Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I know that, when $\mathcal{F}$ is a coherent sheaf on a smooth algebraic variety $X$ over a field $k$ equipped with a connection $$ \nabla: \mathcal{F} \to \mathcal{F} \otimes \Omega^1_X, $$ then $\mathcal{F}$ is automatically locally free.

Does this result remain valid when one has a connection with logarithmic singularities along some normal crossings divisor?

Any reference, examples, counter-examples will be appreciated.

share|improve this question
add comment

1 Answer 1

I may be misunderstanding something here, but isn't the following a counterexample?

Consider $X=\mathbb A^1 = Spec(\mathbb C[z])$, with the divisor $D= \{0\}$. Let $\mathcal F$ be the 1-dimensional skyscraper sheaf at $0$. Let $\delta$ denote a non-zero section of the stalk at $0$ (so $z\delta = 0$). The sheaf $\mathcal F$ admits a connection with logarithmic singularities defined by:

$$ \nabla (\delta) = - \delta \otimes \frac{dz}{z}.$$

But $\mathcal F$ is certainly not locally free (I am really thinking of $\mathcal F$ as a lattice inside the $D$-module of delta functions at $0$).

share|improve this answer
2  
Yes, this is the standard counterexample. –  S. Carnahan Jun 28 '13 at 2:42
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.