Verra fourfolds have a distinct small quantum cohomology ring implying they are never birational to very general cubic or Gushel-Mukai fourfolds, with primitive cohomology matching a K3 surface when birational.
Quantum cohomology and irrationality of Gushel-Mukai fourfolds
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abstract
We compute the small quantum cohomology of Gushel-Mukai fourfolds. Following [13], our computations imply that the very general ones are not rational. Following [8], and thanks to a suitable deformation of the small quantum cohomology ring, we also deduce that a rational Gushel-Mukai fourfold has the same rational cohomology as some K3 surface.
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Quantum cohomology and birational geometry of Verra fourfolds
Verra fourfolds have a distinct small quantum cohomology ring implying they are never birational to very general cubic or Gushel-Mukai fourfolds, with primitive cohomology matching a K3 surface when birational.