Consider smooth complex manifold $X$ of complex dimension $n$ and its smooth submanifold $Z$ of codimension $k$. Denote the normal bundle of the inclusion $Z\subset X$ by $\nu.$ One can blow up $X$ along $Z,$ acquiring $\tilde{X}.$ Now let $A$ and $B$ be tubular neighbourhoods of $Z$ in $X$ and $\mathbb{P}(\nu)$ in $\tilde{X},$ respectively. If one glues $-A$ and $B$ along their diffeomorphic boundaries (bundles of odd-dimensional spheres over $Z$), he acquires $\mathbb{P}(\nu\oplus\underline{\mathbb{C}})$ as differentiable manifolds. The question is what stably complex structure is glued on that manifold from $-A$ and $B,$ and how the restriction giving the particular stably complex structure appears here.
My hypothetic answer is that for odd $k$ it is set by an isomorphism $$ T\mathbb{P}(\nu\oplus\underline{\mathbb{C}})\cong p^{*}\nu\otimes\bar{\eta}\oplus\bar{\eta}\oplus p^{*}TZ, $$ where $\eta$ is tautological bundle over $\mathbb{P}(\nu\oplus\underline{\mathbb{C}})$ and $p$ is the projection $\mathbb{P}(\nu\oplus\underline{\mathbb{C}})\rightarrow Z.$ For even $k$ it is set by $$ T\mathbb{P}(\nu\oplus\underline{\mathbb{C}})\cong p^{*}\nu\otimes\eta\oplus\bar{\eta}\oplus p^{*}TZ. $$ Actually the above specified stably complex manifold has complex cobordism class equal to $[\tilde{X}]-[X].$ In case of $Z=pt$ we also have $[\tilde{X}]=[X\sharp\bar{\mathbb{C}P^{n}}].$ So I am also interested in formula for connected sums with non-standard stably complex manifolds in complex cobordism.
