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Period Maps and Torelli Problems

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2072)

Abstract

In this section we take a more geometric point of view on the orthogonal decomposition

$$\mathbf{\tilde{H}}_{-l_{\Gamma }} =\displaystyle\bigoplus _{ p=0}^{l_{\Gamma }-1}{\mathbf{H}}^{p}$$

resulting from (2.51).

Keywords

  • Closed Point
  • Natural Projection
  • Local Section
  • Relative Differential
  • Relative Derivative

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    According to our convention in §2.7 the relative tangent sheaf is denoted by \(\mathcal{T}_{\pi }\).

  2. 2.

    The vertical arrows in the diagram (5.22) are the natural projection.

  3. 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. 4.

    \(\mathbf{T}_{\overleftarrow{Fl}_{\Gamma }}^{{\ast}}\) stands for the relative cotangent bundle of the structure morphism \(\overleftarrow{Fl}_{\Gamma }\) in (5.30).

  5. 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. 6.

    We assume \(l_{\Gamma } \geq 3\), see Remark 5.18.

References

  1. I. Reider, Nonabelian Jacobian of smooth projective surfaces. J. Differ. Geom. 74, 425–505 (2006)

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© 2013 Springer-Verlag Berlin Heidelberg

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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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