• Cycles et Périodes, Strasbourg, France
• Journées de Géométrie Algébrique Réelle, Angers, France
• Workshop on Complex Geometry and Geometric Group Theory, Karlsruhe, Germany

Teaching

• Algebraic Geometry II (May 2024 - July 2024)
• In Hannover, I have taught Algebraic Geometry II. For the course notes, see the above link.
• The geometry and arithmetic of cubic hypersurfaces (October 2023 - February 2024)
• In Hannover, I taught a course on cubic hypersurfaces. For the course notes, see the above link. For the exam, see the link below.
• Exam: The geometry and arithmetic of cubic hypersurfaces (March 2024)
• Complex Analysis, École Normale Supérieure de Paris, Assistant teacher, Fall 2022.
  Notes: Exercices in Complex Analysis (September 2022).
• Elliptic Curves (Teacher, 2020), Working seminar, M1, ENS Paris, Autumn/Winter 2020
• Galois Theory (Assistant teacher, 2019)
• Geometry (Assistant teacher, 2018)
• Analysis (Assistant teacher, 2018)
• Differential Equations (Assistant teacher, 2017)
• Calculus (Assistant teacher, 2016-2018)