# Marc Levine-Principal Investigator

**Description of research**

[ML0] modifies the Behrend-Fantechi construction of the fundamental class of the intrinsic normal cone and virtual fundamental class in the case of a perfect obstruction theory for a quasi-projective scheme to give rise to classes in twisted Borel-Moore homology with coefficients in an arbitrary motivic commutative ring spectrum. Applying this suitably will yield quadratic versions of the classical virtual fundamental classes.

[ML1]. The paper [ML1] gives a foundation for many of the applications of motivic homotopy theory to enumerative geometry. In some detail, the paper discusses quadratic refinements of the classical Euler characteristic of a topological space and the Euler classes of a vector bundle. We develop tools for computing these invariants and apply these to give quadratic refinements of a number of classical results: comparing the Euler class of a bundle with that of its dual, finding formulas for the Euler class of a symmetric power and a tensor product for rank two bundles and a formula for the Euler class of the twist of a vector bundle by a line bundle. Some of these results were obtained in an earlier version of this paper (*Toward an enumerative geometry with quadratic forms*), but we have used the progress in ``motivic'' enumerative geometry since that paper was written to refine and extend many results. As applications we prove a quadratic refinement of the classical Riemann-Hurwitz formula (again an extended version of the formula obtained in *Toward an enumerative geometry ...*}) and compute the quadratic Euler characteristic of generalized Fermat hypersurfaces.

[ML2] The main goal of this paper is to compute the coefficient ring of algebraic special linear cobordism: MSL. Besides having to resolve a number of complicated technical issues, we filled in a gap in the literature by supplying a proof of a result appearing in a 1962 paper by Novikov, which result forms a central pillar of our arguments. The proof was not given in Novikov's paper and the result does seem to have been taken up anywhere in the literature since. Our main result gives a description of the image of the \(\eta, \ell\) completion of \(\text{MSL}^{2*,*}\), for \(\ell\) an odd prime different from the characteristic, using a motivic version of the Adams spectral sequence.

[ML3] This paper has three aspects. The first is to make the computation of the quadratic Euler characteristic as the trace form on Hodge cohomology (as proven in our paper with A. Raksit) completely concrete in the case of a smooth hypersurface in a projective space or a weighted projective space. This is accomplished by making explicit the slight difference between this trace form and the bilinear form on a suitable part of the Jacobian ring. Having this explicit formula, the second part considers the question of a quadratic refinement of classical and modern conductor formulas, measuring the difference between the Euler characteristic of the general (smooth) member in a proper family and the Euler characteristic of some special (singular) fiber. Using our computation for hypersurfaces, we could make the explicit computation in some special cases (all having only single singularity in the singular fiber) and find that the naive generalization of the classical formulas are not correct. The third aspect of the paper is to give a more abstract discussion of the conductor formulas we found, in terms of motivic vanishing and nearby cycles functors.

[ML5] This paper develops the foundations of Atiyah-Bott localization in a setting relevant for computing quadratic refinements of degree of characteristic classes and virtual fundamental classes. The main idea is that the classical version for localization with respect to a torus action is doomed to failure because the quadratic invariants for the classifying space of a torus all vanish. We find two suitable replacements for a torus action: replacing \(\mathbb{G}_m\) with \(\text{SL}_2\) or with the normalizer of the torus in \(\text{SL}_2\). With some restrictions, one achieves a results for localization of equivariant cohomology with values in the sheaf of Witt rings roughly parallel to classical torus localization to the fixed points. We hope to apply this to compute the quadratic Donaldson-Thomas invariants for the Hilbert scheme of points on \(\mathbb{P}^3\), \(\mathbb{P}^1\times \mathbb{P}^2\) ,\(\mathbb{P}^1)^3\) and hopefully other smooth toric threefolds.

[ML6] This paper continues from [ML6], developing a quadratic virtual localization theorem for an action of the normaliser of the torus in \(\text{SL}_2\), along the lines of the virtual localization theorem for a torus action, by Graber-Pandharipande.

**Project related publications**

[ML1] Marc Levine. * Aspects of enumerative geometry with quadratic forms*. Doc. Math. 25 (2020), 2179-2239. doi 10.25537/dm.2020v25.2179

**Project related preprints**

[ML0] Marc Levine. * The intrinsic stable normal cone*. Algebr. Geom. 8 (2021), no. 5, 518–561.

[ML2] Marc Levine, Yaping Yang, Gufang Zhao. *Algebraic elliptic cohomology and flops II: SL-cobordism*. Adv. Math. 384 (2021), Paper No. 107726, 66 pp.

**Preprints**

[ML3] Marc Levine, Simon Pepin Lehalleur, Vasudevan Srinivas. * Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces*

https://arxiv.org/abs/2101.00482

[ML5] Marc Levine. * Atiyah-Bott localization in equivariant Witt cohomology* https://arxiv.org/abs/2203.13882

[ML6] Marc Levine * Virtual localization in equivariant Witt cohomology * https://arxiv.org/abs/2203.15887

** Works in progress**

[ML4i] Jesse Kass, Marc Levine, Jacob Solomon, Kirsten WIckelgren. *A quadratically enriched count of rational plane curves*.

work-in-progress.

[ML4ii] 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

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