## The rational homotopy type of a blow-up in the stable case.(English)Zbl 1153.55010

Let $$f\colon V\to W$$ a smooth embedding of closed manifolds of codimension $$2k$$ ($$k>1$$) such that the normal bundle $$\nu$$ of this embedding has the structure of a $$k$$-complex bundle $$\nu\colon \mathbb C^k\to E\buildrel{\pi}\over{\to}V$$.
In this paper, the authors construct a Sullivan model of the projectivization of $$\nu$$, $$\mathbb CP(k-1)\hookrightarrow P\nu\buildrel{\pi'}\over\to V$$, out of a model of the embedding and the Chern classes of $$\nu$$, $$c_i(\nu)\in H^{2i}(V;\mathbb Z)$$.
In fact, if $$A$$ is a particular common model of $$V$$ and of a compact tubular neighborhood of $$V$$ inside $$W$$, a model of $$P\nu$$ is given by $$(A\otimes\Lambda (x,z),D)$$, in which $$x$$ has degree $$2$$, $$y$$ has degree $$2k-1$$, $$Dx=0$$ and $$Dz=\sum_{i=0}^k \gamma_i x^{k-i}$$. Here, the $$\gamma_i$$’s denote representatives of the image of the Chern classes through the map $$H^{2i}(V;\mathbb Z)\to H^{2i}(V;\mathbb Q)$$.
Next the authors give an interesting description of the blow-up construction of McDuff for symplectic manifolds in this more general setting.
Finally, the results above, interesting in their own rite, are combined to prove the main result: if $$dim\,W\geq 2\,dim\,V+3$$ and $$H^1(f)$$ is injective, the authors build a Sullivan model of the blow-up $$\widetilde W$$ of $$W$$ along $$V$$. Explicitly, if $$\phi\colon R\to A$$ is a model of the embedding $$f\colon V\to W$$, the model of $$\widetilde W$$ has the form $$B(R,A)=(R\oplus A\otimes \Lambda^+(x,z),D)$$ in which the algebra structure is given by the one in $$R$$ and $$A\otimes \Lambda^+(x,z)$$, and by the $$R$$-module structure on $$A\otimes \Lambda^+(x,z)$$ via the morphism $$\phi$$. The differential also involves the previous constructions.
This deep result proves in particular that, if $$W$$ is simply connected, the rational homotopy type of the blow-up only depends on the rational homotopy class of the embedding and on the Chern classes of the normal bundle $$\nu$$.
Interesting applications of these results to the behavior of blow-ups of complex projective spaces are also included.

### MSC:

 55P62 Rational homotopy theory 14F35 Homotopy theory and fundamental groups in algebraic geometry 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53D05 Symplectic manifolds (general theory)
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### References:

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