On a generalization of Hilbert’s 21st problem. (English) Zbl 0616.14004

Hain, Richard M.

Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 609-627 (1986).

Let V be a smooth complex algebraic variety and X a smooth compactification of V such that \(X\setminus V\) is a divisor with normal crossings. Any integrable meromorphic gl(n)-valued 1-form on X with logarithmic poles along \(X\setminus V\) defines (via its associated connection) a monodromy representation \(\rho: \pi_ 1(V)\to GL(n).\) The paper deals with the problem of detecting those representations which come from some 1-form as above. Let B be the completion of the group \({\mathbb{C}}\)-algebra of \(\pi_ 1(V)\) with respect to its augmentation ideal. The main result says there exist a closed ideal \(I\subset B\) (which is defined via the ”natural” mixed Hodge structure on B) and a topological algebra \(A\subset B/I\) such that A contains the image of the map \(\pi_ 1(V)\to B/I\) and such that a representation \(\rho\) comes from a 1-form as above if and only if the map \(\pi_ 1(V)\to^{\rho}GL(n)\hookrightarrow gl(n)\) factors through a continuous \({\mathbb{C}}\)-algebra map \(A\to gl(n)\). As opposed to Deligne’s generalisation of the Riemann-Hilbert problem. [P. Deligne, ”Equations différentielles à points singuliers réguliers”, Lect. Notes Math. 163 (1970; Zbl 0244.14004)] where connections on nontrivial bundles were also considered and where no restrictions turned out to exist for a representation to come from a connection, in the setting above (which corresponds to the case of trivial bundles) there are two kinds of restrictions imposed on the representations which come from 1- forms: first a group theoretic restriction (living in \(\ker (\pi_ 1(V)\to B)\) and secondly a Hodge-theoretic restriction (living in I).

Reviewer: A.Buium

MSC:

14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
35Q15	Riemann-Hilbert problems in context of PDEs
14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials

Keywords:

monodromy representation; mixed Hodge structure

Citations:

Zbl 0244.14004

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