\(V\)-filtrations in positive characteristic and test modules. (English) Zbl 1352.13005
According to P. Deligne [Lect. Notes Math. 340, 165–196 (1973; Zbl 0266.14010)], the theory of nearby and vanishing cycles is an important tool to study of hypersurface singularities. Nearby and vanishing cycles are derived functors from the derived category of constructible sheaves on \(X\) (or \(X_{\text{ét}}\)) to the derived category of constructible sheaves on the fiber \(X_\theta\). These come equipped with an action of the (étale) fundamental group. For a complex manifold \(X\), the famous Riemann-Hilbert correspondence states that there is a functor from the bounded derived category of regular holonomic \(D_X\)-modules to the bounded derived category of constructible sheaves inducing an equivalence of category which respects the “six operations”.
By work of B. Malgrange [Astérisque 101–102, 243–267 (1983; Zbl 0528.32007)] and M. Kashiwara [Lect. Notes Math. 1016, 134–142 (1983; Zbl 0566.32022)] one has a direct construction of nearby and vanishing cycles for regular holonomic \(D_{X}\)-modules without passing through the Riemann-Hilbert correspondence. The key ingredient in this construction is the so-called \(V\)-filtration of a (regular holonomic) \(D_{X}\)-module whose associated graded pieces recover nearby and vanishing cycles. The construction of the \(V\)-filtration itself uses so-called Bernstein-Sato polynomials.
The author of the paper under review is concerned with a partial characteristic \(p\) analog of \(V\)-filtrations. The characteristic \(p\)-version of multiplier ideals are so-called test ideals. In work of M. Blickle [J. Algebr. Geom. 22, No. 1, 49–83 (2013; Zbl 1271.13009)] this point of view is further emphasized and the construction of test ideals is extended to modules. The author constructs a V -filtration for Cartier modules and show that this coincides with the test module filtration under some conditions. In particular, this means that the test ideal filtration (say in a polynomial ring) is uniquely determined by discreteness, rationality, the Brian-Skoda property and how the Cartier operator acts on the filtration.
The author uses work of M. Blickle and G. Böckle [J. Reine Angew. Math. 661, 85–123 (2011; Zbl 1239.13007); “Cartier crystals”, Preprint, arXiv:1309.1035] and of M. Emerton and M. Kisin [The Riemann-Hilbert correspondence for unit \(F\)-crystals. Paris: Société Mathématique de France (2004; Zbl 1056.14025)] to drive an analog of a Riemann-Hilbert correspondence. The paper is good for all researchers in the field of the subject.
By work of B. Malgrange [Astérisque 101–102, 243–267 (1983; Zbl 0528.32007)] and M. Kashiwara [Lect. Notes Math. 1016, 134–142 (1983; Zbl 0566.32022)] one has a direct construction of nearby and vanishing cycles for regular holonomic \(D_{X}\)-modules without passing through the Riemann-Hilbert correspondence. The key ingredient in this construction is the so-called \(V\)-filtration of a (regular holonomic) \(D_{X}\)-module whose associated graded pieces recover nearby and vanishing cycles. The construction of the \(V\)-filtration itself uses so-called Bernstein-Sato polynomials.
The author of the paper under review is concerned with a partial characteristic \(p\) analog of \(V\)-filtrations. The characteristic \(p\)-version of multiplier ideals are so-called test ideals. In work of M. Blickle [J. Algebr. Geom. 22, No. 1, 49–83 (2013; Zbl 1271.13009)] this point of view is further emphasized and the construction of test ideals is extended to modules. The author constructs a V -filtration for Cartier modules and show that this coincides with the test module filtration under some conditions. In particular, this means that the test ideal filtration (say in a polynomial ring) is uniquely determined by discreteness, rationality, the Brian-Skoda property and how the Cartier operator acts on the filtration.
The author uses work of M. Blickle and G. Böckle [J. Reine Angew. Math. 661, 85–123 (2011; Zbl 1239.13007); “Cartier crystals”, Preprint, arXiv:1309.1035] and of M. Emerton and M. Kisin [The Riemann-Hilbert correspondence for unit \(F\)-crystals. Paris: Société Mathématique de France (2004; Zbl 1056.14025)] to drive an analog of a Riemann-Hilbert correspondence. The paper is good for all researchers in the field of the subject.
Reviewer: Manouchehr Misaghian (Prairie View)
MathOverflow Questions:
Vanishing cycles of a locally constant sheaf for a smooth morphism in the \(l = p\)-caseMSC:
| 13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |
| 14B05 | Singularities in algebraic geometry |
Keywords:
nearby and vanishing cycles; Riemann-Hilbert correspondence; V-filtration; Cartier modules; test idealsCitations:
Zbl 0266.14010; Zbl 0528.32007; Zbl 0566.32022; Zbl 1271.13009; Zbl 1239.13007; Zbl 1056.14025Software:
MathOverflowReferences:
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