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Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi’s canonical-ampleness conjecture in the smooth compact Kähler setting.
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Let X be a compact connected Kähler manifold of positive complex dimension. If every holomorphic map \C\to X is constant, then K_X is ample. In particular, a positive tensor power of K_X defines a holomorphic embedding of X into a complex projective space.
Reported proof verification
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We prove that every compact connected Kähler manifold of positive complex dimension containing no nonconstant entire curve has ample canonical bundle and is projective. This resolves positively Kobayashi’s canonical-ampleness conjecture in the smooth compact Kähler category.
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Independent comparisons and proof-verification reports
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Papers elsewhere in the release that this family’s papers cite
Papers elsewhere in the release that cite this family’s papers
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State the result in Lean
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Why this passage is shown
Theorem 1.1
In Canonical ampleness of compact hyperbolic Kähler manifolds, 23 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.