Tokyo Journal of Mathematics

$\mathcal{D}$-Modules and Arrangements of Hyperplanes

Francisco James LEÓN TRUJILLO

Full-text: Open access

Abstract

Let $\mathcal{A}$ be a central arrangement of hyperplanes in $\mathbb{C}^n$ defined by the homogeneous polynomial $d_{\mathcal{A}}$. Let $D_n$ be the Weyl algebra of rank $n$ over $\mathbb{C}$ and let $P=\mathbb{C}[x_1,\ldots ,x_n,d_{\mathcal{A}}^{-1}]$ be the algebra of rational functions on the variety $Y_{\mathcal{A}}=\mathbb{C}^n\setminus \bigcup_{H\in \mathcal{A}}H$. Studying the structure of $P$ as a $D_n$-module we obtain a sequence of new $D_n$-modules. These modules allow us to define useful complexes that determine the De Rham cohomology of $Y_{\mathcal{A}}=\mathbb{C}^n\setminus \bigcup_{H\in \mathcal{A}}H$. Finally we compute the Poincaré series of $P$.

Article information

Source
Tokyo J. Math., Volume 29, Number 2 (2006), 429-444.

Dates
First available in Project Euclid: 1 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1170348177

Digital Object Identifier
doi:10.3836/tjm/1170348177

Mathematical Reviews number (MathSciNet)
MR2284982

Zentralblatt MATH identifier
1140.32020

Subjects
Primary: 32S22: Relations with arrangements of hyperplanes [See also 52C35]
Secondary: 52C35: Arrangements of points, flats, hyperplanes [See also 32S22] 13N10: Rings of differential operators and their modules [See also 16S32, 32C38] 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series

Citation

LEÓN TRUJILLO, Francisco James. $\mathcal{D}$-Modules and Arrangements of Hyperplanes. Tokyo J. Math. 29 (2006), no. 2, 429--444. doi:10.3836/tjm/1170348177. https://projecteuclid.org/euclid.tjm/1170348177


Export citation

References

  • Björk, J. E., Rings of differential operators, North Holland Mathematics Library $\mathbf{21}$, Amsterdam (1979).
  • Brion, M., Vergne, M., Arrangement of hyperplanes I. Rational functions and Jeffrey-Kirwan residue. Ann. Sceint. Éc. Norm. Sup. $\mathbf{32}$ (1999), 715–741.
  • Coutinho, S. C., A Primer of Algebraic $\mathcal{D}$-modules, London Math. Soc. Student Texts $\mathbf{33}$ (1995).
  • Dold A., Lectures on Algebraic Topology, Classics in Mathematics, Springer-Verlag, Berlin (1980).
  • Horiuchi, H., Terao, H., The Poincaré series of the algebra of rational functions which are regular outside hyperplanes, J. Algebra $\mathbf{266}$ no. 1 (2003), 169–179.
  • Grothendieck, A., On the De Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. $\mathbf{29}$ (1966), 95–103.
  • León Trujillo, F. J., PhD Thesis, University of Rome “La Sapienza” (2003).
  • Orlik, P., Solomon, L., Combinatorics and topology of complements of hyperplanes, Invent. Math. $\mathbf{56}$ (1980), 167–189.
  • Orlik, P., Terao, H., Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften $\mathbf{300}$, Springer-Verlag, Berlin (1992).
  • Terao, H., Algebras generated by reciprocals of linear forms, J. Algebra $\mathbf{250}$ no. 2 (2002), 549–558.