Bott-Chern currents, excess normal bundles and the Chern character
Let \(i:Y\to X\) be an embedding of complex manifolds with normal bundle \(N_{Y/X}\). Let \(\eta\) be a holomorphic vector bundle on \(Y\) and \((\xi,v)\) be a holomorphic complex vector bundle on \(X\) which provide a resolution of the sheaf \(i_ *{\mathcal O}_ Y(\eta)\), and let \(h^ \xi\), \(g^ NY/X\), \(g^ \eta\) be Hermitian metrics on \(\xi\), \(N_{Y/X}\) and \(\eta\). It is then known that under certain compatibility assumptions on these metrics that the Quillen superconnection Chern character forms \(\alpha_ t\) for \(t\geq 0\) converge at infinity to the current \(Td^{- 1}(N_{Y/X},g^ NY/X)ch(\eta,g^ \eta)\delta_ Y\). In an earlier work, the author in collaboration with H. Gillet and C. Soulé constructed a current \(T^ X(\xi,h^ \xi)\) on \(X\) which satisfies the equation \[ (2\pi i)^{-1}T^ X(\xi,h^ \xi)=Td^{-1}(N_{Y/X},g^ NY/X)ch(\eta,g^ \eta)\delta_ Y-ch(\xi, h^ \xi). \] Suppose now that \(i':Y'\to X\) is another embedding of complex manifolds with corresponding vector bundles and Hermitian metrics as above. If \(U=Y\cap Y'\), then one has the diagram \[ \begin{tikzcd} U \ar[r,"k^\prime"]\ar[d,"k"'] \ar[dr,"j"] & Y' \ar[d,"i^\prime"] \\ Y \ar[r,"i"'] & X \end{tikzcd} \] It is known that if \(Y\) and \(Y'\) are transversal, then \(T^ X(\xi,h^ \xi)\) has a well defined restriction to \(Y'\), \(i^{'*}T^ X(\xi,h^ \xi)\) and \(T^{Y'}(\xi|_{Y'},h^{\xi|_{Y'}})=i^{'*}\) \(T^ X(\xi,h^ \xi)\). Furthermore, one has the identity \begin{multline*} T^ X\xi\hat\otimes\xi',h^{\xi'}\hat\otimes h^{\xi'})=ch(\xi,h^ \xi)T^ X(\xi',h^{\xi'})+ \\ +i^{'*}(Td^{-1}(N_{Y'/X},g^ NY'/X)ch(\eta',g^{\eta'})T^{Y'}(\xi|_{Y'},h^{\xi |_{Y'}})\quad\text{modulo } (\partial,\overline\partial\text{ coboundaries}).\end{multline*} The author generalizes this identity to the non-transversal case. The expression involves in addition the excess normal bundle \(\tilde N\).
scientific article; zbMATH DE number 93434