Projective models of Nikulin orbifolds

Abstract

We study projective fourfolds of K3 [2] -type with a symplectic involution and the deformations of their quotients, called orbifolds of Nikulin types; they are IHS orbifolds. We compute the Riemann-Roch formula for Weil divisors on such orbifolds and describe the first complete family of orbifolds of Nikulin type with a polarization of degree 2 as double covers of special complete intersections (3, 4) in P 6 .

Second Chern class and Fujiki constants of hyperkähler manifolds

“…Let M ′ be the irreducible symplectic orbifold of dimension 4 with second Betti number b 2 (M ′ ) = 16, also known as a Nikulin orbifold (see [7,Sec. 5.11] and [3]). It has 28 isolated quotient singularities of order 2, i.e., a 2 = 28.…”

“…4 is a linear combination of c 2 2 and c 4 , so the same may be said for the class c 4 . Then we can use Proposition 2.4 to determine that c 4 (OG 6 ) = q 2 , which then allows us to also compute C(c 4 c 2 ) and C(c 3 2 ). Finally we can use C(td 6 ) = 4 to solve the Euler characteristic C(c 6 ).…”

“…, c 10 lie in the Verbitsky component. We have c 2 (OG 10 ) = 3 2 q, c 4 (OG 10 ) = 15 16 q 2 , c 6 (OG 10 ) = 21 64 q 3 , c 8 (OG 10 ) = 237 3328 q 4 , c 10 (OG 10 ) = 27 2560 q 5 . Proof.…”

“…For a generic X in the moduli space, the (special) Mumford-Tate algebra is the maximal possible and is isomorphic to so (3,21). Using the branching rules, we get the following decompositions of so(3, 21)-modules/Hodge structures (H 12 and H 14 are omitted by symmetry)…”