Abstract:In this paper, we study relations between automorphism groups of cubic fourfolds and Kuznetsov components. Firstly, we characterize automorphism groups of cubic fourfolds as subgroups of autoequivalence groups of Kuznetsov components using Bridgeland stability conditions. Secondly, we compare automorphism groups of cubic fourfolds with automorphism groups of their associated K3 surfaces. Thirdly, we note that the existence of a non-trivial symplectic automorphism on a cubic fourfold is related to the existence… Show more
“…Computing the weights in the corresponding tangent spaces, we find that β(X) equals [4,24,31,22]+ [28,24,19,10]+ [24,12,7,34]+ [9,5,17,29]+[14, 26,…”
We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions.
“…Computing the weights in the corresponding tangent spaces, we find that β(X) equals [4,24,31,22]+ [28,24,19,10]+ [24,12,7,34]+ [9,5,17,29]+[14, 26,…”
We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions.
Given an action of a finite group G on the derived category of a smooth projective variety X, we relate the fixed loci of the induced G-action on moduli spaces of stable objects in D b (Coh(X)) with moduli spaces of stable objects in the equivariant category D b (Coh(X)) G . As an application, we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence and yields a general framework for describing fixed loci of symplectic group actions on moduli spaces of stable objects on symplectic surfaces.
We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174, 960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds.
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