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.


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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