## Harmonic metrics and connections with irregular singularities.(English)Zbl 0947.32019

Let $$X$$ be a compact Riemann surface, and $$D\subset X$$ be a finite set of points. Let $${\mathcal M}$$ be a locally free $${\mathcal O}_X[^*D]$$-module of finite rank equipped with a connection $$\nabla: {\mathcal M}\to {\mathcal M} \otimes_{{\mathcal O}_X}\Omega^1_X$$, which may have regular or irregular singularities at each point of $$D$$. Therefore, $${\mathcal M}$$ is a holonomic module over the ring $${\mathcal D}_X$$ of holomorphic differential operators on $$X$$. There exist a unique holonomic $${\mathcal D}_X$$-submodule $${\mathcal M}_{\min} \subset {\mathcal M}$$ such that $${\mathcal O}_X [^*D] \otimes_{{\mathcal O}_X} {\mathcal M}_{\min}= {\mathcal M}$$ and $${\mathcal M}_{\min}$$ has no quotient supported on a subset of $$D$$. Fix a complete metric on $$X^*=X-D$$ which, at each point of $$D$$, is locally equivalent to the Poincaré metric on the punctured disk.
The author shows that there exists a hermitian metric $$k$$ on the flat bundle $$C^\infty_{X^*} \otimes_{{\mathcal O}_{X^*}} {\mathcal M}|_{X^*}$$ such that the $$L^2$$ complex $${\mathcal L}^\bullet_{ (2)}({\mathcal M})$$ is defined depending on the metric on $$X^*$$ and the metric $$k$$ is quasi-isomorphic to the de Rham complex $$({\mathcal M}_{\min} \otimes_{{\mathcal O}_X} \Omega^1_X, \nabla)$$.
Note that the result is given without any regularity condition on the connection.

### MSC:

 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 35A20 Analyticity in context of PDEs
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### References:

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