An asymptotic vanishing theorem for the cohomology of thickenings

Abstract

Let X be a closed equidimensional local complete intersection subscheme of a smooth projective scheme Y over a field, and let X t denote the t-th thickening of X in Y . Fix an ample line bundle O Y (1) on Y . We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer c, such that for all integers t ≥ 1, the cohomology group H k (X t , O X t ( j)) vanishes for k < dim X and j < −ct. Note that there are no restrictions on the characteristic of the field, or on the singular locus of X . We also construct examples illustrating that a linear bound is indeed the best possible, and that the constant c is unbounded, even in a fixed dimension.

Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension

“…Moreover, the Hodge filtration 𝐹 • H 𝑞 𝑍 (𝜔 𝑋 ) is obtained as the image of the pushforward of a natural filtration 𝐹 • 𝐴 • , also described in §2. 4. This description parallels the birational definition of Hodge ideals of hypersurfaces in [36].…”

“…L. Ma has pointed out that when Z is not Cohen-Macaulay, it can happen that Z is not Du Bois, but 𝐹 0 H 𝑞 𝑍 (𝒪 𝑋 ) = 𝐸 0 H 𝑞 𝑍 (𝒪 𝑋 ) for all q. For example, this is the case if 𝑍 = Spec C[𝑠 4 , 𝑠 3 𝑡, 𝑠𝑡 3 , 𝑡 4 ] ↩→ A 4 . We leave the argument for Chapter 5, in which we discuss some basic facts about Du Bois complexes (see Example 5.4).…”

“…The connection may have to do with properties of the (local) cohomology of Du Bois complexes yet to be discovered. (see [4] and [54]). For example, this is used in [4] in order to give another proof of Ogus' characterisation of local cohomological dimension.…”

“…(see [4] and [54]). For example, this is used in [4] in order to give another proof of Ogus' characterisation of local cohomological dimension. Remark 4.14 (Analytic setting).…”

“…Note that Z is an integral variety, with a unique singular point 0, defined by the ideal 𝔪 0 . The normalisation morphism 𝜋 : 𝑍 → 𝑍 corresponds to the inclusion 𝑆 ↩→ 𝑆 = C[𝑠 4 , 𝑠 3…”