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Correction to: A characterization of symplectic Grassmannians

  • Gianluca OcchettaEmail author
  • Luis E. Solá Conde
  • Kiwamu Watanabe
Correction
  • 27 Downloads

1 Correction to: Math Z  https://doi.org/10.1007/s00209-016-1807-6

We refer to our original paper, using the same notation.

We thank Jun-Muk Hwang for pointing out that the homomorphism of algebraic groups \(\eta :{\mathrm{GL}}({\mathcal {F}}^{\,\vee }_{2,x})_{{\mathcal {Q}}^\vee _x}\rightarrow {\mathrm{GL}}(T_{X,x})\) has a nontrivial kernel (isomorphic to \({{\mathbb {Z}}}_2\)), preventing us from defining, in Section 3.3, the vector bundles \({\mathcal {Q}}^\vee \), \({\mathcal {F}}_1^\vee \), \({\mathcal {F}}_2^\vee \). However, the main Theorem of the paper may still be proved by slightly modifying our arguments as follows.

1. In Section 3.3, instead of defining the vector bundles \({\mathcal {Q}}^\vee \), \({\mathcal {F}}_1^\vee \), \({\mathcal {F}}_2^\vee \) over X, we may only define the corresponding projective bundles \({\mathcal {Z}}\), \({\mathcal {U}}_1\) and \({\mathcal {U}}_2\) over X, and vector bundles \({\mathcal {G}}\), \(\mathrm{H}{\mathcal {G}}\) over \({\mathcal {U}}_1\), whose projectivizations give \({\mathcal {U}}\) and \(\mathrm{H}{\mathcal {U}}\), fitting in a sequence:
$$\begin{aligned} 0 \rightarrow {\mathcal {K}}\longrightarrow {\mathcal {G}}\longrightarrow \mathrm{H}{\mathcal {G}}\rightarrow 0, \end{aligned}$$
with \( {\mathcal {K}}\) a line bundle.
On the other hand, the desired vector bundles \({\mathcal {Q}}^\vee \), \({\mathcal {F}}_1^\vee \), \({\mathcal {F}}_2^\vee \) may still be constructed over a rational curve \(\ell \) of the family \({\mathcal {M}}\). By choosing an appropriate twist, they fit in the commutative diagramfor \(d=0\) or 1.

2. The computations in Section 4 can then be carried out in the same way, and they provide \({\mathcal {O}}(\mathrm{H}{\mathcal {U}}) \cdot \ell = d=0\).

3. To prove Corollary 5.2 we had used a result of Fujita, which no longer applies, since we do not have a divisor of degree one on the fibers of \(\rho \); however the conclusion of the Corollary is still true, since \(\rho \) is equidimensional, by applying [1, Theorem 1.3].

4. The main goal of Section 5 was to prove the existence of an everywhere nondegenerate skew-symmetric form on the vector bundle \({\mathcal {Q}}\), whose existence is now not clear. However, the arguments of the section provide, verbatim, the existence of an everywhere nondegenerate skew-symmetric form on the bundle \(\mathrm{H}{\mathcal {G}}\).

5. In section 6, we may now consider the \(\mathrm{A}_{r-1}\)-bundle \(\overline{{\mathcal {U}}}_1\rightarrow X\) associated to \({\mathcal {U}}_1\rightarrow X\) and the fiber productSince \(\mathrm{H}{\mathcal {U}}\rightarrow {\mathcal {U}}_1\) is given by a cocycle with values in \({\mathrm{Sp}}({\mathcal {Q}}^{\,\vee }_x)\), by the results of Section 5, the same holds for \(\overline{{\mathcal {U}}}_1\times _{{\mathcal {U}}_1}{\mathrm{H}{\mathcal {U}}}\rightarrow \overline{{\mathcal {U}}}_1\), and we may construct its associated \(\mathrm{C}_{n-r}\)-bundle, \(\overline{\mathrm{H}{\mathcal {U}}}\rightarrow \overline{{\mathcal {U}}}_1\). Then the composition
$$\begin{aligned} \overline{q}:\overline{\mathrm{H}{\mathcal {U}}}\longrightarrow \overline{{\mathcal {U}}}_1\longrightarrow X \end{aligned}$$
is an \((\mathrm{A}_{r-1}\sqcup \mathrm{C}_{n-r})\)-bundle over X. From this point on, the proof goes on verbatim.

Notes

Reference

  1. 1.
    Höring, A., Novelli, C.: Mori contractions of maximal length. Publ. Res. Inst. Math. Sci. 49(1), 215–228 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TrentoPovo di TrentoItaly
  2. 2.Course of Mathematics, Programs in Mathematics, Electronics and Informatics, Graduate School of Science and EngineeringSaitama UniversitySakura-kuJapan

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