On Lyubeznik type invariants

Abstract

We discuss for an affine variety Y embedded in affine space X two sets of integers attached to Y ⊆ X via local and de Rham cohomology spectral sequences. We give topological interpretations, study them in small dimension, and investigate to what extent one can attach them to projective varieties.

Local cohomology -- an invitation

“…Earlier, Dao and Takagi, inspired by remarks of Varbaro, showed that over any field, Serre's condition S 3 implies that λ d´1,d " 0, [DT16], while in increasing generality it was shown in [Wal01b,Lyu06a,Zha07] that rκpAq ě dimpAq ´1s ô rλ d,d pAq " 1s. In [RWZ21] are some other results on the effect of Serre's conditions pS t q on ΛpAq. 4.4.2.…”

“…We note in passing that there are algorithmic methods that can compute the pages of this spectral sequence as vector spaces over C, see [OT99, Wal00, Wal01a, OT01]. In the sequence (4.2.1.1), the Tor-groups involved vanish for the index exceeding dim X, and so the spectral sequence operates clearly inside the rectangle 0 ď i ď lcd R pIq, 0 ď j ď n. However, it is actually limited to a much smaller, triangular region, compare [RWZ21].…”

“…The construction of [RSW21] starts with a variety whose Picard number is greater than one, and from it constructs a suitable X. In [RWZ21] it is shown that if the rational Picard group of X is Q then almost all Lyubeznik numbers of X are projective. In particular, this applies to determinantal varieties so that the Lörincz-Raicu computation in [LR20] determines the vast majority of the entries of the Lyubeznik tables for such varieties under all embeddings.…”

“…Using algebraic de Rham cohomology, this is actually shown over all fields of characteristic zero. These Čech-de Rham numbers are further investigated in [RWZ21] from the viewpoint of projectivity since, if I is homogeneous, one can ask to what extent these numbers are defined by the associated projective variety (rather than the affine cone). [RWZ21] studies their behavior under Veronese maps, and the degeneration of the spectral sequence.…”