부산대학교 도서관

For every $n>2$, W. C. Hsiang, C. T. C. Wall, D. Montgomery andC. T. Yangshowed that there are infinitely many distinctsmooth structures on complex projective $n$-space, $\bold C{\roman P}^n$, distinguished by their Pontryaginclasses. On the other hand, results of F. Hirzebruch,K. Kodaira and S.-T. Yaushowed that a Kähler manifoldwith 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 proveTheorem 1: A Kähler manifold homotopy equivalent to$\bold C{\roman P}^n$ for $n\leq 6$ is analyticallyequivalent to $\bold C{\roman P}^n$.\parIn Theorem 2 they give a similar result for the quadrichypersurfaces $X_3(2)$ 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 Kählermanifold $V$ 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 complexmanifold $V$, the Chern number $c_{n-1}c_1[V]$ is determined by$\chi_y(V)$ and hence by the Hodge number.\parThis result and various integrality theorems, togetherwith Iskov-\break skikh's work on Fano 3-folds, suffice to check that$c_1(V)$ is uniquely determined. The proof is thencompleted by a result of Kobayashi and Ochiai.