AMS :: Sugaku Expositions

Topics of motivic characteristic classes
HTML articles powered by AMS MathViewer

by Shoji Yokura
Translated by: the author

Sugaku Expositions 33 (2020), 57-84

DOI: https://doi.org/10.1090/suga/448

Published electronically: May 8, 2020

Abstract:

This is a survey about motivic Hirzebruch classes which in a sense ‘unifies’ the distinguished three theories of characteristic classes of singular varieties and some related topics.

References

Paolo Aluffi, Weighted Chern-Mather classes and Milnor classes of hypersurfaces, Singularities—Sapporo 1998, Adv. Stud. Pure Math., vol. 29, Kinokuniya, Tokyo, 2000, pp. 1–20. MR 1819626, DOI 10.2969/aspm/02910001
Aravind Asok and Jean Fasel, Secondary characteristic classes and the Euler class, Doc. Math. Extra vol.: Alexander S. Merkurjev’s sixtieth birthday (2015), 7–29. MR 3404374
J. Baez, Euler Characteristic versus Homotopy Cardinality, Fields Institute Program on Homotopy Theory and its Applications, 2003, http://math.ucr.edu/home/ baez/cardinality/cardinality.pdf
J. Baez, The Mysteries of Counting:Euler Characteristic versus Homotopy Cardinality, Categories in Algebra Geometry and Mathematical Physics, 2005, http://math. ucr.edu/home/baez/counting/
J. Baez, This Week’s Finds in Mathematical Physics, http://math.ucr.edu/home/baez/twfcontents.html
M. Banagl, J. Schürmann and S. Yokura, Fiberwise bordism groups and their bivariant analogues, in preparation.
Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 101–145. MR 412190
Kai Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), no. 3, 1307–1338. MR 2600874, DOI 10.4007/annals.2009.170.1307
Kai Behrend, Grégory Ginot, Behrang Noohi, and Ping Xu, String topology for stacks, Astérisque 343 (2012), xiv+169 (English, with English and French summaries). MR 2977576
Franziska Bittner, The universal Euler characteristic for varieties of characteristic zero, Compos. Math. 140 (2004), no. 4, 1011–1032. MR 2059227, DOI 10.1112/S0010437X03000617
Raoul Bott and Clifford Taubes, On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989), no. 1, 137–186. MR 954493, DOI 10.1090/S0894-0347-1989-0954493-5
Jean-Paul Brasselet, Existence des classes de Chern en théorie bivariante, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 7–22 (French). MR 737926
J.-P. Brasselet and M.-H. Schwartz, Sur les classes de Chern d’un ensemble analytique complexe, The Euler-Poincaré characteristic (French), Astérisque, vol. 82, Soc. Math. France, Paris, 1981, pp. 93–147 (French). MR 629125
Jean-Paul Brasselet, Jörg Schürmann, and Shoji Yokura, Classes de Hirzebruch et classes de Chern motiviques, C. R. Math. Acad. Sci. Paris 342 (2006), no. 5, 325–328 (French, with English and French summaries). MR 2201957, DOI 10.1016/j.crma.2005.12.022
Jean-Paul Brasselet, Jörg Schürmann, and Shoji Yokura, On the uniqueness of bivariant Chern class and bivariant Riemann-Roch transformations, Adv. Math. 210 (2007), no. 2, 797–812. MR 2303240, DOI 10.1016/j.aim.2006.07.014
Jean-Paul Brasselet, Jörg Schürmann, and Shoji Yokura, On Grothendieck transformations in Fulton-MacPherson’s bivariant theory, J. Pure Appl. Algebra 211 (2007), no. 3, 665–684. MR 2344222, DOI 10.1016/j.jpaa.2007.03.004
Jean-Paul Brasselet, Jörg Schürmann, and Shoji Yokura, Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal. 2 (2010), no. 1, 1–55. MR 2646988, DOI 10.1142/S1793525310000239
Vittoria Bussi and Shoji Yokura, Naive motivic Donaldson-Thomas type Hirzebruch classes and some problems, J. Singul. 10 (2014), 26–53. MR 3300284, DOI 10.5427/jsing.2014.10b
Sylvain E. Cappell and Julius L. Shaneson, Stratifiable maps and topological invariants, J. Amer. Math. Soc. 4 (1991), no. 3, 521–551. MR 1102578, DOI 10.1090/S0894-0347-1991-1102578-4
Sylvain E. Cappell and Julius L. Shaneson, Genera of algebraic varieties and counting of lattice points, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 62–69. MR 1217352, DOI 10.1090/S0273-0979-1994-00436-7
Sylvain E. Cappell, Laurentiu Maxim, Jörg Schürmann, and Julius L. Shaneson, Characteristic classes of complex hypersurfaces, Adv. Math. 225 (2010), no. 5, 2616–2647. MR 2680178, DOI 10.1016/j.aim.2010.05.007
Sylvain E. Cappell, Laurentiu Maxim, Jörg Schürmann, Julius L. Shaneson, and Shoji Yokura, Characteristic classes of symmetric products of complex quasi-projective varieties, J. Reine Angew. Math. 728 (2017), 35–63. MR 3668990, DOI 10.1515/crelle-2014-0114
Sylvain Cappell, Laurentiu Maxim, Toru Ohmoto, Jörg Schürmann, and Shoji Yokura, Characteristic classes of Hilbert schemes of points via symmetric products, Geom. Topol. 17 (2013), no. 2, 1165–1198. MR 3070522, DOI 10.2140/gt.2013.17.1165
V. I. Danilov and A. G. Khovanskiĭ, Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 5, 925–945 (Russian). MR 873655
Ben Davison, Motivic Donaldson-Thomas theory and the role of orientation data, Glasg. Math. J. 58 (2016), no. 1, 229–262. MR 3426438, DOI 10.1017/S0017089515000178
Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 498551
Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 498552
Philippe Du Bois, Complexe de de Rham filtré d’une variété singulière, Bull. Soc. Math. France 109 (1981), no. 1, 41–81 (French). MR 613848
Heath Emerson and Ralf Meyer, Bivariant $K$-theory via correspondences, Adv. Math. 225 (2010), no. 5, 2883–2919. MR 2680187, DOI 10.1016/j.aim.2010.04.024
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
William Fulton and Robert MacPherson, Categorical framework for the study of singular spaces, Mem. Amer. Math. Soc. 31 (1981), no. 243, vi+165. MR 609831, DOI 10.1090/memo/0243
José Luis González and Kalle Karu, Bivariant algebraic cobordism, Algebra Number Theory 9 (2015), no. 6, 1293–1336. MR 3397403, DOI 10.2140/ant.2015.9.1293
Mark Goresky and Robert MacPherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162. MR 572580, DOI 10.1016/0040-9383(80)90003-8
A. Grothendieck, Récoltes et Semailles – Réflexions et Témoignages sur un passé de mathématicien, Preprint, 1985.
Thomas C. Hales, What is motivic measure?, Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 2, 119–135. MR 2133307, DOI 10.1090/S0273-0979-05-01053-0
F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and modular forms, Aspects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils-Peter Skoruppa and by Paul Baum. MR 1189136, DOI 10.1007/978-3-663-14045-0
M. Kapranov, The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups, arXiv AG/0001005.
Gary Kennedy, Clint McCrory, and Shoji Yokura, Natural transformations from constructible functions to homology, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 9, 969–973 (English, with English and French summaries). MR 1302800
Shun-ichi Kimura, Fractional intersection and bivariant theory, Comm. Algebra 20 (1992), no. 1, 285–302. MR 1145334, DOI 10.1080/00927879208824340
S.-i. Kimura, Rationality and irrationality of Motivic Chow Groups (in Japanese), Lecture Notes of Kinosaki Algebraic Geometry 2013 , (2013), 5–14.
Frances Kirwan and Jonathan Woolf, An introduction to intersection homology theory, 2nd ed., Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2207421, DOI 10.1201/b15885
Steven L. Kleiman, The development of intersection homology theory, A century of mathematics in America, Part II, Hist. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1989, pp. 543–585. MR 1003155, DOI 10.24033/bsmf.2027
Maxim Kontsevich, Notes on motives in finite characteristic, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progr. Math., vol. 270, Birkhäuser Boston, Boston, MA, 2009, pp. 213–247. MR 2641191, DOI 10.1007/978-0-8176-4747-6_{7}
M. Levine, Motivic Cohomology and Algebraic Cycles: a categorical approach, http://www.math.uiuc.edu/ K-theory/0107/mca.pdf.
M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2286826
M. Levine and R. Pandharipande, Algebraic cobordism revisited, Invent. Math. 176 (2009), no. 1, 63–130. MR 2485880, DOI 10.1007/s00222-008-0160-8
Eduard Looijenga, Motivic measures, Astérisque 276 (2002), 267–297. Séminaire Bourbaki, Vol. 1999/2000. MR 1886763
I. G. Macdonald, The Poincaré polynomial of a symmetric product, Proc. Cambridge Philos. Soc. 58 (1962), 563–568. MR 143204
R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432. MR 361141, DOI 10.2307/1971080
Robert MacPherson, Characteristic classes for singular varieties, Proceedings of the Ninth Brazilian Mathematical Colloquium (Poços de Caldas, 1973) Inst. Mat. Pura Apl., São Paulo, 1977, pp. 321–327. MR 534464
Carlo Mazza, Vladimir Voevodsky, and Charles Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006. MR 2242284
Laurentiu Maxim, Morihiko Saito, and Jörg Schürmann, Hirzebruch-Milnor classes of complete intersections, Adv. Math. 241 (2013), 220–245. MR 3053711, DOI 10.1016/j.aim.2013.04.001
Laurentiu Maxim, Morihiko Saito, and Jörg Schürmann, Hirzebruch-Milnor classes and Steenbrink spectra of certain projective hypersurfaces, Arbeitstagung Bonn 2013, Progr. Math., vol. 319, Birkhäuser/Springer, Cham, 2016, pp. 265–287. MR 3618053, DOI 10.1007/978-3-319-43648-7_{9}
Laurenţiu Maxim and Jörg Schürmann, Characteristic classes of mixed Hodge modules and applications, Schubert varieties, equivariant cohomology and characteristic classes—IMPANGA 15, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2018, pp. 159–202. MR 3754192
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
Boudewijn Moonen, Das Lefschetz-Riemann-Roch-Theorem für singuläre Varietäten, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 106, Universität Bonn, Mathematisches Institut, Bonn, 1978 (German). Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn. MR 544020
Serge Ochanine, Sur les genres multiplicatifs définis par des intégrales elliptiques, Topology 26 (1987), no. 2, 143–151 (French). MR 895567, DOI 10.1016/0040-9383(87)90055-3
Toru Ohmoto, Equivariant Chern classes of singular algebraic varieties with group actions, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 1, 115–134. MR 2197579, DOI 10.1017/S0305004105008820
Toru Ohmoto, Generating functions of orbifold Chern classes. I. Symmetric products, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 2, 423–438. MR 2405899, DOI 10.1017/S0305004107000898
Toru Ohmoto, Enumerative theory of singularities and equivariant Chern classes, Sūgaku 61 (2009), no. 1, 21–39 (Japanese). MR 2560143
Toru Ohmoto, Singularities of maps and characteristic classes, School on real and complex singularities in São Carlos, 2012, Adv. Stud. Pure Math., vol. 68, Math. Soc. Japan, [Tokyo], 2016, pp. 191–265. MR 3585782, DOI 10.2969/aspm/06810191
T. Ohmoto and S. Yokura, in preparation.
C. A. M. Peters, Tata Lecture on Motivic Aspects of Hodge Theory, Lecture Notes at the Tata Institute of Fundamental Research at Mumbay, December 2007.
Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 52, Springer-Verlag, Berlin, 2008. MR 2393625
Bjorn Poonen, The Grothendieck ring of varieties is not a domain, Math. Res. Lett. 9 (2002), no. 4, 493–497. MR 1928868, DOI 10.4310/MRL.2002.v9.n4.a8
Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989) (French). MR 1000123, DOI 10.2977/prims/1195173930
Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. MR 1047415, DOI 10.2977/prims/1195171082
Morihiko Saito, Induced $\scr D$-modules and differential complexes, Bull. Soc. Math. France 117 (1989), no. 3, 361–387 (English, with French summary). MR 1020112
Morihiko Saito, ${\scr D}$-modules on analytic spaces, Publ. Res. Inst. Math. Sci. 27 (1991), no. 2, 291–332. MR 1095238, DOI 10.2977/prims/1195169840
Morihiko Saito, Mixed Hodge complexes on algebraic varieties, Math. Ann. 316 (2000), no. 2, 283–331. MR 1741272, DOI 10.1007/s002080050014
Morihiko Saito, Arithmetic mixed sheaves, Invent. Math. 144 (2001), no. 3, 533–569. MR 1833893, DOI 10.1007/s002220100133
Jörg Schürmann, Characteristic classes of mixed Hodge modules, Topology of stratified spaces, Math. Sci. Res. Inst. Publ., vol. 58, Cambridge Univ. Press, Cambridge, 2011, pp. 419–470. MR 2796417
Jörg Schürmann and Shoji Yokura, A survey of characteristic classes of singular spaces, Singularity theory, World Sci. Publ., Hackensack, NJ, 2007, pp. 865–952. MR 2342943, DOI 10.1142/9789812707499_{0}037
Jörg Schürmann and Shoji Yokura, Motivic bivariant characteristic classes and related topics, J. Singul. 5 (2012), 124–152. MR 2928939, DOI 10.5427/jsing.2012.5j
Jörg Schürmann and Shoji Yokura, Motivic bivariant characteristic classes, Adv. Math. 250 (2014), 611–649. MR 3122179, DOI 10.1016/j.aim.2013.09.024
Marie-Hélène Schwartz, Classes caractéristiques définies par une stratification d’une variété analytique complexe. I, C. R. Acad. Sci. Paris 260 (1965), 3262–3264 (French). MR 212842
V. Srinivas, The Hodge characteristic, Lectures in Jet Schemes Seminar, MSRI, December 2002, Manuscript, preprint (2002).
Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858
D. Sullivan, Combinatorial invariants of analytic spaces, Proceedings of Liverpool Singularities—Symposium, I (1969/70), Lecture Notes in Mathematics, Vol. 192, Springer, Berlin, 1971, pp. 165–168. MR 0278333
Tatsuo Suwa, Characteristic classes of singular varieties, Sūgaku 52 (2000), no. 4, 376–393 (Japanese). MR 1802958
Clifford Henry Taubes, $S^1$ actions and elliptic genera, Comm. Math. Phys. 122 (1989), no. 3, 455–526. MR 998662
R. Thom, Les classes caractéristiques de Pontrjagin des variétés triangulées, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 54–67 (French). MR 0102071
Charles B. Thomas, Elliptic cohomology, The University Series in Mathematics, Kluwer Academic/Plenum Publishers, New York, 1999. MR 1742240
Edward Witten, The index of the Dirac operator in loop space, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986) Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 161–181. MR 970288, DOI 10.1007/BFb0078045
Nobuaki Yagita, Chern classes and the Rost cohomological invariant, Kodai Math. J. 36 (2013), no. 1, 174–178. MR 3043408
Shoji Yokura, Polar classes and Segre classes on singular projective varieties, Trans. Amer. Math. Soc. 298 (1986), no. 1, 169–191. MR 857438, DOI 10.1090/S0002-9947-1986-0857438-5
Shoji Yokura, A generalized Grothendieck-Riemann-Roch theorem for Hirzebruch’s $\chi _y$-characteristic and $T_y$-characteristic, Publ. Res. Inst. Math. Sci. 30 (1994), no. 4, 603–610. MR 1308959, DOI 10.2977/prims/1195165791
Shoji Yokura, On Cappell-Shaneson’s homology $L$-classes of singular algebraic varieties, Trans. Amer. Math. Soc. 347 (1995), no. 3, 1005–1012. MR 1283567, DOI 10.1090/S0002-9947-1995-1283567-6
Shoji Yokura, A singular Riemann-Roch for Hirzebruch characteristics, Singularities Symposium—Łojasiewicz 70 (Kraków, 1996; Warsaw, 1996) Banach Center Publ., vol. 44, Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 257–268. MR 1677403
S. Yokura, Constructible functions and Hodge polynomials, unpublished note, 2003.
Shoji Yokura, Characteristic classes of proalgebraic varieties and motivic measures, Algebr. Geom. Topol. 12 (2012), no. 1, 601–641. MR 2916288, DOI 10.2140/agt.2012.12.601
Shoji Yokura, Oriented bivariant theories. I, Internat. J. Math. 20 (2009), no. 10, 1305–1334. MR 2574317, DOI 10.1142/S0129167X09005777
Shoji Yokura, Motivic Milnor classes, J. Singul. 1 (2010), 39–59. MR 2671765, DOI 10.5427/jsing.2010.1c
Shoji Yokura, Motivic characteristic classes, Topology of stratified spaces, Math. Sci. Res. Inst. Publ., vol. 58, Cambridge Univ. Press, Cambridge, 2011, pp. 375–418. MR 2796416
Shoji Yokura, Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class, Singularities in geometry and topology, IRMA Lect. Math. Theor. Phys., vol. 20, Eur. Math. Soc., Zürich, 2012, pp. 285–343. MR 3074940, DOI 10.4171/118-1/15
Boris Youssin, Witt groups of derived categories, $K$-Theory 11 (1997), no. 4, 373–395. MR 1451761, DOI 10.1023/A:1007741027370
Don Bernard Zagier, Equivariant Pontrjagin classes and applications to orbit spaces. Applications of the $G$-signature theorem to transformation groups, symmetric products and number theory, Lecture Notes in Mathematics, Vol. 290, Springer-Verlag, Berlin-New York, 1972. MR 0339202