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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.

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
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