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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.

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  • $\begingroup$ Crossposted on MSE. $\endgroup$ May 16, 2015 at 22:13
  • $\begingroup$ If you glue to smooth manifolds along their boundary, you do not have canonical structure of smooth manifold on this topological space. If you glue two complex curves then you do have canonical smooth structure due to conformal welding. But is there conformal welding for higher dimensions? $\endgroup$
    – quinque
    May 19, 2015 at 23:05
  • $\begingroup$ @quinque I believe that here it is not important (I've presented the corresponding differential manifold). Moreover you can check lemma 2.1 from arxiv.org/abs/math/0003240 . The question is actually about stably complex structure on the projective bundle. I see two possible ways of determining it. Either through the comparison of Chern numbers of $\tilde{X}$ and $X,$ or with help of K-theory (one can understand stably complex structure as an element of reduced K-group). Actually the complex structure is $\sum_{i=0}^{k}(-1)^{k}\lambda^{i}[E-E^{-1}],$ where $E$ is st. comp. structure on bundle $\endgroup$ May 20, 2015 at 18:09
  • $\begingroup$ Sorry, last passage is wrong. The idea is to see what is the image of difference bundle in the corresponding K-group of the projectivisation. $\endgroup$ May 20, 2015 at 22:11
  • $\begingroup$ So the answer $A\cup_{M} B$ is $\mathbb{P}(\nu\oplus\overline{\underline{\mathbb{C}}}),$ because one can cut out $D(\nu)$ from it using "affine chart fibration" given by zero section in trivial summand of $\nu\oplus\overline{\underline{\mathbb{C}}}$ and get $\mathbb{P}(\nu).$ $-A\cup_{M} B$ isn't corectly defined. The desired formula is $[\tilde{X}]-[X]=-\mathbb{P}(\nu\oplus\overline{\underline{\mathbb{C}}}).$ It can be proven by applying the aforementioned lemma from Totaro's paper to $A,B$ and $C=-X\setminus A$ (prop,; actually I'd like to choose signs $A,B,C$ as +1,-1,-1 corr.). That's it. $\endgroup$ Jun 20, 2015 at 13:31

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