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

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

Zbl 0244.14004
Full Text:

### References:

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