# Constructible sheaves are holonomic

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## Abstract

We develop the theory of singular support for étale sheaves on algebraic varieties over an arbitrary base field.

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## Notes

1. The terminology of [5] 5.4.12 is “h is non-characteristic for C.”

2. Here is a simple proof of this fact due to Kuznetsov: By Theorem 1.3(ii) it is enough to prove that $$SS({\mathcal {F}})$$ is isotropic. We use induction by $$\dim Y$$ where Y is the support of $${\mathcal {F}}$$. By Hironaka one finds a proper map. $$r : Z\rightarrow X$$ and a dense open $$V \subset Y$$ such that Z is smooth, $$r (Z)=Y$$, $$r^{-1}(V) \buildrel \sim \over \rightarrow V$$ (so we have the embedding $$j : V \hookrightarrow Z$$), $$D:=Z\smallsetminus V$$ is a divisor with normal crossings in Z, and $${\mathcal {F}}|_V$$ is locally constant. The cone $${\mathcal {G}}$$ of the canonical map $${\mathcal {F}} \rightarrow r_* j_* ({\mathcal {F}} |_V )$$ is supported on $$Y\smallsetminus V$$, so $$SS({\mathcal {G}})$$ is isotropic by the induction assumption. Since $$SS({\mathcal {F}} )\subset SS({\mathcal {G}}) \cup SS (r_* j_* ({\mathcal {F}} |_V ))$$, it remains to show that $$SS(r_* j_* ({\mathcal {F}} |_V ))$$ is isotropic. Now $$SS( j_* ({\mathcal {F}} |_V ))$$ is isotropic: indeed, it is the union of the zero section and of the conormals to intersections of the components of D (use 2.2(ii) applied to a local covering of Z ramified at D on which $${\mathcal {F}} |_V$$ trivializes to reduce the assertion to the case of constant $${\mathcal {F}}$$, then use 1.4(ii)). We are done since $$SS(r_* j_* ({\mathcal {F}} |_V ))\subset r_\circ SS(j_* ({\mathcal {F}} |_V ))$$ (see 2.2(ii)) and $$r_\circ$$ sends isotropic cones to isotropic cones.

3. Deligne shows that every C that is not a conormal can be identified étale locally at the generic point with one of $$C_n$$’s from the next example.

4. Proof The map $$r_n$$ is finite, so $$SS({{\mathcal {F}}}_n )\subset r_{n\circ } SS (\mathbb {Z}/\ell _V)$$ by 2.2(ii). Since $$SS(\mathbb {Z}/\ell _V)$$ is the zero section of $$T^* V$$ (see 2.1(iii)) and $$r_n$$ is étale over the complement to the x-axis, an immediate computation shows that $$r_{n\circ } SS (\mathbb {Z}/\ell _V )$$ is the union of $$C_n$$ and the zero section of $$T^* V$$. Now $$SS({{\mathcal {F}}}_n )$$ contains the zero section of $$T^* V$$ since $${{\mathcal {F}}}_n$$ is nonzero at the generic point of V, and $$SS({{\mathcal {F}}}_n )$$ is not equal to the zero section at the generic point $$\eta _x$$ of the x-axis since $${{\mathcal {F}}}_n$$ is not locally constant there (use 2.1(iii)). We are done since every closed subcone of $$C_n$$ other than $$C_n$$ is contained in the zero section at $$\eta _x$$.

5. I.e., $$\{ {{\mathcal {F}}}_\alpha \}$$ are irreducible perverse sheaves that can be realized as subquotients of some perverse sheaf cohomology $${}^p {{\mathcal {H}}}^a {{\mathcal {F}}}$$.

6. To check that $$C_1 ,C_2 \in {{\mathcal {C}}}' ({{\mathcal {F}}})$$ implies $$C_1 \cap C_2 \in {{\mathcal {C}}}' ({{\mathcal {F}}})$$ notice that if a test pair (hf) as in 1.1 with $$\dim Y=1$$ is $$C_1 \cap C_2$$-transversal, then locally on U it is either $$C_1$$- or $$C_2$$-transversal. The latter assertion need not be true if $$\dim Y>1$$ (consider cones $$C_1$$, $$C_2$$ that are nonzero with zero intersection at the generic point of X and $$(h,f)=(\text {id}_X ,\text {id}_X )$$).

7. With modification as above in case k is finite. Indeed, the modification is needed to ensure that $$SS^w ({{\mathcal {F}}})$$ for $${{\mathcal {F}}}$$ a skyscraper sheaf at $$x\in X$$ equals $$T^*_x X$$.

8. Since we live on a projective space this means that the cohomology sheaves come from Spec$$\, k$$.

9. Theorem 1.3(i) asserts that $${{\mathcal {C}}^{\text {min}}}({{\mathcal {F}}})$$ has a single element. But we did not prove it yet.

10. Use the fact that for any $$\nu \ne 0$$ in $$\mathbb {A}^n$$, the set $$\{ A(\nu ), A\in \text {Mat}_{n,m-n}(k)\}$$ is dense in $$\mathbb {A}^{m-n}$$.

11. Which is always true if the base field k is perfect.

12. So a geometric point of $$Q^{(a )}_P$$ is a collection $$(q_1,\ldots , q_a )$$ of pairwise distinct geometric points of Q such that $$\pi (q_1 )=\cdots =\pi (q_a )$$.

13. In other words, $$\rho _{(x_1 ,\ldots , x_m )}^{(m)}$$ is the restriction map $$\Gamma (\mathbb {P}, {{\mathcal {O}}}(d ))_{\bar{k}} \rightarrow \Gamma (\sqcup x_i^{(1)},{{\mathcal {O}}}(d))$$.

14. We use the fact that on a smooth variety every perverse subquotient of a locally constant perverse sheaf is locally constant.

15. Notice that this evidently excludes the situation when $$\dim Y_1=\dim Y_2 =0$$.

16. Here we assume that k is infinite; otherwise $$U\subset \mathbb {P}_{k'}$$ for a finite extension $$k'$$ of k, see 1.5.

17. Use the fact that for any subset $$A\subset T^* X$$ one has $$\pi ^\circ \overline{A}=\overline{\pi ^\circ A}$$ where $$\bar{\,\,}$$ means the closure (which follows since $$\pi$$ is open and $$d\pi : T^* X\times _X Z \rightarrow T^* Z$$ is a closed embedding).

18. Here U is a neighborhood of x where f is defined.

19. Inspired by a discussion with Deligne.

20. I.e., $$\kappa$$ vanishes on $$TC_{\text {reg} }\subset T(T^* \mathbb {P} )$$.

21. The parity of the rank does not depend on the choice of $$(x,\nu )\in C_{\text {reg}}$$: it is odd if and only if $$T_{(x,\nu )}C$$ and $$T^v (T^* \mathbb {P})$$ lie in the different components of $$\text {Gr}^\kappa (T_{(x,\nu )}(T^* \mathbb {P} ))$$.

## References

1. Brylinski, J.-L.: Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques. Astérisque 140–141, 3–134 (1986)

2. Deligne, P.: Théorèmes de finitude en cohomologie $$\ell$$-adique. Cohomologie étale (SGA $$4\frac{1}{2}$$), Lect. Notes in Math. 569, pp. 233–251. Springer, Berlin (1977)

3. Deligne, P.: Notes sur Euler-Poincaré: brouillon projet. Manuscript (2011)

4. Deligne, P.: Letter to the author from July 9, 2015

5. Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Grundlehren der Mathematischen Wissenschaften, vol. 292. Springer, Berlin (1990)

6. Katz, N.: Pinceaux de Lefschetz: théorème d’existence. Groupes de monodromie en géométrie algébrique (SGA 7 II), Lect. Notes in Math., vol. 340, pp. 212–253. Springer, Berlin (1972)

7. Saito, T.: The characteristic cycle and the singular support of a constructible sheaf. arXiv:1510.03018 (2015)

8. Sato, M., Kawai, T., Kashiwara M.: Microfunctions and Pseudo-Differential Equations, Lecture Notes in Math., vol. 287, pp. 265–529. Springer, Berlin (1973)

## Author information

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Correspondence to A. Beilinson.

To Joseph Bernstein

The author was supported in part by NSF Grant DMS-1406734.

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Beilinson, A. Constructible sheaves are holonomic. Sel. Math. New Ser. 22, 1797–1819 (2016). https://doi.org/10.1007/s00029-016-0260-z