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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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