Cohomology of regular differential forms for affine curves
| dc.creator | Bonnet, Philippe | |
| dc.date | 2006-02-13 | |
| dc.date.accessioned | 2026-07-07T07:03:23Z | |
| dc.date.available | 2026-07-07T07:03:23Z | |
| dc.description | Let $C$ be a complex affine reduced curve, and denote by $H^1(C)$ its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant $μ'(C,x)$ that measures the complexity of the singularity of $C$ at the point $x$. Then, if $H_1(C)$ denotes the first singular homology group of $C$ with complex coefficients, we establish the following formula: $$ dim H^1(C)=dim H_1(C) + \sum_{x\in C} μ'(C,x) $$ Second we consider a family of curves given by the fibres of a dominant morphism $f:X\to \mathbb{C}$, where $X$ is an irreducible complex affine surface. We analyze the behaviour of the function $y\mapsto dim H^1(f^{-1}(y))$. More precisely, we show that it is constant on a Zariski open set, and that it is lower semi-continuous in general. | |
| dc.description | 16 pages | |
| dc.identifier | https://arxiv.org/abs/math/0602275 | |
| dc.identifier | http://arxiv.org/abs/math/0602275 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/108950 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.subject | 14F40; 14D99 | |
| dc.title | Cohomology of regular differential forms for affine curves | |
| dc.type | text |