Holonomic D-modules on abelian varieties

Schnell, Christian

Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 1-55.

We study the Fourier-Mukai transform for holonomic D-modules on complex abelian varieties. Among other things, we show that the cohomology support loci of a holonomic D-module are finite unions of linear subvarieties, which go through points of finite order for objects of geometric origin; that the standard t-structure on the derived category of holonomic complexes corresponds, under the Fourier-Mukai transform, to a certain perverse coherent t-structure in the sense of Kashiwara and Arinkin-Bezrukavnikov; and that Fourier-Mukai transforms of simple holonomic D-modules are intersection complexes in this t-structure. This supports the conjecture that Fourier-Mukai transforms of holonomic D-modules are “hyperkähler perverse sheaves”.

Mots clés : Line Bundle, Abelian Variety, Coherent Sheave, Coherent Sheaf, Distinguished Triangle

[Ara92] Arapura, D. Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves, Bull., New Ser., Am. Math. Soc., Volume 26 (1992), pp. 310-314 | DOI | MR | Zbl

[AB10] Arinkin, D.; Bezrukavnikov, R. Perverse coherent sheaves, Mosc. Math. J., Volume 10 (2010), pp. 3-29 | MR | Zbl

[BS94] Bando, S.; Siu, Y.-T. Stable Sheaves and Einstein-Hermitian Metrics (1994), pp. 39-50

[BBD82] Beĭlinson, A. A.; Bernstein, J.; Deligne, P. Faisceaux pervers, Analysis and Topology on Singular Spaces, I (1982), pp. 5-171

[Bon10] Bonsdorff, J. Autodual connection in the Fourier transform of a Higgs bundle, Asian J. Math., Volume 14 (2010), pp. 153-173 | DOI | MR | Zbl

[Dim04] Dimca, A. Sheaves in Topology (2004) (xvi+236) | DOI | Zbl

[FK00] Franecki, J.; Kapranov, M. The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties, Duke Math. J., Volume 104 (2000), pp. 171-180 | DOI | MR | Zbl

[GL87] Green, M.; Lazarsfeld, R. Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math., Volume 90 (1987), pp. 389-407 | DOI | MR | Zbl

[GL91] Green, M.; Lazarsfeld, R. Higher obstructions to deforming cohomology groups of line bundles, J. Am. Math. Soc., Volume 1 (1991), pp. 87–103-103 | MR

[Hac08] Hacon, C. A derived category approach to generic vanishing, J. Reine Angew. Math., Volume 575 (2004), pp. 173-187 | MR | Zbl

[HTT08] Hotta, R.; Takeuchi, K.; Tanisaki, T. D-Modules, Perverse Sheaves, and Representation Theory (2008) (xii+407) | DOI | Zbl

[Jar02] Jardim, M. Nahm transform and spectral curves for doubly-periodic instantons, Commun. Math. Phys., Volume 225 (2002), pp. 639-668 | DOI | MR | Zbl

[Kas04] Kashiwara, M. t-structures on the derived categories of holonomic D-modules and coherent 𝒪-modules, Mosc. Math. J., Volume 4 (2004), pp. 847-868 | MR | Zbl

[KW11] T. Krämer and R. Weissauer, Vanishing theorems for constructible sheaves on abelian varieties, | arXiv

[Lau96] G. Laumon, Transformation de Fourier généralisée, | arXiv

[Mal04] Malgrange, B. On irregular holonomic D-modules, Éléments de la théorie des systèmes différentiels géométriques (2004), pp. 391-410

[MM74] Mazur, B.; Messing, W. Universal Extensions and One Dimensional Crystalline Cohomology (1974) | Zbl

[Moc10] T. Mochizuki, Holonomic D-module with Betti structure, | arXiv

[Moc11] T. Mochizuki, Wild harmonic bundles and wild pure twistor D-modules, Astérisque, 340 (2011).

[Moc13] T. Mochizuki, Asymptotic behaviour and the Nahm transform of doubly periodic instantons with square integrable curvature, | arXiv

[Muk81] Mukai, S. Duality between D(X) and $D\left(\widehat{X}\right)$ with its application to Picard sheaves, Nagoya Math. J., Volume 81 (1981), pp. 153-175 | MR | Zbl

[PR01] Polishchuk, A.; Rothstein, M. Fourier transform for D-algebras. I, Duke Math. J., Volume 109 (2001), pp. 123-146 | DOI | MR | Zbl

[Pop12] Popa, M. Generic vanishing filtrations and perverse objects in derived categories of coherent sheaves, Derived Categories in Algebraic Geometry (2012), pp. 251-278

[PS14] Popa, M.; Schnell, C. Kodaira dimension and zeros of holomorphic one-forms, Ann. Math., Volume 179 (2014), pp. 1-12 | DOI | MR

[PS13] M. Popa and C. Schnell, Generic vanishing theory via mixed Hodge modules, Forum Math., Sigma, 1, e1, 60pp (2013). doi:. | DOI

[Rot96] Rothstein, M. Sheaves with connection on abelian varieties, Duke Math. J., Volume 84 (1996), pp. 565-598 | DOI | MR | Zbl

[Sab13] Sabbah, C. Théorie de Hodge et correspondance de Kobayashi-Hitchin sauvages (d’après T. Mochizuki), Astérisque, Volume 352 (2013), pp. 201-243 (Séminaire Bourbaki. Vol. 2011/2012, Exposés 1043–1058)

[Sai88] Saito, M. Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., Volume 24 (1988), pp. 849-995 | DOI | MR | Zbl

[Sai90] Saito, M. Mixed Hodge modules, Publ. Res. Inst. Math. Sci., Volume 26 (1990), pp. 221-333 | DOI | MR | Zbl

[Sai91] Saito, M. Hodge conjecture and mixed motives. I, Complex Geometry and Lie Theory (1991), pp. 283-303 | DOI

[Sch13] C. Schnell, Torsion points on cohomology support loci: from D-modules to Simpson’s theorem, 2013, to appear in Recent Advances in Algebraic Geometry (Ann Arbor, 2013). | arXiv

[Sim93] Simpson, C. Subspaces of moduli spaces of rank one local systems, Ann. Sci. Éc. Norm. Super., Volume 26 (1993), pp. 361-401 | Numdam | Zbl

[Wat60] Watts, C. E. Intrinsic characterizations of some additive functors, Proc. Am. Math. Soc., Volume 11 (1960), pp. 5-8 | DOI | MR | Zbl

[Wei12] R. Weissauer, Degenerate perverse sheaves on abelian varieties, | arXiv