# Correction to: A characterization of symplectic Grassmannians

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## 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.

*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:

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 \(\mathrm{H}{\mathcal {G}}\).

*X*. From this point on, the proof goes on verbatim.

## Notes

## Reference

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