2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74973First steps towards a classification of irreducible symplectic 4-folds whose integral 2-cohomology with 4-tuple cup product is isomorphic to that of Hilb^2(K3). We prove that any such 4-fold deforms to an irreducible symplectic 4-fold of Type A or Type B. A 4-fold of Type A is a double cover of a (singular) sextic hypersurface and a 4-fold of Type B is birational to a hypersurface of degree at most 12. We conjecture that 4-folds of Type B do not exist.We eliminated the last section because we have solved (in another paper) the problem that was discussed in that section. We added Propositions (3.7) and (4.8)Algebraic GeometryDifferential Geometry14J15text