Let $\mathcal{M}$ be a holonomic D-module on a complex analytic (or alternatively, algebraic) manifold $X$. One can attach to it (using a good filtration) a characteristic cycle $Ch(\mathcal{M})$ which is an analytic cycle on $T^*X$ whose components are conic Lagrangian subvarieties and their multiplicities are positive integers (see e. g. Def. 1.8.5 in the book “Analytic D-modules and applications” by Bjoern).

Is it true that for any short exact sequence of holonomic D-modules $$0\to \mathcal{M}_1\to \mathcal{M}_2\to \mathcal{M}_3\to 0$$ one has $$Ch(\mathcal{M}_2)=Ch(\mathcal{M}_1)+Ch(\mathcal{M}_3).$$

An answer for algebraic D-modules would be equally useful.

Apologies if the question is too basic. I am not a specialist.


1 Answer 1


Isn’t this Björk’s 3.1.4 (p. 130)?


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