Ampleness of Schur powers of cotangent bundles and k-hyperbolicity

Abstract

In this paper, we study a variation in a conjecture of Debarre on positivity of cotangent bundles of complete intersections. We establish the ampleness of Schur powers of cotangent bundles of generic complete intersections in projective manifolds, with high enough explicit codimension and multi-degrees. Our approach is naturally formulated in terms of flag bundles and allows one to reach the optimal codimension. On complex manifolds, this ampleness property implies intermediate hyperbolic properties. We give a natural application of our main result in this context.

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Fig. 1

Notes

  1. 1.

    Let XY be varieties, and let \(p_{1}:X\times Y \rightarrow X\), \(p_{2}:X \times Y \rightarrow Y\) be the first and second projection. If \(E_{1}\) is a vector bundle over X, and \(E_{2}\) is a vector bundle over Y, we denote \(E_{1}\boxtimes E_{2} \rightarrow X\times Y\) the vector bundle

    $$\begin{aligned} E_{1} \boxtimes E_{2} :=pr_{1}^{*}E_{1} \otimes pr_{2}^{*}E_{2}. \end{aligned}$$

    This definition generalizes to an arbitrary finite product of varieties.

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Acknowledgements

I would like to thank my supervisor Erwan Rousseau as well as my co-supervisor Lionel Darondeau for their help and support. This work owes a lot to Lionel’s insights on the subject: I could never thank him enough for the time he spent sharing it with me.

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Correspondence to Antoine Etesse.

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Etesse, A. Ampleness of Schur powers of cotangent bundles and k-hyperbolicity. Res Math Sci 8, 7 (2021). https://doi.org/10.1007/s40687-020-00243-2

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Keywords

  • Ampleness
  • Cotangent bundle
  • Complete intersection
  • Flag manifolds
  • Hyperbolicity

Mathematics Subject Classification

  • 14M10
  • 14M12
  • 32Q45