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Sabatini show that if $M$ satisfies an extra \"positivity condition\" then the isotropy weights at the fixed points of $M$ agree with those of some linear action on $\\mathbb{CP}^4$. Therefore, the (equivariant) cohomology rings and the (equivariant) Chern classes of $M$ and $\\mathbb{CP}^4$ agree; in particular, $H^*(M;\\mathbb{Z}) \\simeq \\mathbb{Z}[y]/y^5$ and $c(TM) = (1+y)^5$. 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