The connecting homomorphism for Hermitian $K$-theory

Abstract

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 $\textit{buffalo-check}$ 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.

Figures (5)

• Figure 1: Young diagram $\Lambda$ in the $(d \times m)$-frame $\Xi$. The boundary $b(\Lambda)$ is thickened.
• Figure 2: Five examples of $K$-even Young diagrams with centers hatched and boundaries thickened.
• Figure 3: An example of the buffalo-check pattern of size $(6\times 6)$. The centers of $K$-even Young diagrams can only sit on those non-white boxes. A black box (resp. grey box) is the location of the center of a buffalo-check Young diagram, if the corresponding basis element lands in the even (resp. odd) twist.
• Figure 4: The indexing set for the additive basis of $\mathrm{GW}^{[n]}(\mathrm{Gr}_2(2))^{\mathrm{tot}}$. The first line consists of even Young diagrams in $(2\times 2)$-frame with thickened inner-frame boundary segments; The second line (resp. third line) includes buffalo-check Young diagrams in $\mathfrak{B}^{2}_{2,2}$ (resp. $\mathfrak{B}^{3}_{2,2}$) with center hatched.
• Figure 5: The indexing set for the additive basis of $\mathrm{GW}^{[n]}(\mathrm{Gr}_3(3))^{\mathrm{tot}}$. The first line consists of even Young diagrams in $(3\times 3)$-frame with thickened inner-frame boundary segments; The second line (resp. third line) includes buffalo-check Young diagrams in $\mathfrak{B}^{3}_{3,3}$ (resp. $\mathfrak{B}^{4}_{3,3}$) with center hatched.

Theorems & Definitions (185)

• Theorem 1.1
• Theorem 1.2: Theorem \ref{['theo:GW-Grassmannian']}
• Definition 2.2: hartshorne1966residues
• Definition 2.3: hartshorne1966residues
• Remark 2.4
• Proposition 2.5
• proof
• Definition 2.6: schlichting2017hermitian
• Definition 2.7
• Theorem 2.8: Localization