Vector bundles over elliptic fibrations. (English) Zbl 0937.14004
Let \(\pi : Z\to B\) be an elliptic fibration with section. Vector bundles based on a spectral cover of \(B\) and based on extensions of certain bundles \(V\) over \(Z\) are determined. Concepts from string theory are investigated in this process. First, facts concerning bundles over an elliptic curve \(E\) are introduced. Then, a tautological and “universal” bundle over \(\mathcal{P}^{n-1}\times E\) are introduced and studied. The results obtained are extended to the map \(\pi : Z\to B\). A typical result follows (theorem 6.10 with detail):
Suppose that \(\dim B = 2\) and \(a\) is an integer. Let \(\gamma_{n} = \pi_{*}(\mathcal{O}_{Z}(na))\) and \(\mathcal{P}_{n}={\mathbb P}\gamma_{n}\). Consider a quasi-section \(A: B\to \mathcal{P}_{n-1}\) and set \(V_{A,a}= (A,\text{Id})^{*}\mathcal{U}_{a}\) where \[ \mathcal{U}_{a}= (\nu\times I)_{*}{\mathcal O}_{{\mathcal T}\times_{B}Z}(\Delta - {\mathcal G}-a(r^{*}\sigma\times_{B}Z)), \] \(\nu\) is the spectral map, \(r: {\mathcal T}={\mathbb P}{\mathcal E}\to Z\) is the projection with \({\mathcal E}\) defined by the exact sequence \(0\to {\mathcal E}\to \pi^{*}\pi_{*}{\mathcal O}_{Z}(n\sigma)\to {\mathcal O}_{Z}(n\sigma)\to 0\), \(\Delta = (r\times \text{Id})^{*}(\Delta_{0})\) with \(\Delta_{0}\) the diagonal in \(Z\times_{B}Z\), \(\sigma\) is the section of \(\pi\) under consideration and finally \({\mathcal G} = (r\times \text{Id})^{*}p_{2}^{*}\sigma\) with \(p_{2}: Z\times_{B}Z\to Z\) the projection onto the second factor of \(Z\times_{B}Z\). Suppose that \(A\) is smooth, \(A\) is the blowup of \(B\) at a finite number of points \(b_{1}, \cdots ,b_{r}\) and the image of the exceptional \({\mathbb P}^{1}\) is a generic line in the fiber \({\mathbb P}^{n-1}\). Then, the \(\text{rank } n\) bundle \(V_{A,a}\) (\(a\not\equiv 1 \bmod n\)) which is defined on \(Z-\bigcup_{i}E_{b_{i}}\) extends to a vector bundle over \(Z\), denoted again by \(V_{A,a}\). The restriction of \(V_{A,a}\) to a fiber \(E_{b_{i}}\) is the unstable bundle \(W_{d}^{\vee}\otimes W_{n-d}\) where \(a = a'+nk\) with \(-(n-2)\leq a'\leq 1\) and \(d = 1-a'\). Here, \(W_{d}\) is the stable bundle of rank \(d\) on \(E\) whose determinant is isomorphic to \(\mathcal{O}_{E}(P_{0})\), unique up to isomorphism.
Suppose that \(\dim B = 2\) and \(a\) is an integer. Let \(\gamma_{n} = \pi_{*}(\mathcal{O}_{Z}(na))\) and \(\mathcal{P}_{n}={\mathbb P}\gamma_{n}\). Consider a quasi-section \(A: B\to \mathcal{P}_{n-1}\) and set \(V_{A,a}= (A,\text{Id})^{*}\mathcal{U}_{a}\) where \[ \mathcal{U}_{a}= (\nu\times I)_{*}{\mathcal O}_{{\mathcal T}\times_{B}Z}(\Delta - {\mathcal G}-a(r^{*}\sigma\times_{B}Z)), \] \(\nu\) is the spectral map, \(r: {\mathcal T}={\mathbb P}{\mathcal E}\to Z\) is the projection with \({\mathcal E}\) defined by the exact sequence \(0\to {\mathcal E}\to \pi^{*}\pi_{*}{\mathcal O}_{Z}(n\sigma)\to {\mathcal O}_{Z}(n\sigma)\to 0\), \(\Delta = (r\times \text{Id})^{*}(\Delta_{0})\) with \(\Delta_{0}\) the diagonal in \(Z\times_{B}Z\), \(\sigma\) is the section of \(\pi\) under consideration and finally \({\mathcal G} = (r\times \text{Id})^{*}p_{2}^{*}\sigma\) with \(p_{2}: Z\times_{B}Z\to Z\) the projection onto the second factor of \(Z\times_{B}Z\). Suppose that \(A\) is smooth, \(A\) is the blowup of \(B\) at a finite number of points \(b_{1}, \cdots ,b_{r}\) and the image of the exceptional \({\mathbb P}^{1}\) is a generic line in the fiber \({\mathbb P}^{n-1}\). Then, the \(\text{rank } n\) bundle \(V_{A,a}\) (\(a\not\equiv 1 \bmod n\)) which is defined on \(Z-\bigcup_{i}E_{b_{i}}\) extends to a vector bundle over \(Z\), denoted again by \(V_{A,a}\). The restriction of \(V_{A,a}\) to a fiber \(E_{b_{i}}\) is the unstable bundle \(W_{d}^{\vee}\otimes W_{n-d}\) where \(a = a'+nk\) with \(-(n-2)\leq a'\leq 1\) and \(d = 1-a'\). Here, \(W_{d}\) is the stable bundle of rank \(d\) on \(E\) whose determinant is isomorphic to \(\mathcal{O}_{E}(P_{0})\), unique up to isomorphism.
Reviewer: P.Cherenack (Rondebosch)
MSC:
| 14D06 | Fibrations, degenerations in algebraic geometry |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
