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Integral representations for solutions of exponential Gauß-Manin systems. (English) Zbl 1171.32019
The Gauss-Manin systems of a function $$f$$ on a smooth affine complex variety $$U$$ is generalised by considering an additional regular function $$g$$. Consider on $$\mathcal O_U$$ the flat connection $$\nabla u = du+dg\cdot u$$ and let $$\mathcal O_U e^g$$ be the associated holonomic $$\mathcal D_U$$-module. The exponential Gauss-Manin systems are defined to be the cohomology sheaves of the direct image $$f_+(\mathcal O_U e^g)$$. If $$g\neq 0$$, the connection is irregular singular at infinity.
The authors show that holomorphic solutions admit representations in terms of period integrals over topological chains in an appropriate homology theory: chains with possibly closed support and with rapid decay condition.

##### MSC:
 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 14F40 de Rham cohomology and algebraic geometry 32C38 Sheaves of differential operators and their modules, $$D$$-modules
##### Keywords:
Gauss-Manin systems: D-modules
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