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Olivier de Gaay Fortman

Postdoc in Algebraic Geometry

Department of Mathematics
Utrecht University
Budapestlaan 6, 3584 CD
Utrecht, The Netherlands
Room number: 4.08

Email: a.o.d.degaayfortman((at))uu.nl

I am a postdoc in Algebraic Geometry at the University of Utrecht, mentored by Martijn Kool. Before, between 2022 and 2024, I was doing a postdoc at the Institute of Algebraic Geometry in Hannover, mentored by Stefan Schreieder. Between 2019 and 2022, I did my PhD at the ENS in Paris, under the supervision of Olivier Benoist.


Research papers

  1. Matroids and the integral Hodge conjecture for abelian varieties (with Philip Engel and Stefan Schreieder)
    arXiv preprint, 2025. Submitted. [+]
    We prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic threefold. This disproves the integral Hodge conjecture for abelian varieties and shows that very general cubic threefolds are not stably rational. Our proof is motivated by tropical geometry; it relies on multivariable Mumford constructions, monodromy considerations, and the combinatorial theory of matroids.
    Supporting code
  2. Combinatorics and Hodge theory of degenerations of abelian varieties: A survey of the Mumford construction (with Philip Engel and Stefan Schreieder)
    arXiv preprint, 2025. Submitted. [+]
    We survey the Mumford construction of degenerating abelian varieties, with a focus on the analytic version of the construction, and its relation to toric geometry. Moreover, we study the geometry and Hodge theory of multivariable degenerations of abelian varieties associated to regular matroids, and extend some fundamental results of Clemens on 1-parameter semistable degenerations to the multivariable setting.
  3. Descent for algebraic stacks
    arXiv preprint, 2025. Submitted. [+]
    We prove that algebraic stacks satisfy 2-descent for fppf coverings. We generalize Galois descent for schemes to stacks, by considering the case where the fppf covering is a finite Galois covering and reformulating 2-descent data in terms of group actions on the stack.
  4. On the topology of real algebraic stacks (with Emiliano Ambrosi)
    arXiv preprint, 2025. Submitted. [+]
    Motivated by questions arising in the theory of moduli spaces in real algebraic geometry, we develop a range of methods to study the topology of the real locus of a Deligne-Mumford stack over the real numbers. As an application, we verify in several cases the Smith-Thom type inequality for stacks that we conjectured in an earlier work. This requires combining techniques from group theory, algebraic geometry, and topology.
  5. Topological groupoids with involution and real algebraic stacks (with Emiliano Ambrosi)
    arXiv preprint, 2025. Submitted. [+]
    To a topological groupoid endowed with an involution, we associate a topological groupoid of fixed points, generalizing the fixed-point subspace of a topological space with involution. We prove that when the topological groupoid with involution arises from a Deligne-Mumford stack over the real numbers, this fixed locus coincides with the real locus of the stack. This provides a topological framework to study real algebraic stacks, and in particular real moduli spaces. Finally, we propose a Smith-Thom type conjecture in this setting, generalizing the Smith-Thom inequality for topological spaces endowed with an involution.
  6. Non-arithmetic hyperbolic orbifolds attached to unitary Shimura varieties
    arXiv preprint, 2024. Submitted. [+]
    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.
  7. Abelian varieties with no power isogenous to a Jacobian (with Stefan Schreieder)
    arXiv preprint, 2024. To appear in Compositio Mathematica. [+]
    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.
  8. Hyperbolic geometry and real moduli of five points on the line
    arXiv preprint, 2023. To appear in Moduli. [+]
    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.
  9. On the integral Hodge conjecture for real abelian threefolds
    Journal ref.: Crelle, vol. 2024, no. 807, pp. 221-255 (2024). [+]
    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.
  10. Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties (with Thorsten Beckmann)
    Journal ref.: 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.
  11. Real moduli spaces and density of non-simple real abelian varieties
    Journal ref.: 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

Seminar organization

Conference organization: JAVA

Selected Conference Talks

Teaching