Quantum cohomology and irrationality of Gushel-Mukai fourfolds
Vladimiro Benedetti (LJAD), Laurent Manivel (I2M), Nicolas Perrin (CMLS)
TL;DR
This paper computes the small quantum cohomology of Gushel-Mukai fourfolds and uses these computations to show that very general such fourfolds are not rational, linking their cohomology to K3 surfaces.
Contribution
It provides the first detailed computation of the small quantum cohomology for Gushel-Mukai fourfolds and establishes a connection between rationality and cohomology type.
Findings
Very general Gushel-Mukai fourfolds are not rational.
Rational Gushel-Mukai fourfolds share cohomology with K3 surfaces.
Quantum cohomology computations inform rationality properties.
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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Taxonomy
TopicsAlgebraic structures and combinatorial models · Algebraic Geometry and Number Theory · Commutative Algebra and Its Applications