Ran Azouri

My thesis deals with two applications of motivic homotopy theory to refined enumerative geometry. The first of these is presented in the preprint [Az21]. Here I generalize the quadratic conductor formulas found in the paper [LPS21] to more general types of degenerations. This uses the description of the quadratic conductor in terms of the motivic nearby cycles functor combined with a detailed description of the reduced semi-stable reduction one can obtain in the cases to be considered.

A second part of my thesis gives a quadratic/motivic refinement of the approach used by Aluffi [Al06] for constructing Chern-Schwartz-Macpherson classes. Following Aluffi, I define a "limit Borel-Moore motive" (assuming resolution of singularities) and construct a well-defined quadratic Euler class in the corresponding Borel-Moore cohomology group. The hope here is to find a quadratic version of the results of Behrend on virtual fundamental classes for a symmetric obstruction theory, but at present, this has not been accomplished.

References

[Al06] P. Aluffi, Limits of Chow groups, and a new construction of Chern-Schwartz-MacPherson classes. Pure Appl. Math. Q. 2 (2006), no. 4, Special Issue: In honor of Robert D. MacPherson. Part 2, 915–941.

[LPS21] M. Levine, S. Pepin Lehalleur, V. Srinivas, Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces arXiv:2101.00482

Project related preprints

[Az21] Ran Azouri, The Quadratic Euler Characterstic of Nearby Cycles and a Generalized Conductor Formula. Preprint Jan. 2021 (16 pages) arXiv:2101.02686

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This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 832833)