ﻻ يوجد ملخص باللغة العربية
For a smooth variety $Y$ over a perfect field of positive characteristic, the sheaf $D_Y$ of crystalline differential operators on $Y$ (also called the sheaf of $PD$-differential operators) is known to be an Azumaya algebra over $T^*_{Y},$ the cotangent space of the Frobenius twist $Y$ of $Y.$ Thus to a sheaf of modules $M$ over $D_Y$ one can assign a closed subvariety of $T^*_{Y},$ called the $p$-support, namely the support of $M$ seen as a sheaf on $T^*_{Y}.$ We study here the family of $p$-supports assigned to the reductions modulo primes $p$ of a holonomic $mathcal{D}$-module. We prove that the Azumaya algebra of differential operators splits on the regular locus of the $p$-support and that the $p$-support is a Lagrangian subvariety of the cotangent space, for $p$ large enough. The latter was conjectured by Kontsevich. Our approach also provides a new proof of the involutivity of the singular support of a holonomic $mathcal{D}$-module, by reduction modulo $p.$
On a complex manifold, the embedding of the category of regular holonomic D-modules into that of holonomic D-modules has a left quasi-inverse functor $mathcal{M}mapstomathcal{M}_{mathrm{reg}}$, called regularization. Recall that $mathcal{M}_{mathrm{r
We determine five extremal effective rays of the four-dimensional cone of effective surfaces on the toroidal compactification $overline{mathcal A}_3$ of the moduli space ${mathcal A}_3$ of complex principally polarized abelian threefolds, and we conj
We find explicit free resolutions for the $scr D$-modules ${scr D} f^s$ and ${scr D}[s] f^s/{scr D}[s] f^{s+1}$, where $f$ is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi
We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible comp
On a complex symplectic manifold we prove a finiteness result for the global sections of solutions of holonomic DQ-modules in two cases: (a) by assuming that there exists a Poisson compactification (b) in the algebraic case. This extends our previous