The \(\bar\partial\) operator in holomorphic \(K\)-theory (Q2577044)

Let \(X\) be a smooth projective variety \(X\) and let \(\text{Hol}(X; \text{Gr}_n({\mathbb C}^N))\) denote the space of holomorphic maps from the underlying complex manifold \(X\) into the Grassmannian of \(n\)-planes in \({\mathbb C}^N\). \textit{R. L. Cohen} and \textit{P. Lima-Filho} [K-Theory 23, 345--376 (2001; Zbl 1073.14508)] defined the holomophic \(K\)-theory space \(K_{\text{hol}}(X)\) to be the Quillen-Segal group completion of the union of these mapping spaces, which is written as \(K_{\text{hol}}(X)= \text{Hol}(X; {\mathbb Z}\times BU)^+\), and the holomorphic \(K\)-groups to be the homotopy groups \(K^{-q}_{\text{hol}}(X)= \pi_q(K_{\text{hol}}(X))\). In this paper the author defines a family of \(\overline{\partial}\) operators on holomorphic \(K\)-theory in a manner analogous to the construction of \textit{M. F. Atiyah} [Q. J. Math. Oxf. Ser. (2) 19, 113--140 (1968; Zbl 0159.53501)] in topological \(K\)-theory and proves \(K_{\text{hol}}^{-2}(X)\cong K_{\text{hol}}^0(X)\). Multiplication by the Bott class of \(K({\mathbb R}^2)={\mathbb Z}\) induces a homomorphism \(\beta : K(X) \to K^{-2}(X)\). The Bott periodicity theorem asserts that this homomorphism is an isomorphism. To prove this, \textit{M. F. Atiyah} [loc. cit.] defined a homomorphism \(\text{index}_{\overline{\partial}} : K(X\times S^2) \to K(X)\), using the index of a family of \(\overline{\partial}\) operators on \(\mathbb CP^1=S^2\), such that the composition of this and an obvious homomorphism \(\alpha : K^{-2}(X)=K(X\times {\mathbb R}^2) \to K(X\times S^2) \to K(X)\) becomes an inverse of \(\beta\). If one describes \(K\)-theory in terms of mapping spaces, then \(\alpha\) can be represented as \(\pi_0(\text{Map}(X; \text{Map}^\bullet(S^2; BU)))\cong \pi_0(\text{Map}(X; {\mathbb Z}\times BU))\) where \(Map^\bullet(-; -)\) denotes the space of base-point preserving maps. Here the author attempts to realize this isomorphism on the space level. In fact, in Section 2 the author constructs a homotopy equivalence \(\overline{\partial} : \text{Map}(X; \text{Map}^\bullet(S^2; BU))\to \text{Map}(X; {\mathbb Z}\times BU)\) satisfying \(\overline{\partial}_*=-\alpha\). The next three sections are devoted to showing how this construction may be extended to holomorphic mapping spaces and in the last section it is proved that the map obtained defines a homotpy equivalence \(\overline{\partial} : \text{Hol}(X; \text{Hol}^\bullet(\mathbb CP^1; BU))^+ \to \text{Hol}(X; {\mathbb Z}\times BU)^+\) (Theorem 1.3). Clearly this map produces the desired isomorphism above. But the author remarks that this theorem does not ensure that \(K_{\text{hol}}^{-q-2}(X)\cong K_{\text{hol}}^{-q}(X)\) holds for any integer \(q\).