A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

Perverse obstructions to flat regular compactifications

“…(Some authors say "very general" for the above notion and "general" when the property holds outside of a closed proper subscheme.) This is the way that the term is used by Voisin in [13, p. 93], and I believe that it conforms to the usage by Laza, Saccà and Voisin in [9].…”

“…In Section 4, I will give examples where Corollary 1.7 can be used to rule out the existence of a flat, regular compactification, or a flat, regular compactification with irreducible fibers. I will also give a consequence (Corollary 4.3) of the main result of [9] and state a conjecture (Conjecture 4.4) about palindromicity partially motivated by the results of [9].…”

“…Then [9, Proposition 5.1] states that Prym C B /C B → B is flat when B is a certain Fano variety F 0 of lines. Section 5 of [9] descends the family of Prym varieties over F 0 (which maps surjectively to P 5 ) to P ∨ = P 5 . As explained in the paragraph between Lemmas 5.3 and 5.4 of [9], the result is a familyJ → P ∨ whose pullback to F 0 is the above family of compactified Prym varieties.…”

“…In other words, when does there exists a flat, proper morphismπ : W →S extending π with W regular? One interesting recent example of this occurs in the preprint [9] of Laza, Saccà and Voisin where π is a family of abelian 5-folds over a Zariski open subset S ofS = P 5 . In that paper, the authors construct W using the theory of compactified Prym varieties and show that it is a holomorphic symplectic manifold (deformation equivalent to O'Grady's 10-dimensional example).…”

“…Then I observe that, in some cases of interest, results of Brylinski, Beilinson and Schnell can be used to compute the intersection cohomology [1,4,12]. I also give examples involving cubic 4-folds motivated by [9] and ask a question about palindromicity of hyperplane sections. …”