Two vector bundles $E$ and $F$, are said two be S-equivalent if they have isomorphic gradients. My question is: Is it possible to caracterise this properity using transition functions? Thanks
1 Answer
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Yes. Suppose for simplicity that $E$ and $F$ are extensions of two stable bundles with the same slope, say of ranks $r$ and $s$. Then you can choose their transition matrices of the form $\pmatrix{ (g_{ij}) & (a_{ij})\\ 0 & (h_{ij})}$, with $(g_{ij}) \in GL_r(\mathcal{O})$ and $(h_{ij}) \in GL_s(\mathcal{O})$, and same for $F$. S-equivalence means that the cocycles $(g_{ij})$ for $E$ and $F$ are cohomologous, and same for $(h_{ij})$.