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# Motives Seminar

## 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.

We will meet on Tuesdays, 16:15-18 Uhr, in WSC-N-U-4.03.

### Program

Lecture 0. October 11: Marc Levine. Introduction and overview
Lecture 0 Notes
Lecture 0 Video
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.

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

Lecture 2. October 25: Alessandro D'Angelo.  Special linear projective bundle formula, II [An15].
Lecture 2 Notes
Lecture 2 Video
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 1: holiday

Lecture 3. November 8: Sabrina Pauli.  Milnor-Witt sheaves and the Chow-Witt groups
Lecture 3 Video
Lecture 3 Notes
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.

Lecture 4. November 15: Herman Rohrbach.  Chow-Witt rings of symplectic classifying spaces, I
Lecture 4 Notes
Lecture 4 Video
[HW] sections 1-5

Lecture 5. November 22: Herman Rohrbach.  Chow-Witt rings of symplectic classifying spaces, II
Lecture 5 Notes
Lecture 5 Video
[HW] sections 6-9

Lecture 6. November 29: Chirantan Chowhury.  Chow-Witt rings of Grassmannians, I
Lecture 6 Video (part 1)
Lecture 6 Video (part 2)
Lecture 6 Notes
[W18], sections 4, 5, 6.

Lecture 7. December 6: Pietro Gigli.  Chow-Witt rings of Grassmannians, II
Lecture 7 Notes
Lecture 7 Video
[W20], sections 2, 3, 4 (section 2 is mainly a recollection of results from [W18], so you need only present the new material).

Lecture 8. December 13: Jan Hennig.  Chow-Witt rings of Grassmannians, III
Lecture 8 Notes
Lecture 8 Video
[W20], sections 5, 6, 7

Lecture 9. December 20: Marc Levine.  Chow-Witt rings of Grassmannians, IV
Lecture 9 Notes
Lecture 9 Video
[W20], sections 8, 9

Lecture 10. January 10:  Anneloes Viergever(presented by Levine).  Witt-valued Euler classes
Lecture 10 Notes
Lecture 10 Video
[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.

Lecture 11. January 17:  Stephen McKean (via Zoom). Explicit formulas for local Euler classes, I
Lecture 11 Notes
Lecture 11 Video
[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.

Lecture 12. January 24: Andrés Jaramillo Puentes.  Applications: lines on hypersurfaces and the quadratic Bézout theorem
Lecture 12 Video
[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 $\mathbb{P}^n$.

Lecture 13. January 31:  Anneloes Viergever.  Applications: Computing quadratic Euler characteristics 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 [LPS], mentioning the Gauß-Bonnet formula from [LR].

Lecture 14. 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. Link

[An20] Ananyevskiy, Alexey SL-oriented cohomology theories. Motivic homotopy theory and refined enumerative geometry, 1–19, Contemp. Math., 745, Amer. Math. Soc., [Providence], RI, [2020]. Link

[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. Link

[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. Link

[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. Link

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

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

[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 arXiv:2206.05729

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

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

[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. Link

[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. Link

[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] Link