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GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

  • MIHNEA POPA (a1) and CHRISTIAN SCHNELL (a2)
Abstract
Abstract

We extend most of the results of generic vanishing theory to bundles of holomorphic forms and rank-one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated with irregular varieties. Our main tools are Saito’s mixed Hodge modules, the Fourier–Mukai transform for $\mathscr{D}$ -modules on abelian varieties introduced by Laumon and Rothstein, and Simpson’s harmonic theory for flat bundles. In the process, we also discover two natural categories of perverse coherent sheaves.

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References
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Arapura D., ‘Higgs line bundles, Green–Lazarsfeld sets, and maps of Kähler manifolds to curves’, Bull. Amer. Math. Soc. (N.S.) 26(2) (1992), 310314.
Arinkin D. and Bezrukavnikov R., ‘Perverse coherent sheaves’, Mosc. Math. J. 10(1) (2010), 329.
Beilinson A., Bernstein J. and Deligne P., ‘Faisceaux pervers’, Astérisque 100 (1982), 3171.
Chen J. and Hacon C., ‘Kodaira dimension of irregular varieties’, Invent. Math. 186(3) (2011), 481500.
Clemens H. and Hacon C., ‘Deformations of the trivial line bundle and vanishing theorems’, Amer. J. Math. 124(4) (2002), 769815.
de Cataldo M. A. A. and Migliorini L., ‘The Hodge theory of algebraic maps’, Ann. Sci. Éc. Norm. Supér. (4) 38(5) (2005), 693750.
Deligne P., ‘Théorie de Hodge. II’, Publ. Math. Inst. Hautes Études Sci. 40 (1971), 557.
Ein L. and Lazarsfeld R., ‘Singularities of theta divisors and the birational geometry of irregular varieties’, J. Amer. Math. Soc. 10(1) (1997), 243258.
Eisenbud D., The Geometry of Syzygies: a Second Course in Commutative Algebra and Algebraic Geometry, Graduate Texts in Mathematics, 229 (Springer, New York, 2005).
Eisenbud D., Fløystad G. and Schreyer F.-O., ‘Sheaf cohomology and free resolutions over the exterior algebra’, Trans. Amer. Math. Soc. 355(11) (2003), 43974426.
Green M. and Lazarsfeld R., ‘Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville’, Invent. Math. 90(2) (1987), 389407.
Green M. and Lazarsfeld R., ‘Higher obstructions to deforming cohomology groups of line bundles’, J. Amer. Math. Soc. 1(4) (1991), 87103.
Griffiths P. and Harris J., Principles of Algebraic Geometry (Wiley-Interscience, 1978).
Hacon C., ‘A derived category approach to generic vanishing’, J. Reine Angew. Math. 575 (2004), 173187.
Kashiwara M., ‘ inline-graphic $t$ -structures on the derived categories of holonomic inline-graphic $\mathscr{D}$ -modules and coherent inline-graphic $\mathscr{O} $ -modules’, Mosc. Math. J. 4(4) (2004), 847868.
Kollár J., ‘Higher direct images of dualizing sheaves II’, Ann. of Math. (2) 124 (1986), 171202.
Krämer T. and Weissauer R. (2011), Vanishing theorems for constructible sheaves on abelian varieties, available at arXiv:1111.4947v1.
Laumon G., ‘Transformations canoniques et spécialisation pour les inline-graphic $\mathscr{D}$ -modules filtrés. In Differential Systems and Singularities (Luminy, 1983). Astérisque (130) (1985), 56129.
Laumon G. (1996), Transformation de Fourier généralisée, available at arXiv:alg-geom/9603004.
Lazarsfeld R. and Popa M., ‘Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds’, Invent. Math. 182(3) (2010), 605633.
Lazarsfeld R., Popa M. and Schnell C., ‘Canonical cohomology as an exterior module’, Pure Appl. Math. Q. 7(4) (2010), 15291542.
Libgober A., ‘First order deformations for rank-one local systems with a non-vanishing cohomology’, Topology Appl. 118(1–2) (2002), 159168.
MacDonald I., ‘Symmetric products of an algebraic curve’, Topology 1 (1962), 319343.
McCleary J., A User’s Guide to Spectral Sequences, 2nd edn. Cambridge Studies in Advanced Mathematics, 58 (Cambridge University Press, Cambridge, 2001).
Mukai S., ‘Duality between inline-graphic $D(X)$ and inline-graphic $D(\hat {X} )$ with its application to Picard sheaves’, Nagoya Math. J. 81 (1981), 153175.
Pareschi G. and Popa M., ‘Strong generic vanishing and a higher dimensional Castelnuovo–de Franchis inequality’, Duke Math. J. 150(2) (2009), 269285.
Pareschi G. and Popa M., ‘GV-sheaves, Fourier–Mukai transform, and generic vanishing’, Amer. J. Math. 133(1) (2011), 235271.
Popa M., ‘Generic vanishing filtrations and perverse objects in derived categories of coherent sheaves. In Derived Categories in Algebraic Geometry, Tokyo, 2011 (European Mathematical Society, 2012), 251277.
Roberts P., Homological Invariants of Modules Over Commutative Rings, Séminaire de Mathématiques Supérieures, 72 (Presses de l’Université de Montréal, Montreal, Que., 1980).
Rothstein M., ‘Sheaves with connection on abelian varieties’, Duke Math. J. 84(3) (1996), 565598.
Sabbah C., ‘Polarizable twistor inline-graphic $\mathscr{D}$ -modules’, Astérisque 300 (2005).
Sabbah C., ‘Wild twistor inline-graphic $\mathscr{D}$ -modules. In Algebraic Analysis and Around, Advanced Studies in Pure Mathematics, 54 (Math. Soc. Japan, Tokyo, 2009), 293353.
Saito M., ‘Modules de Hodge polarisables’, Publ. Res. Inst. Math. Sci. 24(6) (1988), 849995.
Saito M., ‘Mixed Hodge modules’, Publ. Res. Inst. Math. Sci. 26(2) (1990), 221333.
Saito M., ‘On the Theory of Mixed Hodge Modules’, Amer. Math. Soc. Transl. Ser. 2 160 (1994), 4761, Originally published in Japanese.
Schnell C., ‘Local duality and polarized Hodge modules’, Publ. Res. Inst. Math. Sci. 47(3) (2011), 705725.
Schnell C. (2012), Holonomic complexes on abelian varieties, Part I, available at arXiv:1112.3582.
Simpson C., ‘Higgs bundles and local systems’, Publ. Math. Inst. Hautes Études Sci. 75 (1992), 595.
Simpson C., ‘Subspaces of moduli spaces of rank one local systems’, Ann. Sci. Ec. Norm. Supér. 26 (1993), 361401.
Weissauer R. (2008), Brill–Noether sheaves, available at arXiv:math/o610923v4.
Wells R. O.Jr, Differential Analysis on Complex Manifolds, 3rd edn, with a new appendix by Oscar Garcia-Prada. Graduate Texts in Mathematics, 65 (Springer, New York, 2008).
Zucker S., ‘Hodge theory with degenerating coefficients. inline-graphic ${L}_{2} $ cohomology in the Poincaré metric’, Ann. of Math. (2) 109(3) (1979), 415476.
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