부산대학교 도서관

For every $n>2$, \n W. C. Hsiang, C. T. C. Wall, D. Montgomery and C. T. Yang\en showed that there are infinitely many distinct smooth structures on complex projective $n$-space, $\bold C {\roman P}^n$, distinguished by their Pontryagin classes. On the other hand, results of \n F. Hirzebruch, K. Kodaira and S.-T. Yau\en showed that a Kähler manifold with the same homotopy type and Pontryagin classes as $\bold C{\roman P}^n$ is analytically equivalent to $\bold C{\roman P}^n$. Here the authors prove Theorem 1: A Kähler manifold homotopy equivalent to $\bold C{\roman P}^n$ for $n\leq 6$ is analytically equivalent to $\bold C{\roman P}^n$. \par In Theorem 2 they give a similar result for the quadric hypersurfaces $X_3(2)$ and $X_4(2)$, if $c_1\neq -4x$, where $x$ is the generator of $H^2$. One ingredient in the proof is the fact that for a Kähler manifold $V$ homotopy equivalent to $\bold C{\roman P}^n$ on $X_n(2)$ the Hodge numbers and the $\chi_y$-genus of $V$ are uniquely determined. Theorem 3: For a compact complex manifold $V$, the Chern number $c_{n-1}c_1[V]$ is determined by $\chi_y(V)$ and hence by the Hodge number. \par This result and various integrality theorems, together with Iskov-\break skikh's work on Fano 3-folds, suffice to check that $c_1(V)$ is uniquely determined. The proof is then completed by a result of Kobayashi and Ochiai.