Chern character for infinity vector bundles (Q6661612)

The Atiyah-Singer index theorem [\textit{M. F. Atiyah} and \textit{I. M. Singer}, Ann. Math. (2) 87, 484--530 (1968; Zbl 0164.24001); Ann. Math. (2) 87, 546--604 (1968; Zbl 0164.24301)] and the theory of elliptic and pseudoelliptic differential operators are to be thought of a far-reaching generalization of the celebrated Hirzebruch-Riemann-Roch theorem [\textit{F. Hirzebruch}, Proc. Natl. Acad. Sci. USA 40, 110--114 (1954; Zbl 0055.38803)] and other high-powered theorems, such as the Gauss-Bonnet theorem, to a much vaster domain. Unfortunately, such techniques, found in [\textit{M. Atiyah} et al., Matematika, Moskva 17, No. 6, 3--48 (1973; Zbl 0364.58016)] as well as [\textit{P. B. Gilkey}, Adv. Math. 11, 311--325 (1973; Zbl 0285.53044)], use differential geometric methods that heavily rely on an auxiliary choice of a Hermitian metric on the manifold as well as the bundle.\N\NToledo et al. [\textit{D. Toledo} and \textit{Y. L. L. Tong}, Topology 15, 273--301 (1976; Zbl 0355.58014); Math. Ann. 237, 41--77 (1978; Zbl 0391.32008); Ann. Math. (2) 108, 519--538 (1978; Zbl 0413.32006); \textit{N. R. O'Brian} et al., Bull. Am. Math. Soc., New Ser. 5, 182--184 (1981; Zbl 0495.14010); Am. J. Math. 103, 253--271 (1981; Zbl 0474.14009); Am. J. Math. 103, 225--252 (1981; Zbl 0473.14008); \textit{D. Toledo} and \textit{Y. L. L. Tong}, Contemp. Math. 58, 261--275 (1986; Zbl 0612.32009)] made several remarkable conceptual breakthroughs by providing local Čech cohomological proofs of the Hirzebruch-Riemann-Roch theorem and Grothendieck-Riemann-Roch [\textit{N. R. O'Brian} et al., Bull. Am. Math. Soc., New Ser. 5, 182--184 (1981; Zbl 0495.14010)]. Through the modern lens, one may interpret their work as a hands-on theory of infinity stacks, which only much more recently has been made into a full-fledged mathematical theory. One of the key constructions by \textit{N. R. O'Brian} et al. [Am. J. Math. 103, 225--252 (1981; Zbl 0473.14008)] is that of the Chern class for a coherent analytic sheaf on a complex manifold. This paper takes the first step in providing a homotopy-theoretic framework for some of Toledo and Tong's mathematical objects. By simply finding the appropriate homotopy-theoretic setting, their constructions extend far beyond what they had intended and point to new and exciting advances.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] defines two simplicial presheaves on the site of complex manifolds, namely, \(\boldsymbol{IVB}:\mathrm{CMan}^{\mathrm{op}}\rightarrow\mathrm{sSet}\), which later gives rise to infinity vector bundles, and the presheaf of holomorphic froms \(\boldsymbol{\Omega}:\mathrm{CMan}^{\mathrm{op}}\rightarrow\mathrm{sSet}\).\N\N\item[\S 3] defines the Chern map \(\boldsymbol{Ch}:\boldsymbol{IVB} \rightarrow\boldsymbol{\Omega}\). It is established (Theorem 3.14) that\N\NTheorem. The Chern map \(\boldsymbol{Ch}:\boldsymbol{IVB}\rightarrow\boldsymbol{\Omega}\) is a map of simplicial presheaves.\N\N\item[\S 4] constructs what is called the Čech sheafification \(\boldsymbol{Ch}^{\overset{\vee}{\dag}}:\boldsymbol{IVB}^{\overset{\vee}{\dag}}\rightarrow\boldsymbol{\Omega}^{\overset{\vee}{\dag}}\) of the Chern map, establishing (Theorem 4.18) that the Chern map \(\boldsymbol{Ch}^{\overset {\vee}{\dag}}\) recovers the Chern character form.\N\N\item[\S 5] upgrades the results of the previous section to statements about (hyper)sheaves, particularly offering Theorem 5.13 as an upgrade of Theorem 4.18 to sheaves.\N\end{itemize}