The characteristic cycle and the singular support of a constructible sheaf
This paper was inspired by Kashiwara-Schapira's analogous theory over the field of complex numbers \(\mathbb{C}\) and various conjectures of Deligne. Using Belinson's singular supports, the author first constructs the characteristic cycles for constructible étale sheaves. Let \(X\) be a smooth and projective variety of dimension \(d\) over a perfect field of characteristic \(p\), and \(\mathcal F\) a constructible étale sheaf on \(X\). The singular support \(SS(\mathcal{F})\) of \(\mathcal{F}\), defined by \textit{A. Beilinson} [Sel. Math., New Ser. 22, No. 4, 1797--1819 (2016; Zbl 1375.14058)], is a closed subscheme of \(T^*X\), the cotangent bundle of \(X\). All its irreducible components \(C_1, \dots, C_k\) are of dimension \(d\). The characteristic cycle \(CC(\mathcal{F}) \) of \(\mathcal{F}\) is a cycle supported on \(SS(\mathcal{F})\): \[ CC(\mathcal {F}) = \sum_{i=1}^{n}n_iC_i \] where \(n_i\in \mathbb{Z}\). The coefficients \(n_i\) are defined using a generalization of Milnor's formula for the number of vanishing cycles. Its main interest is that it can be used to compute the Euler-Poincaré characteristic of \(\mathcal{F}\) by the following index formula: \[ \chi(X, F) = (CC(\mathcal{F}), T^*_XX)_{T^*X} \] where \(T^*_XX\) is the zero section. This provides a generalization to higher dimension of the Grothendieck-Ogg-Shafarevich formula. The main ingredients of this paper are: Radon and Legendra transforms after Brylinski, geometric theory of Lefschetz pensils, ramifications theory for imperfect residue fields and Deligne's theory of vanishing cycles over general bases.
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