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Semistability and Restrictions of tangent bundle to curves

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Abstract

We consider all complex projective manifolds X that satisfy at least one of the following three conditions:

  1. (1)

    There exists a pair \({(C\,,\varphi)}\) , where C is a compact connected Riemann surface and

    $$\varphi\,:\, C\,\longrightarrow\, X$$

    a holomorphic map, such that the pull back \({\varphi^* {\it TX}}\) is not semistable.

  2. (2)

    The variety X admits an étale covering by an abelian variety.

  3. (3)

    The dimension dim X ≤ 1.

We prove that the following classes are among those that are of the above type.

  • All X with a finite fundamental group.

  • All X such that there is a nonconstant morphism from \({{\mathbb C}{\mathbb P}^1}\) to X.

  • All X such that the canonical line bundle K X is either positive or negative or \({c_1(K_X)\,\in\,H^2(X,\, {\mathbb Q})}\) vanishes.

  • All X with \({{\rm dim}_{\mathbb C} X\, =\,2}\).

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Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay, 400005, India

    Indranil Biswas

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  1. Indranil Biswas
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Correspondence to Indranil Biswas.

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Biswas, I. Semistability and Restrictions of tangent bundle to curves. Geom Dedicata 142, 37–46 (2009). https://doi.org/10.1007/s10711-009-9356-3

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  • DOI: https://doi.org/10.1007/s10711-009-9356-3

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