In this paper, we study structures of smooth complex projective polarized manifolds (X, H) of dimension n ≥ 2 which are rationally connected with respect to a family of H-degree four. Under the assumption (-K<inf>X</inf> · ) ≥ n + 3, we prove that, with two kinds of exceptions, the Picard number of X is at most four and X is covered by rational curves of H-degree one. In addition, we provide a classification in case n = 2.
- covered by lines
- Picard number
- rational curves
- Rationally connected manifolds
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