Preprints
[9] The connecting homomorphism for Hermitian K-theory: Projective bundles and Grassmannians, (Joint with T. Huang), preprint (2024).
We provide a geometric interpretation for the connecting homomorphism in the localization sequence of Hermitian K-theory. As an application, we compute the Hermitian K-theory of projective bundles and Grassmannians in the regular case. We provide an explicit basis for Hermitian K-theory of Grassmannians, which is indexed by even Young diagrams together with another special class of Young diagrams, that we call buffalo-check Young diagrams. To achieve this, we develop pushforwards and pullbacks in Hermitian K-theory using Grothendieck’s residue complexes, and we establish fundamental theorems for those pushforwards and pullbacks, including base change, projection, and excess intersection formulas.
We discuss a connection between coherent duality and Verdier duality via a Gersten-type complex of sheaves on real schemes, and show that this construction is compatible with the exceptional inverse image functor f^!. The hypercohomology of this complex coincides with cohomology of the Gersten-Witt complex, which in some cases can be related to topological or semialgebraic Borel-Moore homology. This generalizes smooth schemes in [5] to singular schemes.
The preprint is split into two parts due to the referee's suggestion that the Witt kernel of function fields is a classical problem, which should be the main topic of the first part (see [2] below). For the second part, I am going to include more new results on Grothendieck-Witt groups of quadrics with twisted line bundles and affine spheres.
Published Articles
[6] Witt groups of spinor varieties, (Joint with T. Hudson and A. Martirosian), Proceedings of the London Mathematical Society (accepted). An Appendix by H. Xie on Euler classes and projective bundles.
We compute Witt groups of maximal isotropic Grassmannians, aka. spinor varieties. They are examples of type D homogenuous varieties. Our method relies on the Blow-up setup of Balmer-Calmès, and we investigate the connecting homomorphism via the projective bundle formula of Walter-Nenashev, the projection formula of Calmès-Hornbostel and the excess intersection formula of Fasel. The computation in the Type D case can be presented by so called “even shifted young diagrams”.
[5] The real cycle class map, (Joint with J. Hornbostel, M. Wendt and M. Zibrowius), Annals of K-theory (2) 6 (2021), 239-317.
For smooth varieties over real numbers, we study the real cycle class map from the I-cohomology ring to singular cohomology induced by the signature. We prove that the real cycle class maps respect pull-backs, cup products, Thom classes, and localization sequences. As an application, we show the real cycle class maps are isomorphisms for smooth cellular varieties.
[4] A transfer morphism for Hermitian K-theory of schemes with involution, Journal of Pure and Applied Algebra (4) 224 (2020), 26 pages.
We consider the Hermitian K-theory of schemes with involution, for which we construct a transfer morphism and prove a version of the dévissage theorem. This theorem is then used to compute the Hermitian K-theory of the projective line with involution given by sending [X:Y] to [Y:X]. We also prove the C2-equivariant homotopy invariance of Hermitian K-theory, which confirms the representability of Hermitian K-theory in the C2-equivariant motivic homotopy category of Heller, Krishna, and Østvær.
[3] Chow-Witt rings of split quadrics, (Joint with J. Hornbostel and M. Zibrowius), In Motivic homotopy theory and refined enumerative geometry, Contemporary Mathematics, AMS series, 745 (2020), 123-162.
In this paper, we compute the I-cohomology of split quadrics (those quadrics corresponding to the hyperbolic forms) via the blow-up setup of Balmer-Calmès. We also study the ring structure by using projection formulas and base-change formulas. Our explicit calculations confirm that the I-cohomology ring of a split quadric over the reals is isomorphic to the singular cohomology ring of the space of its real points.
In the 1970s, researchers want to understand the Witt kernel of the function field of the projective quadrics. Only certain cases are known. In this paper, we construct some exact sequences connecting Witt groups of projective quadrics and Clifford algebras, which give an alternative way to understand the Witt kernel via the trace map of Witt groups of Clifford algebras.
[1] An application of Hermitian K-theory: Sums-of-squares formulas, Documenta Mathematica 19 (2014), 195-208.
The paper studies an old problem of Hurwitz on sums-of-squares formulas via Hermitian K-theory. A numerical condition (some powers of 2 dividing some binomial coefficients) of Dugger-Isaksen for sums-of-squares formulas is improved.
See [1], [2] and [6] above.