OAI.ReverseLogKodaira.SmoothProjectiveVariety.veryGenerally_fiber_negative_branch

The theorem states that, for smooth projective complex varieties X and Y, each equipped with a reduced simple-normal-crossing boundary (E on X, D on Y, given by Cartier components with transverse intersections and Cartier union), and for a stratum-smooth fibration f from (X,E) to (Y,D), the following holds for very general points of Y. Here f is a surjective morphism over ℂ with connected fibers over complex points, whose preimage of the support of D lies in the support of E, and which is smooth over the complement of D both on X and on every stratum, meaning each intersection of a nonempty set of components of E. Very general means there is a countable family of proper closed subsets Zₙ of Y such that every complex point y of Y outside all the Zₙ satisfies the property. The property is that y lies in the complement of D and, for every fiber model F of f over y whose boundary has log Kodaira dimension equal to ⊥ (the value meaning there is no nonzero log pluriform section of any positive degree m on the fiber), the log Kodaira dimension of E equals that of D plus that of the fiber boundary F.boundary, and moreover every log m-pluriform rational section of X with poles allowed along E, for every integer m>0, is zero. A fiber model is a smooth projective variety mapping to X as the fiber product of f with the point y, together with an SNC boundary whose ideal is the pullback of E's ideal.