Symplectic involutions of holomorphic symplectic four-folds (Q2903271)

\textit{V. V. Nikulin} [Trans. Mosc. Math. Soc. 2, 71--135 (1980; Zbl 0454.14017)] proved that the fixed locus of a symplectic involution of a \(K3\) surface is 8 isolated points. As higher-dimensional generalizations, we are interested in the study of the fixed locus of a symplectic involution of a irreducible holomorphic symplectic manifold. The paper under review deals with the particular case of irreducible holomorphic symplectic manifolds of dimension 4 with second Betti number 23 (e.g. the deformations of \(K3^{[2]}\) type manifolds). By general arguments, each connected component of the fixed locus is a symplectic submanifold, thus in the 4-dimensional case, the only possibilities are: isolated points, \(K3\) surfaces and abelian surfaces.NEWLINENEWLINEHere is the main result: for a symplectic involution of a irreducible holomorphic symplectic manifold of dimension 4 and second Betti-number 23, there are three possibilities for the fixed locus: (i) 12 isolated points, no \(K3\) surfaces and at least one abelian surface; (ii) 36 isolated points, no \(K3\) surfaces and at least one abelian surface; (iii) 28 isolated points, one \(K3\) surface and possibly some abelian surfaces.NEWLINENEWLINEThe key ingredient of the proof is the holomorphic Lefschetz-Riemann-Roch fixed point formula (cf. [\textit{P. Donovan}, Bull. Soc. Math. Fr. 97, 257--273 (1969; Zbl 0185.49401)]), which is a generalization of the Hirzebruch-Riemann-Roch formula to the equivariant case, and can also be seen as a special case of the equivariant Atiyah-Singer index theorem. More precisely, given a equivariant vector bundle, this formula relates on one hand the alternating sum of the traces of the induced action on cohomology groups of the vector bundle, and on the other hand some characteristic classes of bundles naturally arising from this action (e.g. normal bundles of the fixed locus, eigen-subbundles, etc.).NEWLINENEWLINEThe author also conjectures that in the above stated theorem, the fixed locus cannot contain any abelian surfaces. In particular, only the case (iii) can occur with no abelian surfaces: i.e. the fixed locus is exactly 28 isolated points and one \(K3\) surface. In the last three sections, by explicit geometric considerations, the author verified this conjecture in the following three classes of examples: (1) the natural symplectic involutions on \(K3^{[2]}\); (2) the natural symplectic involutions on the Fano variety of lines of a cubic 4-fold; (3) the natural symplectic involutions on the double cover of EPW-sexitics, Here 'natural' means the symplectic involution comes from an involution of the \(K3\) surface, cubic 4-fold, or the 6-dimensional vector space respectively.