Good formal structure for meromorphic flat connections on smooth projective surfaces. (English) Zbl 1183.14027

Mochizuki, Takuro

Miwa, Tetsuji (ed.) et al., Algebraic analysis and around in honor of Professor Masaki Kashiwara’s 60th birthday. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-51-8/hbk). Advanced Studies in Pure Mathematics 54, 223-253 (2009).

Let \(X\) be a smooth complex projective surface, and let \(D\) be a divisor with normal crossings on \(X.\) Let \(({\mathcal E},\nabla)\) be a flat meromorphic connection on \((X,D),\) where \({\mathcal E}\) is a locally free \({\mathcal O}_X(\ast D)\)-module, and \(\nabla: {\mathcal E} \rightarrow {\mathcal E}\otimes \Omega^1_{X/\mathbb C}\) denotes a flat connection.
The author proves the following algebraic version of a conjecture of C. Sabbah [Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2. Astérisque. 263. Paris: Societe Mathematique de France, Paris (2000; Zbl 0947.32005)]: there exists a regular birational morphism \(\pi: \widetilde{X} \rightarrow X\) such that \(\pi^{-1}({\mathcal E}, \nabla)\) has the good formal structure. To be more precise, it is proved that after some blowing up the induced connection has no turning points, that is, it has the good formal structure.
The author remarks that recently it was announced a proof of Sabbah’s conjecture without the algebraic assumption [K. Kedlaya, “Good formal structures on flat meromorphic connections, I: Surfaces”, arXiv:0811.0190].
For the entire collection see [Zbl 1160.32002].

Reviewer: Aleksandr G. Aleksandrov (Moskva)

MSC:

14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C55	The Levi problem in complex spaces; generalizations
14J10	Families, moduli, classification: algebraic theory

Keywords:

meromorphic flat connection, projective surfaces; irregular singularities, \(p\)-curvature, resolution of turning points

Citations:

Zbl 0947.32005