Using the motivic stable homotopy category over a field \(k\), a smooth variety \(X\) over \(k\) has an Euler characteristic \(\chi(X/k)\) in the Grothendieck-Witt ring \(\operatorname{GW}(k)\). The rank of \(\chi(X/k)\) is the classical \(\mathbb{Z}\)-valued Euler characteristic, defined using singular cohomology or étale cohomology, and the signature of \(\chi(X/k)\) under a real embedding \(\sigma:k\to \mathbb{R}\) gives the topological Euler characteristic of the real points \(X^\sigma(\mathbb{R})\). \par We develop tools to compute \(\chi(X/k)\), assuming \(k\) has characteristic \(\neq 2\) and apply these to refine some classical formulas in enumerative geometry, such as formulas for the top Chern class of the dual, symmetric powers and tensor products of bundles, to identities for the Euler classes in Chow-Witt groups. We also refine the classical Riemann-Hurwitz formula to an identity in \(\operatorname{GW}(k)\) and compute \(\chi(X/k)\) for hypersurfaces in \(\mathbb{P}^{n+1}_k\) defined by a polynomial of the form \(\sum_{i=0}^{n+1}a_iX_i^m\); this latter includes the case of an arbitrary quadric hypersurface. \par This paper is a revision of [\textit{M. Levine},``Toward an enumerative geometry with quadratic forms'', Preprint, \url{arXiv:1703.03049v3}].

14C17, 14F42

Euler characteristics, Euler classes, Chow-Witt groups, Grothendieck-Witt ring

• 1. H. Abelson, On the Euler characteristic of real varieties. Michigan Math. J. 23 (1976), no. 3, 267-271 (1977). DOI 10.1307/mmj/1029001721; zbl 0346.14021; MR0424828.
• 2. A. Ananyevskiy, The special linear version of the projective bundle theorem. Compos. Math. 151 (2015), no. 3, 461-501. DOI 10.1112/S0010437X14007702; zbl 1357.14026; MR3320569; arxiv 1205.6067.
• 3. A. Ananyevskiy, SL-oriented cohomology theories. Motivic homotopy theory and refined enumerative geometry, 1-19, Contemp. Math., 745, Amer. Math. Soc., Providence, RI, 2020. DOI 10.1090/conm/745/15020; zbl 1442.14078; MR4071210.
• 4. A. Asok, J. Fasel, Comparing Euler classes. Q. J. Math. 67 (2016), no. 4, 603-635. DOI 10.1093/qmath/haw033; zbl 1372.14013; MR3609848; arxiv 1306.5250.
• 5. M. F. Atiyah, Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. (4) 4 (1971), 47-62. DOI 10.24033/asens.1205; zbl 0212.56402; MR0286136.
• 6. T. Bachmann, K. Wickelgren, \( \mathbb A^1\)-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections. Preprint 2020. arxiv 2002.01848.
• 7. J. Barge, F. Morel, Groupe de Chow des cycles orientés et classe d'Euler des fibrés vectoriels. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 4, 287-290. DOI 10.1016/S0764-4442(00)00158-0; zbl 1017.14001; MR1753295.
• 8. C. Bethea, J. Kass, K. Wickelgren, Examples of wild ramification in an enriched Riemann-Hurwitz formula. Motivic homotopy theory and refined enumerative geometry, 69-82, Contemp. Math., 745, Amer. Math. Soc., Providence, RI, 2020. DOI 10.1090/conm/745/15022; zbl 1441.14076; MR4071212; arxiv 1812.03386.
• 9. B. Calmès, J. Fasel, Finite Chow-Witt correspondences. Preprint 2014. arxiv 1412.2989.
• 10. B. Calmès, J. Hornbostel, Push-forwards for Witt groups of schemes. Comment. Math. Helv. 86 (2011), no. 2, 437-468. DOI 10.4171/CMH/230; zbl 1226.19003; MR2775136; arxiv 0806.0571.
• 11. F. Déglise, F. Jin, A. A. Khan, Fundamental classes in motivic homotopy theory. Preprint 2018. arxiv 1805.05920.
• 12. A. Dold, D. Puppe, Duality, trace, and transfer, 81-102, in Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), edited by K. Borsuk and A. Kirkor, PWN, Warsaw. zbl 0473.55008; MR0656721.
• 13. D. Dugger, Coherence for invertible objects and multigraded homotopy rings. Algebr. Geom. Topol. 14 (2014), no. 2, 1055-1106. DOI 10.2140/agt.2014.14.1055; zbl 1312.18002; MR3180827; arxiv 1302.1465.
• 14. J. Fasel, Groupes de Chow-Witt. Mém. Soc. Math. Fr. (N.S.), 113 (2008). DOI 10.24033/msmf.425; zbl 1190.14001; MR2542148.
• 15. J. Fasel, V. Srinivas, Chow-Witt groups and Grothendieck-Witt groups of regular schemes. Advances in Mathematics 221 (2009), no. 1, 302-329. DOI 10.1016/j.aim.2008.12.005; zbl 1167.13006; MR2509328.
• 16. N. Feld, Milnor-Witt cycle modules. J. Pure Appl. Algebra 224 (2020), no. 7, 106298, 44 pp. DOI 10.1016/j.jpaa.2019.106298; zbl 1442.14026; MR4058234; arxiv 1811.12163.
• 17. N. Feld, Morel homotopy modules and Milnor-Witt cycle modules. Preprint 2019. arxiv 1912.12680.
• 18. S. Gille, S. Scully, C. Zhong, Milnor-Witt \(K\)-groups of local rings. Adv. Math. 286 (2016), 729-753. DOI 10.1016/j.aim.2015.09.014; zbl 1335.11030; MR3415696; arxiv 1501.07631.
• 19. J. Hornbostel, M. Wendt, Chow-Witt rings of classifying spaces for symplectic and special linear groups, J. Topol. 12 (2019), no. 3, 916-966. DOI 10.1112/topo.12103; zbl 1444.14018; MR4072161; arxiv 1703.05362.
• 20. M. Hoyois, A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula. Algebr. Geom. Topol. 14 (2014), 3603-3658. DOI 10.2140/agt.2014.14.3603; zbl 1351.14013; MR3302973; arxiv 1309.6147.
• 21. M. Hoyois, The six operations in equivariant motivic homotopy theory. Adv. Math. 305 (2017), 197-279. DOI 10.1016/j.aim.2016.09.031; zbl 1400.14065; MR3570135; arxiv 1509.02145.
• 22. P. Hu, On the Picard group of the stable \(\mathbb A^1\)-homotopy category. Topology 44 (2005), no. 3, 609-640. DOI 10.1215/kjm/1250518550; zbl 0893.14010; MR2122218.
• 23. J. F. Jardine, Motivic symmetric spectra. Doc. Math. 5 (2000), 445-553. https://elibm.org/article/10000497; zbl 0969.19004; MR1787949.
• 24. J. L. Kass, K. Wickelgren, The class of Eisenbud-Khimshiashvili-Levine is the local \(\mathbb A^1\)-Brouwer degree. Duke Math. J. 168 (2019), no. 3, 429-469. DOI 10.1215/00127094-2018-0046; zbl 1412.14014; MR3909901; arxiv 1608.05669.
• 25. J. L. Kass, K. Wickelgren, An arithmetic count of the lines on a smooth cubic surface. Notices Amer. Math. Soc. 65 (2018), no. 4, 404-405. DOI 10.1090/noti1663; zbl 1390.14111; MR3752384; arxiv 1708.01175.
• 26. M. Kerz, The Gersten conjecture for Milnor K-theory. Invent. Math. 175 (2009), no. 1, 1-33. DOI 10.1007/s00222-008-0144-8; zbl 1188.19002; MR2461425.
• 27. M. Levine, Toward an enumerative geometry with quadratic forms. Preprint 2018. arxiv 1703.03049.
• 28. M. Levine, Motivic Euler characteristics and Witt-valued characteristic classes. Nagoya Math. J. 236 (2019), 251-310. DOI 10.1017/nmj.2019.6; zbl 07209518; MR4094419; arxiv 1806.10108.
• 29. M. Levine, A. Raksit, Motivic Gauß-Bonnet formulas. Algebra Number Theory 14 (2020), no. 7, 1801-1851. DOI 10.2140/ant.2020.14.1801; zbl 07248673; MR4150251; arxiv 1808.08385.
• 30. M. Levine, Y. Yang, G. Zhao, J. Riou, Algebraic elliptic cohomology theory and flops. I. Math. Ann. 375 (2019), no. 3-4, 1823-1855. DOI 10.1007/s00208-019-01880-x; zbl 1433.14015; MR4023393; arxiv 1311.2159.
• 31. J. P. May, The additivity of traces in triangulated categories, Adv. Math. 163 (2001), no. 1, 34-73. DOI 10.1006/aima.2001.1995; zbl 1007.18012; MR1867203.
• 32. J. Milnor, Singular points of complex hypersurfaces. Annals of Mathematics Studies, 61. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo 1968. DOI 10.1515/9781400881819; zbl 0184.48405; MR0239612.
• 33. F. Morel, Sur les puissances de l'idéal fondamental de l'anneau de Witt. Comment. Math. Helv. 79 (2004), no. 4, 689-703. DOI 10.1007/s00014-004-0815-z; zbl 1061.19001; MR2099118.
• 34. F. Morel, Introduction to \(\mathbb A^1\)-homotopy theory. Lectures given at the School on Algebraic \(K\)-Theory and its Applications, ICTP, Trieste. 8-19 July, 2002. zbl 1081.14029.
• 35. F. Morel, \( \mathbb A^1\)-algebraic topology over a field. Lecture Notes in Mathematics, 2052. Springer, Heidelberg, 2012. DOI 10.1007/978-3-642-29514-0; zbl 1263.14003; MR2934577.
• 36. F. Morel, V. Voevodsky, \( \mathbb A^1\)-homotopy theory of schemes. Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45-143 (2001). DOI 10.1007/BF02698831; zbl 0983.14007; MR1813224.
• 37. D. Mumford, Theta characteristics of an algebraic curve, Ann. Sci. École Norm. Sup. (4) 4 (1971), 181-192. DOI 10.24033/asens.1209; zbl 0216.05904; MR0292836.
• 38. D. Orlov, A. Vishik, V. Voevodsky, An exact sequence for \(K^M_*/2\) with applications to quadratic forms. Ann. of Math. (2) 165 (2007), no. 1, 1-13. DOI 10.4007/annals.2007.165.1; zbl 1124.14017; MR2276765.
• 39. I. Panin, Oriented cohomology theories of algebraic varieties. II (After I. Panin and A. Smirnov). Homology Homotopy Appl. 11 (2009), no. 1, 349-405. DOI 10.4310/HHA.2009.v11.n1.a14; zbl 1169.14016; MR2529164.
• 40. I. Panin, K. Pimenov, O. Röndigs, On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory. Invent. Math. 175 (2009), no. 2, 435-451. DOI 10.1007/s00222-008-0155-5; zbl 1205.14023; MR2470112; arxiv 0709.4124.
• 41. I. Panin, C. Walter, On the motivic commutative ring spectrum \(\mathbf{BO} \). Algebra i Analiz 30 (2018), no. 6, 43-96; reprinted in St. Petersburg Math. J. 30 (2019), no. 6, 933-972. DOI 10.1090/spmj/1578; zbl 1428.14011; MR3882540; arxiv 1011.0650.
• 42. S. Pauli, Quadratic types and the dynamic Euler number of lines on a quintic threefold. Preprint 2020. arxiv 2006.12089.
• 43. D. Quillen, Higher algebraic K-theory. I. Algebraic \(K\)-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147. Lecture Notes in Math., 341. Springer, Berlin 1973. DOI 10.1007/BFb0067053; zbl 0292.18004; MR0338129.
• 44. J. Riou, Dualité de Spanier-Whitehead en géomťrie algébrique. C. R. Math. Acad. Sci. Paris 340 (2005), no. 6, 431-436. DOI 10.1016/j.crma.2005.02.002; zbl 1068.14021; MR2135324.
• 45. G. Scheja, U. Storch, Über Spurfunktionen bei vollständigen Durchschnitten. J. Reine Angew. Math. 278(279) (1975), 174-190. zbl 0316.13003; MR0393056.
• 46. G. Scheja, U. Storch, Residuen bei vollständigen Durchschnitten. Math. Nachr. 91 (1979), 157-170. DOI 10.1002/mana.19790910113; zbl 0455.13002; MR0563607.
• 47. M. Schlichting, Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem. J. Pure Appl. Algebra 221 (2017), no. 7, 1729-1844. DOI 10.1016/j.jpaa.2016.12.026; zbl 1360.19008; MR3614976; arxiv 1209.0848.
• 48. M. Schlichting, Hermitian \(K\)-theory of exact categories. J. K-Theory 5 (2010), 105-165. DOI 10.1017/is009010017jkt075; zbl 1328.19009; MR2600285.
• 49. M. Schlichting, G. S. Tripathi, Geometric models for higher Grothendieck-Witt groups in \(mathbb A^1\)-homotopy theory. Math. Ann. 362 (2015), no. 3-4, 1143-1167. DOI 10.1007/s00208-014-1154-z; zbl 1331.14028; MR3368095; arxiv 1309.5818.
• 50. B. Totaro, The Chow ring of a classifying space. Algebraic K-theory (Seattle, WA, 1997), 249-281, Proc. Sympos. Pure Math., 67. Amer. Math. Soc., Providence, RI, 1999. zbl 0967.14005; MR1743244; arxiv math/9802097.
• 51. V. Voevodsky, Motivic cohomology with \(\mathbb Z/2\)-coefficients. Publ. Math. Inst. Hautes Études Sci., 98 (2003), 59-104. DOI 10.1007/s10240-003-0010-6; zbl 1057.14028; MR2031199.
• 52. V. Voevodsky, \( \mathbb A^1\)-homotopy theory. Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math. 1998, Extra Vol. I, 579-604. https://elibm.org/article/10011738; zbl 0907.19002; MR1648048.
• 53. M. Wendt, Chow-Witt rings of Grassmannians. Preprint 2018, rev. 2020. arxiv 1805.06142.

Fakultät Mathematik, Universität Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany