On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

Abstract

We extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon’s relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds.

Second Chern class and Fujiki constants of hyperkähler manifolds

“…The proof is exactly the same as in Section 2, so we only indicate the key ingredients. We follow the paper by Fu-Menet [7] and the notation therein.…”

“…Using these ingredients and repeating the proof in Section 2, we obtain Theorem 1.1 for primitively irreducible symplectic orbifolds in dimension 4. We apply it to examine the examples listed in [7,Sec. 5].…”

“…Example 3.2. 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]).…”

Triple Covers of K3 Surfaces

“…The notion of primitive symplectic variety appears in several papers and the definition is not always equivalent to Definition 1.3. An important example of this is given by the notion of primitive symplectic orbifold that appears in [12], which is more restrictive than that of primitive symplectic variety: following Definition 1.3, a primitive symplectic orbifold, as defined in [12], is an orbifold that is a primitive symplectic variety with terminal singularities.…”

Projective models of Nikulin orbifolds

“…An orbifold Y is said irreducible holomorphic symplectic if Y \ Sing(Y ) is simply connected and admits a unique, up to a scalar multiple, non-degenerate holomorphic 2-form. Such manifolds are intensively studied [BL,Me1,MaT,FuMe] because they can be seen as a natural generalization of smooth irreducible hyper-Kähler manifolds. There are only a few known families of these orbifolds, see [Me2,Pe].…”