## View abstract

### 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. document.getElementById('cloakbd5329772c3d3f0619317fbb9e5d6ac3').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addybd5329772c3d3f0619317fbb9e5d6ac3 = 'th&#111;m&#97;s.b&#105;t&#111;&#117;n' + '&#64;'; addybd5329772c3d3f0619317fbb9e5d6ac3 = addybd5329772c3d3f0619317fbb9e5d6ac3 + '&#117;c&#97;lg&#97;ry' + '&#46;' + 'c&#97;'; var addy_textbd5329772c3d3f0619317fbb9e5d6ac3 = 'th&#111;m&#97;s.b&#105;t&#111;&#117;n' + '&#64;' + '&#117;c&#97;lg&#97;ry' + '&#46;' + 'c&#97;';document.getElementById('cloakbd5329772c3d3f0619317fbb9e5d6ac3').innerHTML += '<a ' + path + '\'' + prefix + ':' + addybd5329772c3d3f0619317fbb9e5d6ac3 + '\'>'+addy_textbd5329772c3d3f0619317fbb9e5d6ac3+'<\/a>';

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).