### Session S07 - Differential operators in algebraic geometry and commutative algebra

Monday, July 19, 16:30 ~ 17:00 UTC-3

## The length of $\mathcal{D}\frac{1}{f}.$

### Thomas Bitoun

#### University of Calgary , Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $f$ be an absolutely irreducible multivariate polynomial with rational coefficients. We consider the $\mathcal{D}$-submodule of the complex local cohomology module $H^1_f(\mathcal{O})$ generated by the class of $\frac{1}{f}.$ Its $\mathcal{D}$-module length is closely related to that of the local cohomology of the reduction modulo $p$ of $f, H^1_{f_p}(\mathcal{O}_p),$ for large primes $p.$ We compute the lengths in the case of $f$ quasi-homogeneous with an isolated singularity and present a conjecture for the general isolated singularity case.

Partly joint work with Travis Schedler (Imperial College, United Kingdom).