# Quadratic refinements in algebraic geometry

## Periodic Reporting for period 2 - QUADAG (Quadratic refinements in algebraic geometry)

Reporting period: 2021-03-01 to 2022-08-31

At its root, algebra is the science of solving equations. Enumerative geometry is a branch of mathematics that is concerned with counting the number of solutions of equations that arise from geometric situations, rather than finding the explicit solutions themselves. In many situations, this number of solutions remains constant under a change in the input parameters of the problem, which points to an intrinsic underlying aspect of the original geometric problem. As an elementary example, the number of solutions in the complex numbers of a degree 5 equation is always at most 5, and if counted with the proper "multiplicity” is always equal to 5.
The choice of number system for finding solutions to the geometric problem often has a large effect on the number of solutions and can even affect the invariance-of-parameter phenomenon mentioned above. The main thrust of the project QUADAG is to replace the simple integer count of solutions with an object known as a quadratic form. In contrast with a simple numerical count, the quadratic form can itself reflect the difference in the number system under consideration, and thus gives a way to unify the disparate counting results that one finds when working over different number systems, such as the rational, real or complex numbers.
Supporting and closely allied to this central objective is the development of effective methods of computing these quadratic invariants, applying these tools to make explicit computations in interesting geometric situations and giving conceptual interpretations of the resulting computations. Although a direct application of this work to pressing societal issues is difficult to see, we hope that, as a part of the overall quest for truth and knowledge that is represented by modern mathematics, this project will make a small contribution to society as a whole.
Here is a resumé of project work performed during the reporting period.

1. [ML0], [ML1], [ML3], [ML4], [ML5] [ML6] develop techniques for construction and computing quadratic characteristic classes, Euler characteristics, and virtual fundamental classes. [ML3] was substantially revised.
2. [Jin20b] extends motivic Euler characteristics to a trace pairing and will be applied to developing quadratic invariants for singularities of maps. [DFJK21], [JX20] and [FJ] allow comparisons of motivic virtual fundamental classes with their real counterparts.
3. [BMP] develops a new method for explicit computations of A1-degrees, useful for all aspects of the project.
4. [Az21] extends the conductor formula computations made in [ML3] to new cases.
5. [R] proves a completion theorem for hermitian K-theory. This will be useful in conjunction with the localization results [ML5], [ML6], [AKLPR].
References
[ML0] M. Levine, The intrinsic stable normal cone, Algebr. Geom. 8 (2021), no. 5,518–561.
[ML1] M. Levine, Aspects of enumerative geometry with quadratic forms. Doc. Math. 25 (2020), 2179-2239.
[ML3] M. Levine, S. Pepin Lehalleur, V. Srinivas, Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces arXiv:2101.00482
[ML5] M. Levine, Atiyah-Bott localization in equivariant Witt cohomology, arXiv 2203.13882
[ML6] M. Levine, Virtual Localization in equivariant Witt cohomology, arXiv 2203.15887
[LYZ] M. Levine, Y. Yang, G. Zhao, Algebraic elliptic cohomology and flops II: SL-cobordism. Adv. Math. 384 (2021), Paper No. 107726, 66 pp.
[BMP] T. Brazelton, S. McKean, S. Pauli, Bézoutians and the A1-degree. https://services.math.duke.edu/~mckean/bezoutian.pdf
[DFJK21] F. Déglise, J. Fasel, F. Jin, A. Khan, On the rational motivic homotopy category, J. Ec. Polytech. Math. 8 (2021), 533-583.
[FJ] F. Jin, On some finiteness results in real étale cohomology, Bulletin of the London Mathematical Society (early view) https://doi.org/10.1112/blms.12563
[JX20] F. Jin, H. Xie, A Gersten complex on real schemes, arXiv:2007.04625 math.AG
[Jin20b] F. Jin, Local terms of the motivic Verdier pairing, arXiv:2010.09292 math.AG
[Az21] R. Azouri, The Quadratic Euler Characteristic of Nearby Cycles and a Generalized Conductor Formula. arXiv:2101.02686
[AP] D. Aranha, P. Pstrągowski, The Intrinsic Normal Cone For Artin Stacks. arXiv 1909.07478.
[R] H. Rohrbach, On Atiyah-Segal completion for T-equivariant Hermitian K-theory, arXiv 2203.15518
The main progress beyond the state of the art is as follows:
1. The works-in-progress [ML4i, ii] give a conceptual approach to quadratic Welschinger invariants, and were substantially revised.
2. The work-in-progress [LP] applies [ML5] to give quadratic counts of twisted cubic curves on hypersurfaces and complete intersections.
4. The work-in-progress [AKLPR] yields torus localization for Artin stacks.
6. The work-in-progress [JPP] opens new ground constructing a quadratic refinement of aspects of tropical intersection theory.
Results expected for the remainder of the project:
1. The localization formulas being developed in [ML5, 6] will be extended to give this method wider application and will be used to compute quadratic Donaldson-Thomas invariants for smooth toric threefolds.
2. The construction of the motivic virtual fundamental class in [ML0] will be extended to a suitable category of stacks.
3. The development of news tools for dealing with quadratic characteristic classes and virtual fundamental classes following (1) and (2) will facilitate quadratic computations in Gromov-Witten theory and Donaldson-Thomas theory.
4. The link between tropic and quadratic methods started in [JPP] will be exploited to compute the quadratic Welschinger invariants constructed in [ML4i,ii].
References
[ML4i] Jesse Kass, Marc Levine, Jacob Solomon, Kirsten Wickelgren, A relative orientation for the moduli space of stable maps to a del Pezzo surface. Work in progress.
[ML4ii] Jesse Kass, Marc Levine, Jacob Solomon, Kirsten Wickelgren, A quadratically enriched count of rational plane curves. Work in progress
[AKLPR] D. Aranha, A. Khan, A. Latyntsev, H. Park, C. Ravi Localization Theorems for Artin stacks, work in progress
[PL] M. Levine, S. Pauli, Quadratic counts of twisted cubic curves, work in progress
[JPP] A. Jaramillo-Puentes, S. Pauli, Tropical methods in A1–enumerative geometry, work in progress