Abstract
In this section we take a more geometric point of view on the orthogonal decomposition
resulting from (2.51).
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Notes
- 1.
According to our convention in §2.7 the relative tangent sheaf is denoted by \(\mathcal{T}_{\pi }\).
- 2.
The vertical arrows in the diagram (5.22) are the natural projection.
- 3.
This vanishing comes from the following two facts: (1) the inclusion \(\mathcal{O}_{\mathbf{\breve{J}}_{\Gamma }}\hookrightarrow \mathbf{\tilde{H}}\) takes the constant section \(1_{\mathbf{\breve{J}}_{\Gamma }}\) of \(\mathcal{O}_{\mathbf{\breve{J}}_{\Gamma }}\) to the section h 0 of \(\mathbf{\tilde{H}}\) whose value \(h_{0}([Z], [\alpha ]) = 1_{Z} \in \mathbf{\tilde{H}}([Z], [\alpha ])\) is the constant function of value 1 on Z, for every \(([Z], [\alpha ]) \in \mathbf{\breve{J}}_{\Gamma }\); (2) D + (t) = 0, for any constant function t on Z.
- 4.
\(\mathbf{T}_{\overleftarrow{Fl}_{\Gamma }}^{{\ast}}\) stands for the relative cotangent bundle of the structure morphism \(\overleftarrow{Fl}_{\Gamma }\) in (5.30).
- 5.
For graded modules we always assume that a graded component is zero, if its degree is not in the range of the grading. Thus, for example, for the graded module \(Gr_{\mathbf{\tilde{H}}_{-\bullet }}^{\bullet }(\tilde{{\mathcal{F}}}^{{\prime}})[1]\), the component \(Gr_{\mathbf{\tilde{H}}_{-\bullet }}^{l_{\Gamma }}(\tilde{{\mathcal{F}}}^{{\prime}})[1] = Gr_{\mathbf{\tilde{H}}_{ -\bullet }}^{l_{\Gamma }+1}(\tilde{{\mathcal{F}}}^{{\prime}}) = 0\).
- 6.
We assume \(l_{\Gamma } \geq 3\), see Remark 5.18.
References
I. Reider, Nonabelian Jacobian of smooth projective surfaces. J. Differ. Geom. 74, 425–505 (2006)
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Reider, I. (2013). Period Maps and Torelli Problems. In: Nonabelian Jacobian of Projective Surfaces. Lecture Notes in Mathematics, vol 2072. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35662-9_5
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DOI: https://doi.org/10.1007/978-3-642-35662-9_5
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