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.
Irreducible holomorphic symplectic (IHS) varieties are fundamental bulding blocks in the classification of varieties with trivial canonical class and possess exceptionally rich moduli and birational geometry. Beyond the classical families -Hilbert schemes of points on a K3 surface, generalized Kummer varieties, and O'Grady examples in dimension six and ten -constructing new deformation types of smooth IHS varieties has proven notoriously difficult.
In the singular setting, however, the situation is more flexible. A particularly fruitful strategy to produce examples of IHS varieties is to study projective terminalizations of quotients of smooth symplectic varieties by groups of symplectic automorphisms. This approach has been used in [BGMM25;FM21;Fuj83;Men22]. Whereas [Men22] gives a complete classification of Fujiki fourfolds (terminalizations of certain quotients of square of K3 surfaces), [BGMM25] classifies all terminalizations of quotients of Hilbert schemes of K3 surfaces or generalized Kummer varieties by symplectic automorphisms induced from the underlying surface. Together, these works already yield at least 38 distinct deformation classes of four-dimensional irreducible symplectic varieties.
In this paper, we continue the classification program designed by Menet in [Men22, Section 1.3] by studying terminalizations of quotients of Fano varieties of lines on smooth cubic fourfolds by finite groups of symplectic automorphisms of the underlying cubic fourfold. Let X be a smooth cubic fourfold and F (X) its Fano variety of lines, which is a smooth four-dimensional IHS variety of K3 [2] -type. An automorphism of X is called symplectic if the induced automorphism of F (X) acts trivially on its holomorphic symplectic form. Although F (X) is deformation equivalent to the Hilbert scheme of two points on a K3 surface S, symplectic automorphisms of F (X) induced by X are in general not deformations of automorphisms of S [2] induced by S previously studied in [BGMM25;Men22]. Therefore, studying terminalizations of the quotients F (X)/G, where G is a group of symplectic automorphisms of X, can provide genuinely new deformation classes of terminalizations.
Theorem 1.1. Let X be a smooth cubic fourfold. For any finite group G acting symplectically on X, the second Betti number and the fundamental group of the regular locus of a projective terminalization Y of the quotient F (X)/G are listed in Table 1.
In [LZ22], Laza and Zheng classified all possible symplectic automorphism groups of smooth cubic fourfolds. Explicit equations for smooth cubic fourfolds realizing each such group as their symplectic automorphism group are known [Adl78; Fu15; HM19; Koi24; Mon13a; Mon13b; YYZ24], and provide a natural starting point for the classification in Theorem 1.1. By [BGMM25, Proposition 8.1], any terminalization in Table 1 whose regular locus has nontrivial fundamental group is a quasi-étale quotient of a terminalization with simply connected regular locus, already appearing in the table. Thus, for purposes of classification, it suffices to focus on IHS varieties with simply connected regular locus.
To compute b 2 (Y ), we establish the following formula.
Theorem 1.2 (Theorem 4.2). Let X be a smooth cubic fourfold, and F (X) its Fano variety of lines. Let G be a finite group acting symplectically on X. Let q : F (X) → F (X)/G be the quotient map, Y a terminalization of F (X)/G, and Σ the reduced singular locus of F (X)/G.
For any g ∈ G, denote by F g ⊂ F (X) the unique component of the fixed locus of g ∈ G of codimension 2 (if any), and by N G (g) (respectively C G (g)) the normaliser (respectively centraliser) of g in G. Denote by
• n 2 the number of components q(F g ) in Sing(F (X)/G) with ord(g) = 2;
• n 31 the number of components q(F g ) in Sing(F (X)/G) with g of order 3 and such that N G (g) \ C G (g) contains elements of even order; • n 32 the number of components q(F g ) in Sing(F (X)/G) with g of order 3 and such that N G (g) \ C G (g) contains no element of even order.
The following identity holds b 2 (Y ) = rk(H 2 (F (X), Z) G ) + n 2 + n 31 + 2n 32 .
In this formula, only the number n 2 depends solely on the abstract structure of G. The quantities n 31 and n 32 , instead, depend on the specific embedding of G in PGL 6 ; in Propositions 3.4 and 4.5 we give alternative descriptions of these terms, following [Fu15] which shows that a symplectic automorphism of X of order 3 fixes a locus of codimension 2 in F (X) if and only if its normal form is diag(1, 1, 1, ω, ω, ω). Such elements not only contribute to the formula for b 2 (Y ), but also imply that the action of G on F (X) cannot arise from an action on the Hilbert scheme S [2] induced by an action on the K3 surface S, even though S [2] is deformation equivalent to F (X). Indeed, the fixed locus of any induced symplectic automorphism of S [2] of order 3 has no components of codimension 2. Therefore, any group action in Table 1 with nontrivial contributi
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