[Paper Review] On the generic vanishing theorem of Cartier modules

Alan Marc Watson, Yuchen Zhang|arXiv (Cornell University)|Apr 10, 2014

Advanced Topics in Algebra12 references3 citations

TL;DR

This paper generalizes the Generic Vanishing theorem to positive characteristic using inverse systems of coherent sheaves and derived category techniques. It establishes equivalent conditions for GV-inverse systems, extending Hacon and Patakfalvi’s results and highlighting pathologies in positive characteristic, such as failure of codimension bounds and non-closed supports in cohomology loci.

ABSTRACT

We generalize the Generic Vanishing theorem by Hacon and Patakfalvi in the spirit of Pareschi and Popa. We give several examples illustrating the pathologies appearing in the positive characteristic setting.

Motivation & Objective

• To extend the Generic Vanishing theorem of Hacon and Patakfalvi to positive characteristic using inverse systems of coherent sheaves.
• To investigate whether the equivalent conditions in the characteristic zero setting (e.g., codimension bounds on cohomology support loci) generalize to positive characteristic.
• To identify pathologies in the positive characteristic setting, such as failure of codimension bounds and non-closed supports in cohomology loci.
• To establish a framework for M-regularity and Generic Vanishing in positive characteristic via derived limits and Fourier-Mukai transforms.
• To clarify the role of the Mittag-Leffler condition in ensuring compatibility between inverse limits and cohomology

Proposed method

• Introduces the notion of a GV-inverse system of coherent sheaves via inverse limits and derived category hocolimits.
• Uses the Fourier-Mukai transform $ R ilde{S} $ and its dual $ R ilde{S}(D_A(ullet)) $ to relate cohomological vanishing to support conditions.
• Applies the Mittag-Leffler condition to ensure compatibility between inverse limits and cohomology groups in derived categories.
• Employs spectral sequences and isomorphisms involving $ H^i(A, ullet imes L^lat) $ to analyze vanishing behavior under ample line bundles.
• Utilizes duality and base change theorems to relate $ H^i(A, ilde{igwedge}_e imes L^lat) $ to $ H^i( ilde{A}, R ilde{S}( ilde{igwedge}_e) imes L^lat) $.
• Applies results from Pareschi and Popa on M-regularity and IT conditions to preserve vanishing under tensoring with locally free sheaves

Experimental results

Research questions

• RQ1 Can the equivalence between cohomological vanishing, support codimension bounds, and M-regularity in characteristic zero be generalized to positive characteristic?
• RQ2 What pathologies arise in the cohomology support loci $ V^i(ullet) $ in positive characteristic, particularly regarding closedness and codimension?
• RQ3 Under what conditions does the inverse system $ igracevert ilde{igwedge}_e igracevert $ satisfy the Mittag-Leffler condition, and how does this affect derived limit behavior?
• RQ4 Can the preservation of IT₀ conditions under tensoring with locally free sheaves be extended to GV-inverse systems?
• RQ5 Is the support of $ ext{im}(R^i ilde{S}( ilde{igwedge}) o R^i ilde{S}( ilde{igwedge}_e)) $ always closed in positive characteristic?

Key findings

• The paper establishes the equivalence of four conditions for a GV-inverse system: cohomological vanishing of $ ilde{igwedge} imes L^lat $, vanishing of $ ilde{igwedge} $-cohomology in derived Fourier-Mukai transform, support codimension bounds, and inverse limit compatibility.
• It proves that if $ ilde{igwedge}_e $ is a Cartier module, then $ ilde{igwedge}_e $ satisfies the conditions of Theorem 1.3, generalizing Hacon-Patakfalvi’s result.
• An example (Example 3.4) shows that the chain of inclusions $ V^0( ilde{igwedge}) riangleright V^1( ilde{igwedge}) riangleright ext{etc.} $ can fail in positive characteristic.
• Another example (Example 3.2) demonstrates that the support of $ ext{im}(R^i ilde{S}( ilde{igwedge}) o R^i ilde{S}( ilde{igwedge}_e)) $ is not necessarily closed, invalidating codimension arguments.
• The paper shows that $ ilde{igwedge} imes H $ satisfies IT₀ if $ ilde{igwedge}_e $ is a GV-inverse system and $ H $ satisfies IT₀.
• It proves that $ ilde{igwedge}_e imes ilde{igwedge} $ is a GV-inverse system if $ ilde{igwedge}_e $ is and $ ilde{igwedge} $ satisfies Generic Vanishing conditions