Terminalizations of quotients of Fano varieties of lines on cubic fourfolds

TL;DR

This paper classifies terminalizations of quotients $F(X)/G$ where $X$ is a smooth cubic fourfold and $G$ is a finite group of symplectic automorphisms, computing the second Betti number $b_2$ and the fundamental group of the regular locus of the terminalization. It introduces a sharp formula $b_2(Y)= ext{rk}(H^2(F(X),Z)^G)+n_2+n_{31}+2n_{32}$ that combines the invariant lattice rank with codimension-2 fixed loci data, distinguishing embeddings of order-3 elements via fixed loci types. The authors identify two new deformation classes of four-dimensional IHS varieties with $b_2=4$, arising from maximal groups $G=L_2(11)$ and $G=A_7$, which are not deformation-equivalent to known Fujiki fourfolds or to terminalizations of other symplectic quotients. Placed in context with prior work by Menet and Bertini–Grossi–Mauri–Mazzon, this work expands the landscape of 4-fold IHS deformation types and highlights genuinely new phenomena beyond previously studied families.

Abstract

We classify projective terminalizations of quotients of Fano varieties of lines on smooth cubic fourfolds by finite groups of symplectic automorphisms of the underlying cubic. We compute the second Betti number and the fundamental group of the regular locus. As a consequence, we identify two new deformation classes of four-dimensional irreducible holomorphic symplectic varieties with second Betti number equal to four and simply connected regular locus.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2: Theorem \ref{['thm:b2']}
• Theorem 1.3: Theorem \ref{['thm:b2 new']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 3.4