Lyubeznik numbers for nonsingular projective varieties

Abstract

Abstract. In this paper, we determine completely the Lyubeznik numbers λ i,j (A) of the local ring A at the vertex of the affine cone over a nonsingular projective variety V , where V is defined over a field of characteristic zero, in terms of the dimensions of the algebraic de Rham cohomology spaces of V . In particular, we prove that these numbers are intrinsic numerical invariants of V , even though a priori their definition depends on an embedding into projective space. This provides supporting evidence for a positive answer to the question of embedding-independence for arbitrary varieties in characteristic zero, which is still open.

Simple D-module components of local cohomology modules

“…Switala [16] in a recent paper about Lyubeznik numbers recovers independently our Theorem 4.8(1) for the vertex of the cone over a nonsingular projective variety. His argument is similar to ours, but uses cohomology instead of homology.…”

Connectedness and Lyubeznik Numbers

“…The answer has been proven to be affirmative if k has prime characteristic [Zha11], or if X is smooth [Swi15]. Additionally, the highest Lyubeznik number is always independent of the choice of embedding [Zha07], as is λ 0,1 (A) (cf.…”

Cohomological dimension, Lyubeznik numbers, and connectedness in mixed characteristic

“…Although our principal motivation for seeking such a theorem is to better understand the topology of a ring's spectrum, cohomological dimension is itself an influential research topic. It has connections with, for instance, the number of equations defining a variety [BS98, ILL + 07], depth [PS73,Lyu06b,Var13,DT15], and the vanishing of singular and algebraic de Rham cohomology [GLS98,Lyu07,Swi15]. The new SVT may be a useful tool for extending results previously known only in equal characteristic to the mixed characteristic setting.…”