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Holonomic \(D\)-modules on abelian varieties. (English) Zbl 1386.14079

Publ. Math., Inst. Hautes Étud. Sci. 121, 1-55 (2015); erratum ibid. 123, 361-362 (2016).
This paper discusses structure theory for holonomic \(D\)-modules on complex abelian varieties in terms of their Fourier-Mukai transforms.
Let \(A\) be a complex abelian variety, and let \(A^\natural\) be the moduli space of pairs \((L,\nabla)\) where \(L\) is a line bundle and \(\nabla\) an integrable connection \(\nabla: L\to \Omega_A^1\otimes L\). Let \(D_A\) be the sheaf of differential operators on \(A\). The Fourier-Mukai transform, denoted \(\text{FM}_A(-)\), is a functor from the bounded derived category of coherent \(D_A\)-modules to the bounded derived category of coherent sheaves on \(A^\natural\).
Let \(\mathcal{M}\) be any \(D_A\)-module, or more generally a complex of \(D_A\)-modules with holonomic cohomology sheaves. Any \((L,\nabla)\) in \(A^\natural\) may be thought of as a \(D_A\)-module. Consider the cohomology of the de Rham complex of \(\mathcal{M}\otimes_{\mathcal{O}_A}(L,\nabla)\). The cohomology support locus \(S^k_m(A,\mathcal{M})\) is the subset of \(A^\natural\) where the \(k\)th cohomology has dimension at least \(m\). The first theorem of the paper proves that the sets \(S^k_m(A,\mathcal{M})\) are finite unions of linear subvarieties of \(A^\natural\) (Definition 2.3).
A subset of \(A^\natural\) is said to definable in terms of \(\text{FM}_A(\mathcal{M})\) if it can be realized as the support of a complex obtained by applying any sheaf-theoretic operatins to \(\text{FM}_A(\mathcal{M})\). The support of any \(\text{FM}_A(\mathcal{M})\) is the union over all \(k\) of \(S^k_1(A,\mathcal{M})\), so by the previous theorem it is a finite union of linear subvarieties. The author proves a stronger result: any subset of \(A^\natural\) definable in terms of some \(\text{FM}_A(\mathcal{M})\) is a finite union of linear subvarieties.
The author proves that \(\text{FM}_A\) sends the standard \(t\)-structure on the bounded derived category of holonomic \(D_A\)-modules to what is known as the \(m\)-perverse \(t\)-structure on the bounded derived category of coherent sheaves on \(A^\natural\). As a consequence of this theorem and the description of the support of \(\text{FM}_A(\mathcal{M})\), holonomic \(D_A\)-modules can be characterized in terms of codimensions of their cohomology support loci. All of the above results are strengthened in the case where \(\mathcal{M}\) is a simple or semisimple holonomic \(D\)-module, and most results have been translated to the language of perverse sheaves via the Riemann-Hilbert correspondence.
Finally, the author asks the question of describing the image of complexes of holonomic \(D_A\)-modules under the functor \(\text{FM}_A(-)\). A conjectural answer is provided.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14D20 Algebraic moduli problems, moduli of vector bundles
14K05 Algebraic theory of abelian varieties
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