On the topological decomposition of the hypersurfaces in projective toric manifolds

In this paper, we want to discuss the topology of the non-singular hypersurface $Y^n$ with complex dimension $n$ in a projective toric manifold $X^{n+1}$. When $n$ is odd, our main results are a decomposition of $Y^{n}\cong Y’\sharp \ s(S^n \times S^n) $asaconnectedsumof$s$copiesof$S^n \times S^n$withadifferentialmanifold$Y’$suchthat$b_n (Y’)=0$or2.When$n$ is even and the degree of $Y$ in $X$ is big enough, we find that $Y$ also admits such a decomposition $Y^{\mathrm{\prime }}\mathrm{♯}s(S^n\times S^n)$, where $Y^{\mathrm{\prime }}$ satisfy $b_n(Y^{\mathrm{\prime }})-\mathrm{\mid }sign(Y^{\mathrm{\prime }})\mathrm{\mid }=b_n(X)\pm sign(H^n(X))$, where $sign(H^n(X))$ is the signature of a certain bilinear form defined on $H^n(X,\text{mz})$.