More Like this

1. The integral Hodge conjecture for two-dimensional Calabi–Yau categories

   https://doi.org/10.1112/S0010437X22007266  

   Perry, Alexander (February 2022, Compositio Mathematica)

   We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi–Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi–Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves. 

   more » « less
2. An Exploration of Degeneracy in Abelian Varieties of Fermat Type

   https://doi.org/10.1080/10586458.2024.2362953  

   Goodson, Heidi (June 2024, Experimental Mathematics)

   The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form $y^2=x^m-1$. We prove that the Jacobian of the curve is degenerate whenever $$m$$ is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur. 

   more » « less
3. Rational points and derived equivalence

   https://doi.org/10.1112/S0010437X21007089  

   Addington, Nicolas; Antieau, Benjamin; Honigs, Katrina; Frei, Sarah (May 2021, Compositio Mathematica)

   null (Ed.)

   We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over $$\mathbb {Q}$$ and $$\mathbb {F}_q(t)$$ , and conclude with a pair of hyperkähler 4-folds over $$\mathbb {Q}$$ . The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article. 

   more » « less
4. An introduction to the algebraic geometry of the Putman–Wieland conjecture

   https://doi.org/10.1007/s40879-023-00637-w  

   Landesman, Aaron; Litt, Daniel (June 2023, European Journal of Mathematics)

   Abstract We give algebraic and geometric perspectives on our prior results toward the Putman–Wieland conjecture. This leads to interesting new constructions of families of “origami” curves whose Jacobians have high-dimensional isotrivial isogeny factors. We also explain how a hyperelliptic analogue of the Putman–Wieland conjecture fails, following work of Marković. 

   more » « less
5. Sato-Tate distributions of y^2 = x^p − 1 and y^2 = x^{2p} − 1

   https://doi.org/10.1016/j.jalgebra.2022.01.002  

   Emory, Melissa; Goodson, Heidi (May 2022, Journal of Algebra)

   We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form y^2 = x^p−1 and y2 = x^{2p}−1, where p is an odd prime. Our results rely on the fact the Jacobians of these curves are nondegenerate, a fact that we prove in the paper. Furthermore, we compute moment statistics associated to the Sato-Tate groups. These moment statistics can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized L-polynomials of the curves. 

   more » « less