On the Betti Numbers of Irreducible Compact Hyperkähler Manifolds of Complex Dimension Four

By D. Guan. 2001

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IRREDUCIBLE SYMPLECTIC 4-FOLDS NUMERICALLY EQUIVALENT TO (K3)[2] (2008, unknown): “(2.1.6) Let M be a numerical (K3)[2]; since b2(M) = 23 we have [10, 23] H3(M ; Q) = 0, (2.1.7) Sym2H2(M ; Q) ∼−→ H4(M ; Q), (2.1.8) where the second isomorphism is given by cup-product.”
Holomorphic symplectic geometry: a problem list (2010, background): “Here we have some more information in the case of fourfolds [16] :”
Symplectic involutions of holomorphic symplectic four‐folds (2010, unknown): “By Guan [12], when X is an irreducible holomorphic symplectic four-fold such that b2 = 23, the canonical map S(2)H(2)(X, C) → H(4)(X, C) is an isomorphism and, as we already said in Section 2, this is the case for the family of Hilbert schemes S([2]) of a K3 surface S and for their deformations.”
Hyper-Kähler fourfolds and Grassmann geometry (2009, methodology): “nk 22 and it is not the whole of H2(Y σ,Q) because it does not contain any Hodge class for a very general σ. As Yσ is an irreducible hyper-Kahler fourfold, the equality b 2(Yσ) = 23 then follows from [Gu], where it is proved that 23 is the maximal possible second Betti number. Remark2.6. It is also ”