## Abstract

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

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## Notes

The terminology of [5] 5.4.12 is “

*h*is non-characteristic for*C*.”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.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.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 \(\eta _x\).I.e., \(\{ {{\mathcal {F}}}_\alpha \}\) are irreducible perverse sheaves that can be realized as subquotients of some perverse sheaf cohomology \({}^p {{\mathcal {H}}}^a {{\mathcal {F}}}\).

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 (

*h*,*f*) 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 \((h,f)=(\text {id}_X ,\text {id}_X )\)).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 \(T^*_x X\).Since we live on a projective space this means that the cohomology sheaves come from Spec\(\, k\).

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

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 \(\mathbb {A}^{m-n}\).

Which is always true if the base field

*k*is perfect.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 \(\pi (q_1 )=\cdots =\pi (q_a )\).In other words, \(\rho _{(x_1 ,\ldots , x_m )}^{(m)}\) is the restriction map \(\Gamma (\mathbb {P}, {{\mathcal {O}}}(d ))_{\bar{k}} \rightarrow \Gamma (\sqcup x_i^{(1)},{{\mathcal {O}}}(d))\).

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

Notice that this evidently excludes the situation when \(\dim Y_1=\dim Y_2 =0 \).

Here we assume that

*k*is infinite; otherwise \( U\subset \mathbb {P}_{k'}\) for a finite extension \(k'\) of*k*, see 1.5.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).

Here

*U*is a neighborhood of*x*where*f*is defined.Inspired by a discussion with Deligne.

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

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 \(\text {Gr}^\kappa (T_{(x,\nu )}(T^* \mathbb {P} ))\).

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To Joseph Bernstein
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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

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DOI: https://doi.org/10.1007/s00029-016-0260-z