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Perverse sheaves on semi-abelian varieties – a survey of properties and applications. (English) Zbl 1453.32037

Summary: We survey recent developments in the study of perverse sheaves on semi-abelian varieties. As concrete applications, we discuss various restrictions on the homotopy type of complex algebraic manifolds (expressed in terms of their cohomology jump loci), homological duality properties of complex algebraic manifolds, as well as new topological characterizations of semi-abelian varieties.

MSC:

32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
14F17 Vanishing theorems in algebraic geometry
14F06 Sheaves in algebraic geometry
55N25 Homology with local coefficients, equivariant cohomology
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[1] Aluffi, P., Mihalcea, L.C., Schürmann, J., Su, C.: Positivity of Segre-MacPherson classes (2019). arXiv:1902.00762
[2] Beilinson, AA; Bernstein, J.; Deligne, P.; Astérisque, I., Faisceaux pervers, Analysis and Topology on Singular Spaces (1982), Paris: Société Mathématique de France, Paris · Zbl 0536.14011
[3] Bhatt, B.; Schnell, C.; Scholze, P., Vanishing theorems for perverse sheaves on abelian varieties, revisited, Selecta Math. (N.S.), 24, 1, 63-84 (2018) · Zbl 1454.14054
[4] Budur, N.; Wang, B., The signed Euler characteristic of very affine varieties, Int. Math. Res. Not. IMRN, 2015, 14, 5710-5714 (2015) · Zbl 1362.14062
[5] Budur, N., Wang, B.: Absolute sets and the decomposition theorem (2017). arXiv:1702.06267. Ann. Sci. École Norm. Sup. (to appear) · Zbl 1453.14054
[6] Bieri, R.; Eckmann, B., Groups with homological duality generalizing Poincaré duality, Invent. Math., 20, 103-124 (1973) · Zbl 0274.20066
[7] de Cataldo, MAA; Migliorini, L., The decomposition theorem, perverse sheaves and the topology of algebraic maps, Bull. Amer. Math. Soc. (N.S.), 46, 4, 535-633 (2009) · Zbl 1181.14001
[8] Chen, JA; Hacon, CD, Characterization of abelian varieties, Invent. Math., 143, 2, 435-447 (2001) · Zbl 0996.14020
[9] Denham, G.; Suciu, AI, Local systems on arrangements of smooth, complex algebraic hypersurfaces, Forum Math. Sigma, 6, # e6 (2018) · Zbl 1400.55002
[10] Denham, G.; Suciu, AI; Yuzvinsky, S., Abelian duality and propagation of resonance., Selecta Math. (N.S.), 23, 4, 2331-2367 (2017) · Zbl 1381.55005
[11] Dimca, A., Sheaves in Topology Universitext (2004), Berlin: Springer, Berlin
[12] Elduque, E., Geske, Chr. Maxim, L.: On the signed Euler characteristic property for subvarieties of abelian varieties. J. Singul. 17, 368-387 (2018) · Zbl 1401.32024
[13] Fernández de Bobadilla, JF; Kollár, J., Homotopically trivial deformations, J. Singul., 5, 85-93 (2012) · Zbl 1292.32003
[14] Franecki, J.; Kapranov, M., The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties, Duke Math. J., 104, 1, 171-180 (2000) · Zbl 1021.14016
[15] Gabber, O.; Loeser, F., Faisceaux pervers \(\ell \)-adiques sur un tore, Duke Math. J., 83, 3, 501-606 (1996) · Zbl 0896.14009
[16] Gelfand, S.; MacPherson, R.; Vilonen, K., Perverse sheaves and quivers, Duke Math. J., 83, 3, 621-643 (1996) · Zbl 0861.32022
[17] Goresky, M.; MacPherson, R., Intersection homology. II, Invent. Math., 72, 1, 77-129 (1983) · Zbl 0529.55007
[18] Goresky, M.; MacPherson, R., Stratified Morse Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (1988), Berlin: Springer, Berlin · Zbl 0639.14012
[19] Green, M.; Lazarsfeld, R., Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math., 90, 2, 389-407 (1987) · Zbl 0659.14007
[20] Green, M.; Lazarsfeld, R., Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc., 4, 1, 87-103 (1991) · Zbl 0735.14004
[21] Huh, J., The maximum likelihood degree of a very affine variety, Compositio Math., 149, 8, 1245-1266 (2013) · Zbl 1282.14007
[22] Huh, J.; Wang, B., Enumeration of points, lines, planes, etc, Acta Math., 218, 2, 297-317 (2017) · Zbl 1386.05021
[23] Iitaka, S., Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23, 3, 525-544 (1976) · Zbl 0342.14017
[24] Jost, J.; Zuo, K., Vanishing theorems for \(L^2\)-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry, Comm. Anal. Geom., 8, 1, 1-30 (2000) · Zbl 0978.32024
[25] Kashiwara, M.: Faisceaux constructibles et systèmes holonômes d’équations aux dérivées partielles linéaires à points singuliers réguliers. In: Séminaire Goulaouic-Schwartz, 1979-1980, Exp. No. 19. École Polytechnique, Palaiseau (1980) · Zbl 0444.58014
[26] Kashiwara, M., The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., 20, 2, 319-365 (1984) · Zbl 0566.32023
[27] Krämer, T., Perverse sheaves on semiabelian varieties, Rend. Semin. Mat. Univ. Padova, 132, 83-102 (2014) · Zbl 1317.14037
[28] Krämer, T.; Weissauer, R., Vanishing theorems for constructible sheaves on abelian varieties, J. Algebraic Geom., 24, 3, 531-568 (2015) · Zbl 1338.14023
[29] Liu, Y., Maxim, L., Wang, B.: Topology of subvarieties of complex semi-abelian varieties (2017). arXiv:1706.07491
[30] Liu, Y., Maxim, L., Wang, B.: Generic vanishing for semi-abelian varieties and integral Alexander modules. Math. Z. 10.1007/s00209-018-2194-y · Zbl 1469.14039
[31] Liu, Y.; Maxim, L.; Wang, B., Mellin transformation, propagation, and abelian duality spaces, Adv. Math., 335, 231-260 (2018) · Zbl 1400.32017
[32] Liu, Y., Maxim, L., Wang, B.: Perverse sheaves on semi-abelian varieties (2018). arXiv:1804.05014
[33] Lück, W., \(L^2\)-Invariants: Theory and Applications to Geometry and \(K\)-Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. A (2002), Berlin: Springer, Berlin
[34] MacPherson, RD, Chern classes for singular algebraic varieties, Ann. Math., 100, 423-432 (1974) · Zbl 0311.14001
[35] MacPherson, R.: Global questions in the topology of singular spaces. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pp. 213-235. PWN, Warsaw (1984)
[36] MacPherson, R.; Vilonen, K., Elementary construction of perverse sheaves, Invent. Math., 84, 2, 403-435 (1986) · Zbl 0597.18005
[37] Maxim, L.: Intersection homology and perverse sheaves, with applications to singularities. https://www.math.wisc.edu/ maxim/book.pdf (in progress) · Zbl 1476.55001
[38] Mebkhout, Z.; Iagolnitzer, D., Sur le problème de Hilbert-Riemann., Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory. Lecture Notes in Physics, 90-110 (1980), Berlin: Springer, Berlin
[39] Mebkhout, Z., Une autre équivalence de catégories, Compositio Math., 51, 1, 63-88 (1984) · Zbl 0566.32021
[40] Palais, RS; Smale, S., A generalized Morse theory, Bull. Amer. Math. Soc., 70, 165-172 (1964) · Zbl 0119.09201
[41] Saito, M., Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., 24, 6, 849-995 (1988) · Zbl 0691.14007
[42] Saito, M., Mixed Hodge modules, Publ. Res. Inst. Math. Sci., 26, 2, 221-333 (1990) · Zbl 0727.14004
[43] Schnell, Chr, Holonomic \({D}\)-modules on abelian varieties, Publ. Math. Inst. Hautes Études Sci., 121, 1-55 (2015) · Zbl 1386.14079
[44] Stanley, RP, The number of faces of a simplicial convex polytope, Adv. Math., 35, 3, 236-238 (1980) · Zbl 0427.52006
[45] Schürmann, J.; Tibar, M., Index formula for MacPherson cycles of affine algebraic varieties, Tohoku Math. J., 62, 1, 29-44 (2010) · Zbl 1187.14013
[46] Weissauer, R., Vanishing theorems for constructible sheaves on abelian varieties over finite fields, Math. Ann., 365, 1-2, 559-578 (2016) · Zbl 1378.14014
[47] Weissauer, R.: Remarks on the nonvanishing of cohomology groups for perverse sheaves on abelian varieties (2016). arXiv:1612.01500
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