Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville

Generic Vanishing Theory via Mixed Hodge Modules

“…https://doi.org/10.1017/fms.2013.1 theorem with some index k. Since each complex C j is an iterated extension of Ra * Ω p X [n − p], the above shows that RΦ P C j ∈ c D k coh (OÂ), and hence that RΦ P (gr F p M i ) = 0 for every p ∈ Z and i > k. Because the Fourier-Mukai transform is an equivalence of categories, we conclude that M i = 0 for i > k, which means that δ(a) k. EXAMPLE 4.8. Our result explains the original counterexample from [11,Section 3]. The example consisted in blowing up an abelian variety A of dimension four along a smooth curve C of genus at least two; if X denotes the resulting variety, then a : X → A is the Albanese mapping, and a short computation shows that…”

“…This allows one to recover the original generic vanishing theorem for ω X of [11], as well as its extension to higher direct images R j a * ω X given in [14]. Indeed, in the proof above note that…”

“…The statement above is the appropriate generalization of the original generic vanishing theorem [11], which dealt with the case p = n. Note that, unlike in [11], the codimension bound necessarily depends on the entire Albanese mapping, not just on the dimension of the generic fiber. Since the inequality is sharp, we obtain the following cohomological formula for the defect of semismallness: δ(a) = max p,q 0 (|p + q − n| − codim V p (Ω q X )), similar to the formula in [20,Remark 2.4] for the dimension of the generic fiber.…”

“…[40,Corollary 12]). The fact that for generic Nakano vanishing it is not sufficient to assume that the Albanese map is generically finite over its image was already pointed out in [11] (see Example 4.8 below). Nevertheless, our method also recovers the stronger statement for the canonical bundle [11] and its higher direct images [14] (see the end of Section 4.3).…”

“…The fact that for generic Nakano vanishing it is not sufficient to assume that the Albanese map is generically finite over its image was already pointed out in [11] (see Example 4.8 below). Nevertheless, our method also recovers the stronger statement for the canonical bundle [11] and its higher direct images [14] (see the end of Section 4.3). This is due to the special properties of the first nonzero piece of the Hodge filtration on mixed Hodge modules, established by Saito. Note also that, since χ(X, Ω p X ) = χ(X, Ω p X ⊗ L) for any L ∈ Pic 0 (X) by the deformation invariance of the Euler characteristic of a coherent sheaf, we have as a consequence the following extension of the fact that χ (X, ω X ) 0 for varieties of maximal Albanese dimension.…”