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Locally Constant Fibrations and Positivity of Curvature
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abstract
Up to finite \'etale cover, any smooth complex projective variety $X$ with nef anti-canonical bundle is a holomorphic fibre bundle over a $K$-trivial variety with locally constant transition functions. We show that this result is optimal by proving that any projective fibre bundle with locally constant transition functions over a $K$-trivial variety has a nef anti-canonical bundle. Moreover, we complement some results on the structure theory of varieties whose tangent bundle admits a singular hermitean metric of positive curvature.
Forward citations
Cited by 2 Pith papers
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On compact K\"ahler manifolds with pseudo-effective tangent bundle
Compact Kähler manifolds with pseudo-effective tangent bundle admit a smooth fibration whose base is an étale quotient of a torus and whose fibers are rationally connected.
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Fundamental groups of compact K\"ahler manifolds with semi-positive holomorphic sectional curvature
A compact Kähler manifold with semi-positive holomorphic sectional curvature is a locally trivial fibration over a finite étale quotient of a torus with rationally connected projective fibers.
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