Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces.(English)Zbl 07823128
In the last decade motivic ideas have been used to establish a quadratic refinement of enumerative geometry [M. Hoyois, Algebr. Geom. Topol. 14, No. 6, 3603–3658 (2014;Zbl 1351.14013);M. Levine, Contemp. Math. 745, 163–198 (2020;Zbl 1505.14054);J. L. Kass andK. Wickelgren, Duke Math. J. 168, No. 3, 429–469 (2019;Zbl 1412.14014)]. For varieties over a field \(k\), these refined invariants live in the Grothendieck-Witt ring of \(k\)-quadratic forms \(GW(k)\). When \(k=\mathbb{C}\), the rank is the only information and this number recovers classical Gromov-Witten invariants. When \(k=\mathbb{R}\), the signature gives another invariant which turns out to be related with the one previously introduced byJ.-Y. Welschinger [Invent. Math. 162, No. 1, 195–234 (2005;Zbl 1082.14052)]; the relation is in the preprint “Toward an algebraic theory of Welschinger invariants”, by Marc Levine. Many more interesting invariants shall be expected over other base fields.
This paper concerns the computation of the quadratic refinement of the Euler characteristic of a smooth hypersurfaces. The expression obtained is based on the previous work [M. Levine andA. Raksit, Algebra Number Theory 14, No. 7, 1801–1851 (2020;Zbl 1458.14029)]. A second goal of the paper is understanding these invariants for slightly singular hypersurfaces. More precisely, given a smooth hypersurface degenerating to a singular one, the authors propose to study how the quadratic Euler characteristic changes. This would be a quadratic refinement of the so called “Deligne-Milnor formula”, which was conjectural at the time that the paper under review was written, but which is now a theorem [D. Beraldo andM. Pippi, “Proof of the Deligne-Milnor conjecture”, Preprint,arXiv:2410.02327]. Already formulating a conjectural quadratic version seems very challenging: if \(K\) and \(k\) are the generic field and the residue field of the base of the degeneration, then the two quadratic Euler characteristics one would like to compare live in two different Grothendieck-Witt rings, namely \(GW(K)\) and \(GW(k)\). A partial result in this direction is proved (Theorem 5.2). The authors explains that in order to have a statement which has the shape of a Deligne-Milnor formula, some artificial corrections are needed. Thoses corrections do not have conceptual explanations for the moment.
This paper concerns the computation of the quadratic refinement of the Euler characteristic of a smooth hypersurfaces. The expression obtained is based on the previous work [M. Levine andA. Raksit, Algebra Number Theory 14, No. 7, 1801–1851 (2020;Zbl 1458.14029)]. A second goal of the paper is understanding these invariants for slightly singular hypersurfaces. More precisely, given a smooth hypersurface degenerating to a singular one, the authors propose to study how the quadratic Euler characteristic changes. This would be a quadratic refinement of the so called “Deligne-Milnor formula”, which was conjectural at the time that the paper under review was written, but which is now a theorem [D. Beraldo andM. Pippi, “Proof of the Deligne-Milnor conjecture”, Preprint,arXiv:2410.02327]. Already formulating a conjectural quadratic version seems very challenging: if \(K\) and \(k\) are the generic field and the residue field of the base of the degeneration, then the two quadratic Euler characteristics one would like to compare live in two different Grothendieck-Witt rings, namely \(GW(K)\) and \(GW(k)\). A partial result in this direction is proved (Theorem 5.2). The authors explains that in order to have a statement which has the shape of a Deligne-Milnor formula, some artificial corrections are needed. Thoses corrections do not have conceptual explanations for the moment.
Reviewer: Giuseppe Ancona (Strasbourg)
MSC:
| 14B05 | Singularities in algebraic geometry |
| 14C15 | (Equivariant) Chow groups and rings; motives |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
| 14F42 | Motivic cohomology; motivic homotopy theory |
Cite
References:
| [1] | Arnol’d, V. I., The index of a singular point of a vector field, the Petrovskiĭ-Oleĭnik inequalities, and mixed Hodge structures, Funkc. Anal. Prilozh., 12, 1, 1-14, 1978 ·Zbl 0407.57025 |
| [2] | Ayoub, J., Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II, Astérisque, 315, 2007, vi+364 pp. (2008) ·Zbl 1153.14001 |
| [3] | Ayoub, J., Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I, Astérisque, 314, 2007, x+466 pp. (2008) ·Zbl 1146.14001 |
| [4] | Azouri, R., Motivic Euler characteristic of nearby cycles and a generalized quadratic conductor formula, 2021, Preprint |
| [5] | Bachmann, T.; Wickelgren, K., \( \mathbb{A}^1\)-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections, J. Inst. Math. Jussieu, 22, 2, 681-746, 2021 ·Zbl 1515.14037 |
| [6] | Beraldo, D.; Pippi, M., Non-commutative intersection theory and unipotent Deligne-Milnor formula, 2022, Preprint |
| [7] | Bloch, S., Cycles on arithmetic schemes and Euler characteristics of curves, (Algebraic Geometry. Algebraic Geometry, Bowdoin, 1985. Algebraic Geometry. Algebraic Geometry, Bowdoin, 1985, Proc. Symp. Pure Math., vol. 46, Part 2, 1987, Am. Math. Soc.: Am. Math. Soc. Providence, RI), 421-450 ·Zbl 0654.14004 |
| [8] | Cisinski, D.-C.; Déglise, F., Triangulated Categories of Mixed Motives, Springer Monographs in Mathematics, 2019 ·Zbl 07138952 |
| [9] | Griffiths, P. A., On the periods of certain rational integrals: I, Ann. Math. (2), 90, 460-495, 1969 ·Zbl 0215.08103 |
| [10] | Carlson, J. A.; Griffiths, P. A., Infinitesimal variations of Hodge structure and the global Torelli problem, (Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry. Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, 1980, Sijthoff & Noordhoff: Sijthoff & Noordhoff Alphen aan den Rijn-Germantown, Md.), 51-76 ·Zbl 0479.14007 |
| [11] | Deligne, P., Cohomologie des intersections completes, Exposé XI, (Deligne, P.; Katz, N., Groupes de Monodromie en Géométrie Algébrique (SGA 7 II). Groupes de Monodromie en Géométrie Algébrique (SGA 7 II), Lect. Notes Math., vol. 340, 1973, Springer: Springer Berlin-New York) ·Zbl 0265.14007 |
| [12] | Delorme, C., Espaces projectifs anisotropes, Bull. Soc. Math. Fr., 103, 2, 203-223, 1975 ·Zbl 0314.14016 |
| [13] | Dolgachev, I., Weighted projective varieties, (Group Actions and Vector Fields. Group Actions and Vector Fields, Vancouver, B.C., 1981. Group Actions and Vector Fields. Group Actions and Vector Fields, Vancouver, B.C., 1981, Lecture Notes in Math., vol. 956, 1982, Springer: Springer Berlin), 34-71 ·Zbl 0516.14014 |
| [14] | Elman, R.; Karpenko, N.; Merkurjev, A., The Algebraic and Geometric Theory of Quadratic Forms, American Mathematical Society Colloquium Publications, vol. 56, 2008, American Mathematical Society: American Mathematical Society Providence, RI ·Zbl 1165.11042 |
| [15] | Elmanto, E.; Khan, A. A., Perfection in motivic homotopy theory, Proc. Lond. Math. Soc. (3), 120, 1, 28-38, 2020 ·Zbl 1440.14123 |
| [16] | Eisenbud, D.; Levine, H. I., An algebraic formula for the degree of a \(C^\infty\) map germ, Ann. Math. (2), 106, 1, 19-44, 1977 ·Zbl 0398.57020 |
| [17] | Gel’fand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Discriminants, Resultants, and Multidimensional Determinants, Math. Theory Appl., 1994, Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA, x+523 pp ·Zbl 0827.14036 |
| [18] | Hornbostel, J., \( \mathbb{A}^1\)-representability of Hermitian K-theory and Witt groups, Topology, 44, 3, 661-687, 2005 ·Zbl 1078.19004 |
| [19] | Hoyois, M., A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula, Algebraic Geom. Topol., 14, 3603-3658, 2014 ·Zbl 1351.14013 |
| [20] | Hoyois, M., The six operations in equivariant motivic homotopy theory, Adv. Math., 305, 197-279, 2017 ·Zbl 1400.14065 |
| [21] | Hoyois, M., From algebraic cobordism to motivic cohomology, J. Reine Angew. Math., 2015, 702, 173-226, 2015 ·Zbl 1382.14006 |
| [22] | Hu, P., On the Picard group of the stable \(\mathbb{A}^1\)-homotopy category, Topology, 44, 3, 609-640, 2005 ·Zbl 1078.14025 |
| [23] | Jin, F.; Yang, E., The quadratic Artin conductor of a motivic spectrum, 2022, arXiv preprint |
| [24] | Jin, F.; Yang, E., Some results on the motivic nearby cycle, 2021, arXiv preprint |
| [25] | Kass, J. L.; Wickelgren, K., The class of Eisenbud-Khimshiashvili-Levine is the local \(\mathbb{A}^1\)-Brouwer degree, Duke Math. J., 168, 3, 429-469, 2019 ·Zbl 1412.14014 |
| [26] | Kato, K.; Saito, T., On the conductor formula of Bloch, Publ. Math. IHÉS, 100, 5-151, 2004 ·Zbl 1099.14009 |
| [27] | Kapranov, M. M., On DG-modules over the de Rham complex and the vanishing cycles functor, (Algebraic Geometry. Algebraic Geometry, Lecture Notes in Mathematics, vol. 1479, 1991) ·Zbl 0757.14010 |
| [28] | Khimshiashvili, G. M., On the local degree of a smooth map, Bull. Georgian Acad. Sci., 85, 309-312, 1977 ·Zbl 0346.55008 |
| [29] | Khimshiashvili, G. M., Signature formulae for topological invariants, Proc. A. Razmadze Math. Inst., 125, 1-121, 2001 ·Zbl 1059.58027 |
| [30] | Knebusch, M., Specialization of quadratic and symmetric bilinear forms, and a norm theorem, Acta Arith., 24, 279-299, 1973 ·Zbl 0287.15010 |
| [31] | Knebusch, M.; Rosenberg, A.; Ware, R., Structure of Witt rings and quotients of Abelian group rings, Am. J. Math., 94, 1, 119-155, 1972 ·Zbl 0248.13030 |
| [32] | Levine, M., Aspects of enumerative geometry with quadratic forms, Doc. Math., 25, 2179-2239, 2020 ·Zbl 1465.14008 |
| [33] | Levine, M.; Pepin Lehalleur, S.; Srinivas, V., Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces, 2022, Preprint |
| [34] | Levine, M.; Raksit, A., Motivic Gauss-Bonnet formulas, Algebra Number Theory, 14, 7, 1801-1851, 2020 ·Zbl 1458.14029 |
| [35] | Levine, M.; Yang, Y.; Zhao, G.; Riou, J., Algebraic elliptic cohomology theory and flops I, Math. Ann., 375, 3-4, 1823-1855, 2019 ·Zbl 1433.14015 |
| [36] | May, J. P., The additivity of traces in triangulated categories, Adv. Math., 163, 1, 34-73, 2001 ·Zbl 1007.18012 |
| [37] | Milnor, J., Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, vol. 61, 1968, Princeton University Press/University of Tokyo Press: Princeton University Press/University of Tokyo Press Princeton, N.J./Tokyo ·Zbl 0184.48405 |
| [38] | Morel, F., \( \mathbb{A}^1\)-Algebraic Topology over a Field, Lecture Notes in Mathematics, vol. 2052, 2012, Springer: Springer Heidelberg ·Zbl 1263.14003 |
| [39] | Morel, F., Introduction to \(\mathbb{A}^1\)-homotopy theory, (Contemporary Developments in Algebraic K-Theory. Lectures given at the School on Algebraic K-Theory and Its Applications, ICTP, Trieste, 8-19 July 2002. ICTP, Trieste, 8-19 July 2002, ICTP Lect. Notes, vol. XV, 2004), 357-441 ·Zbl 1081.14029 |
| [40] | Orgogozo, F., Conjecture de Bloch et nombres de Milnor, Ann. Inst. Fourier, 53, 1739-1754, 2003 ·Zbl 1065.14005 |
| [41] | Pajwani, J.; Pál, A., An arithmetic Yau-Zaslow formula, 2023, arXiv preprint |
| [42] | Riou, J., Dualité de Spanier-Whitehead en géométrie algébrique, C. R. Math. Acad. Sci. Paris, 340, 6, 431-436, 2005 ·Zbl 1068.14021 |
| [43] | Saito, T., Characteristic cycles and the conductor of direct image, J. Am. Math. Soc., 34, 2, 369-410, 2021 ·Zbl 1504.14039 |
| [44] | Saito, T., The discriminant and the determinant of a hypersurface of even dimension, Math. Res. Lett., 19, 4, 855-871, 2012 ·Zbl 1285.14046 |
| [45] | Saito, T., Jacobi sum Hecke characters, de Rham discriminant, and the determinant of ℓ-adic cohomologies, J. Algebraic Geom., 3, 3, 41-434, 1994 ·Zbl 0833.14011 |
| [46] | Saito, T., Determinant representation, Jacobi sum and de Rham discriminant, Surikaisekikenkyusho Kokyuroku, 844, 79-83, 1993 |
| [47] | Sastry, P.; Tong, Y. L., The Grothendieck trace and the de Rham integral, Can. Math. Bull., 46, 3, 429-440, 2003 ·Zbl 1066.14020 |
| [48] | Scheja, G.; Storch, U., Über Spurfunktionen bei vollständigen Durchschnitten, J. Reine Angew. Math., 278, 279, 174-190, 1975 ·Zbl 0316.13003 |
| [49] | Springer, T. A., Quadratic forms over fields with a discrete valuation. I. Equivalence classes of definite forms, Nederl. Akad. Wetensch. Proc. Ser. A 58. Nederl. Akad. Wetensch. Proc. Ser. A 58, Indag. Math., 17, 352-362, 1955 ·Zbl 0067.27605 |
| [50] | Srinivas, V., Gysin maps and cycle classes for Hodge cohomology, Proc. Indian Acad. Sci. Math. Sci., 103, 3, 209-247, 1993 ·Zbl 0816.14003 |
| [51] | The Stacks project |
| [52] | Steenbrink, J., Intersection form for quasi-homogeneous singularities, Compos. Math., 34, 2, 211-223, 1977 ·Zbl 0347.14001 |
| [53] | Szafraniec, Z., A formula for the Euler characteristic of a real algebraic manifold, Manuscr. Math., 85, 1, 1994 ·Zbl 0824.14046 |
| [54] | Takeuchi, D., Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic, 2020, arXiv preprint |
| [55] | Terakado, Y., The determinant of a double covering of the projective space of even dimension and the discriminant of the branch locus, J. Number Theory, 177, 153-169, 2017 ·Zbl 1428.11123 |
| [56] | Toën, B.; Vezzosi, G., Trace and Kunneth formulas for singularity categories and applications, Compos. Math., 158, 3, 483-528, 2022 ·Zbl 1535.14046 |
| [57] | Viergever, A. M., The quadratic Euler characteristic of a smooth projective same-degree complete intersection, 2023, arXiv preprint |
| [58] | Villaflor Loyola, R., Toric differential forms and periods of complete intersections, 2023, arXiv preprint ·Zbl 1553.14008 |
| [59] | Voevodsky, V., Motivic cohomology with \(\mathbb{Z} / 2\)-coefficients, Publ. Math. Inst. Hautes Études Sci., 98, 59-104, 2003 ·Zbl 1057.14028 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
