ترغب بنشر مسار تعليمي؟ اضغط هنا

On the $p$-supports of a holonomic $mathcal{D}$-module

184   0   0.0 ( 0 )
 نشر من قبل Thomas Bitoun
 تاريخ النشر 2010
  مجال البحث
والبحث باللغة English
 تأليف Thomas Bitoun




اسأل ChatGPT حول البحث

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 $p.$



قيم البحث

اقرأ أيضاً

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 $mathcal{M}_{mathrm{r eg}}$ is reconstructed from the de Rham complex of $mathcal{M}$ by the regular Riemann-Hilbert correspondence. Similarly, on a topological space, the embedding of sheaves into enhanced ind-sheaves has a left quasi-inverse functor, called here sheafification. Regularization and sheafification are intertwined by the irregular Riemann-Hilbert correspondence. Here, we study some of their properties. In particular, we provide a germ formula for the sheafification of enhanced specialization and microlocalization.
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 ecture that the cone of effective surfaces is generated by these surfaces. As the surfaces we define can be defined in any genus $gge 3$, we further conjecture that they generate the cone of effective surfaces on the perfect cone toroidal compactification of ${mathcal A}_g$ for any $gge 3$.
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 -homogeneous free divisor is Koszul free, which is also proved in this paper
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 onents. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $mathbb{P}^2$, is used to get a sharp lower bound for the initial degree of the Jacobian module $N(f)$, under a semistability condition.
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 results in which the symplectic manifold was compact. The main tool is a finiteness theorem for R-constructible sheaves on a real analytic manifold in a non proper situation.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا