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Given a smooth proper family $f:X\rightarrow S$, we study the (quasi)-periods of the fibres of f as (germs of) functions on S. We show that the field they generate has the same algebraic closure as that given by the flag variety coordinates parametrizing the corresponding Hodge filtration, together with their derivatives. Moreover, in the more general context of an arbitrary flat vector bundle, we determine the transcendence degree of the function field generated by the flat coordinates of algebraic sections. Our results are inspired by and generalize work of Bertrand–Zudilin.

We present a microlocal version of the Riemann–Hilbert correspondence for regular holonomic $\cal D$-modules. We show that a regular holonomic system of microdifferential equations is associated to a perverse sheaf concentrated in degree 0. Moreover, we show that this perverse sheaf can be recovered from the local system it determines on the complementary of its singular locus. We characterize the classes of perverse sheaves and local systems associated to regular holonomic systems of microdifferential equations.

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PERIODS IN FAMILIES AND DERIVATIVES OF PERIOD MAPS

A Microlocal Riemann–Hilbert Correspondence

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2 results

PERIODS IN FAMILIES AND DERIVATIVES OF PERIOD MAPS

A Microlocal Riemann–Hilbert Correspondence