Fortnightly links (104)

Alexander Perry: The integral Hodge conjecture for two-dimensional Calabi-Yau categories is a must-read (if you have the time and necessary background).

Indranil Biswas, Tomas L. Gomez: On vector bundles over moduli spaces trivial on Hecke curves proves an interesting result for moduli of vector bundles on a curve, whose geometry I find very interesting. It's analogous to the result that if $\mathcal{E}$ is a vector bundle on $\mathbb{P}^n$, such that $\mathcal{E}|_L\cong\mathcal{O}_{\mathbb{P}^1}^{\oplus r}$ for every line on $\mathbb{P}^n$, then $\mathcal{E}$ is itself trivial. For $\mathrm{M}_C(r,\mathcal{L})$, the same result holds, where the role of lines is played by "rational curves of minimal degree", also known as Hecke curves (which have an explicit modular interpretation).

David Favero, Daniel Kaplan, Tyler L. Kelly: A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles gives a counterexample to the Lekili-Ueda conjecture on the existence of a full exceptional collection of line bundles for a Landau–Ginzburg model, which is itself analogous to King's conjecture on the existence of a full exceptional collection of line bundles for a smooth projective toric variety. The latter is known to be false since 2006, by Hille–Perling. Now the Landau--Ginzburg model version for invertible polynomials is also known to be false.