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Register a new userEnhanced nearby and vanishing cycles in dimension one and Fourier transform
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Enhanced ind-sheaves provide a suitable framework for the irregular Riemann-Hilbert correspondence. In this paper, we give some precisions on nearby and vanishing cycles for enhanced perverse objects in dimension one. As an application, we give a topological proof of the following fact. Let $mathcal M$ be a holonomic algebraic $mathcal D$-module on the affine line, and denote by ${}^{mathsf{L}}mathcal M$ its Fourier-Laplace transform. For a point $a$ on the affine line, denote by $ell_a$ the corresponding linear function on the dual affine line. Then, the vanishing cycles of $mathcal M$ at $a$ are isomorphic to the graded component of degree $ell_a$ of the Stokes filtration of ${}^{mathsf{L}}mathcal M$ at infinity.
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We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures and their generalizations. This implies, e.g., semipositivity for the relative canonical divisor of a semistable reduction in positive characteristic and it gives some new strong results generalizing semipositivity even for complex varieties.
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