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Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of Q-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincaré transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GVsubschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.

Chiral magnets endowed with topological spin textures are expected to have promising applications in next‐generation magnetic memories. In contrast to the well‐studied 2D or 3D magnetic skyrmions, the authors report the discovery of 1D nontrivial magnetic solitons in a transition metal dichalcogenide 2H‐TaS2 via precise intercalation of Cr elements. In the synthetic Cr1/3TaS2 (CTS) single crystal, the coupling of the strong spin–orbit interaction from TaS2 and the chiral arrangement of the magnetic Cr ions evoke a robust Dzyaloshinskii–Moriya interaction. A magnetic helix having a short spatial period of ≈25 nm is observed in CTS via Lorentz transmission electron microscopy. In a magnetic field perpendicular to the helical axis, the helical spin structure transforms into a chiral soliton lattice (CSL) with the spin structure evolution being consistent with the chiral sine‐Gordon theory, which opens promising perspectives for the application of CSL to fast‐speed nonvolatile magnetic memories. This work introduces a new paradigm to soliton physics and provides an effective strategy for seeking novel 2D magnets.