Quadratic intersection theory and characteristic classes

The motives seminar this semester will be an introductory course in some of the foundations of quadratic refinements of classical intersection theory. Our main focus will be on the theory given by the Chow-Witt groups and Witt sheaf cohomology, but we will begin with results in a more general setting, mainly about the structure of the cohomology of "Grassmann-like" varieties for SL-oriented theories. We then introduce the Chow-Witt groups and the cohomology of the Witt sheaves. After this, we will study the Chow-Witt ring, and related groups, for special linear and symplective versions of the classical Grassmann varieties, as well as for the classical Grassmann varieties. Next, we will look at quadratic characteristic classes, such as the Pontryagin and Euler classes, and develop their main properties. We conclude with some computations and applications to classical enumerative problems.

Previous Motives Seminars

Follow the Teaching link at the top of this page to find links to previous motives seminars.

Program

October 11: Marc Levine. Introduction and overview We recall the main points of intersections theory with values in the Chow ring, including a discussion of the Schubert calculus for the Chow ring of Grassmann varieties, and some applications. We then give an overview of the main points to be covered in the seminar.

October 18: Dhyan Aranha.  Special linear projective bundle formula, I Cover sections 1-7 of [An15], concentrating on the foundations of SL-oriented theories, Borel, Pontryagin and Euler classes, and their basic properties.

October 25:: Alessandro D'Angelo.  Special linear projective bundle formula, II [An15]. Here you should discuss the main results, computing \(A^*\) for the special linear Grassmannians, and for the classifying spaces \(BSL_n\). If time permits, you can complement this with the main results of [An20] on Pontryagin and Euler classes for \(SL^c\)-oriented bundles.

November 8: Sabrina Pauli.  Milnor-Witt sheaves and the Chow-Witt groups Go over the definitions basic properties of the Milnor-Witt sheaves, and the Chow-Witt groups, following [BCDFØ], Chap. 2, sections 1,2,3. Include the Rost-Schmid resolution for the Milnor-Witt sheaves and derive the same for the Witt sheaves via \(\eta\) localization.

November 1: holiday

November 15: Chow-Witt rings of symplectic classifying spaces, I [HW] sections 1-5

November 11: Chow-Witt rings of symplectic classifying spaces, II [HW] sections 6-9

November 29: Chirantan Chowhury.  Chow-Witt rings of Grassmannians, I [W18], sections 4, 5, 6.

December 6: Pietro Gigli.  Chow-Witt rings of Grassmannians, II [W20], sections 2, 3, 4 (section 2 is mainly a recollection of results from [W18], so you need only present the new material).

December 13: Chow-Witt rings of Grassmannians, III [W20], sections 5, 6, 7

December 20: Chow-Witt rings of Grassmannians, IV [W20], sections 8, 9

January 10  Anneloes Viergever.  Witt-valued Euler classes [L19] state the following results: Theorem 4.1, Remark 4.2, Lemma 4.3. Introduce the group \(N\), the normaliser of the torus in \(SL_2\), and state Prop. 5.5. Introduce the bundles \({\widetilde{\mathcal{O}}}^{\pm}(m)\) from section 6, and state the main results Theorem 7.1, Theorem 8.1 and Prop. 9.1. Give ideas of proofs as you choose.

January 17: Explicit formulas for local Euler classes, I [KW19], [BW]. Discuss the quadratic form on the Jacobian ring given by the Scheja-Storch element and its connection with the \(\mathbb{A}^1\) local degree, as in [KW19], and state the result of [BW] that uses a similar construct to compute the local Euler class in hermitian \(K\)-theory.

January 24: Andrés Jaramillo Puentes.  Applications [KW21], [McK]. For [KW21], mention that Theorem 8.1 of [L19] and the computation of the top power of \(e(E_2^\vee)\) given by [W20] recovers this result and extends to a computation for lines on a smooth hypersurface of degree \(2n-3\) in \(\mathbb{P}^n\).

January 31:  Anneloes Viergever.  Applications From [L20] discuss the quadratic Riemann-Hurwitz formula and show how to use this to compute \(\chi(X/k)\) for \(X\) a generalised Fermat hypersurface. Discuss [LP], mentioning the Gau{\ss}-Bonnet formula from [LR].

Extra meeting (perhaps?) February 7: Applications Discuss the quadratic count of twisted cubics, following [LP].

Bibliography

[An15] Ananyevskiy, Alexey, The special linear version of the projective bundle theorem. Compos. Math. 151 (2015), no. 3, 461–501.

Ananyevskiy, Alexey SL-oriented cohomology theories. Motivic homotopy theory and refined enumerative geometry, 1–19, Contemp. Math., 745, Amer. Math. Soc., [Providence], RI, [2020].

[BCDFØ] Bachmann, Calmès, Déglise, Fasel, Østvær, Milnor-Witt motives arXiv:2004.06634

[BW] Tom Bachmann, Kirsten Wickelgren, \(\mathbb{A}^1\)-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections arXiv:2002.01848

[HW] Hornbostel, Jens; Wendt, Matthias, Chow-Witt rings of classifying spaces for symplectic and special linear groups. J. Topol. 12 (2019), no. 3, 916–966.

[KW21] Kass, Jesse Leo; Wickelgren, Kirsten, An arithmetic count of the lines on a smooth cubic surface. Compos. Math. 157 (2021), no. 4, 677–709.

[KW19] Kass, Jesse Leo; Wickelgren, Kirsten, The class of Eisenbud-Khimshiashvili-Levine is the local \(\mathbb{A}^1\)-Brouwer degree. Duke Math. J. 168 (2019), no. 3, 429–469.

[L19] Marc Levine, Motivic Euler characteristics and Witt-valued characteristic classes. Nagoya Math. J. 236 (2019), 251–310.

[L20] Marc Levine, Aspects of enumerative geometry with quadratic forms. Doc. Math. 25 (2020), 2179–2239.

[LPS] Marc Levine, Simon Pepin Lehalleur, Vasudevan Srinivas, Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces arXiv:2101.00482

[LP] Marc Levine, Sabrina Pauli, Quadratic Counts of Twisted Cubics<\em> arXiv:2206.05729

[LR] Marc Levine, Arpon Raksit, Motivic Gauss-Bonnet formulas. Algebra Number Theory 14 (2020), no. 7, 1801–1851.

[McK] McKean, Stephen, An arithmetic enrichment of Bézout's Theorem. Math. Ann. 379 (2021), no. 1-2, 633–660.

[PW] Pauli, Sabrina; Wickelgren, Kirsten, Applications to \(\mathbb{A}^1\)-enumerative geometry of the \(\mathbb{A}^1\)-degree. Res. Math. Sci. 8 (2021), no. 2, Paper No. 24, 29 pp.

[SW] Srinivasan, Padmavathi; Wickelgren, Kirsten, An arithmetic count of the lines meeting four lines in \(\mathbb{P}^3\). With an appendix by Borys Kadets, Srinivasan, Ashvin A. Swaminathan, Libby Taylor and Dennis Tseng. Trans. Amer. Math. Soc. 374 (2021), no. 5, 3427–3451.

[W18] Wendt, Matthias, Chow-Witt rings of Grassmannians arXiv:1805.06142

[W20] Wendt, Matthias, Oriented Schubert calculus in Chow-Witt rings of Grassmannians. Motivic homotopy theory and refined enumerative geometry, 217–267, Contemp. Math., 745, Amer. Math. Soc., [Providence], RI, [2020]