×

\(t\)-structures on the derived categories of holonomic \({\mathcal D}\)-modules and coherent \({\mathcal O}\)-modules. (English) Zbl 1073.14023

Let \(X\) be a complex manifold. Denote by \(D_c^b({\mathbb C}_X)\) be the derived category of bounded complexes of sheaves on \(X\) with constructible cohomologies, let \(D_c^b({\mathbb C}_X)^{op}\) be the corresponding opposite category, and let \(D_{\text{rh}}^b({\mathcal D}_X)\) be the derived category of bounded complexes of \({\mathcal D}_X\)-modules with regular holonomic cohomologies. One of the motivations of introducing the notion of \(t\)-structures and perverse sheaves by A. Beilinson, J. Bernstein, P. Deligne and O.Gabber [“Faisceaux pervers”, Astérisque 100 (1982; Zbl 0536.14011)] was to understand what the objects corresponding to regular holonomic \({\mathcal D}_X\)-modules by the Riemann-Hilbert correspondence \(R{\mathcal H}om_{{\mathcal D}_X}(-;{\mathcal O}_X): D_{\text{rh}}^b({\mathcal D}_X)\to D_c^b({\mathbb C}_X)^{\text{op}}\) are. By means of the notion of \(t\)-structure (recalled in section 2 of the paper under review) the answer is: the \(t\)-structure of the middle perversity on \(D_c^b({\mathbb C}_X)\) corresponds to the trivial \(t\)-structure on \(D_{\text{rh}}^b({\mathcal D}_X)\).
In this paper, the author answers (in the case where \(X\) is a smooth algebraic variety) the converse question of determining the \(t\)-structure on \(D_{\text{rh}}^b({\mathcal D}_X)\) corresponding to the trivial \(t\)-structure on \(D_c^b({\mathbb C}_X)\). It turns out that this \(t\)-structure can be extended to a \(t\)-structure on the derived category \(D_{\text{qc}}^d({\mathcal D}_X)\) of bounded complexes of \({\mathcal D}_X\)-modules with quasi-coherent cohomologies. He furthermore gives the condition for a decreasing sequence of families of supports to give a \(t\)-structure on the derived category of coherent \({\mathcal O}\)-modules.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32C38 Sheaves of differential operators and their modules, \(D\)-modules
18E30 Derived categories, triangulated categories (MSC2010)

Citations:

Zbl 0536.14011
PDFBibTeX XMLCite
Full Text: arXiv Link