On the topological decomposition of the hypersurfaces in projective toric manifolds

Details On the topological decomposition of the hypersurfaces in projective toric manifolds On the topological decomposition of the hypersurfaces in projective toric manifolds

2011-12-07

arxiv.org/abs/1112.1555v1

Mathematics

Algebraic Topology

15 pages

Scientific paper

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) $ as a connected sum of $s$ copies of $S^n \times S^n$ with a differential manifold $Y'$ such that $b_n (Y')=0$ or 2. When $n$ is even and the degree of $Y$ in $X$ is big enough, we find that $Y$ also admits such a decomposition $Y'\sharp \ s(S^n \times S^n)$, where $Y'$ satisfy $b_n(Y')-|sign(Y')|=b_n(X)\pm sign(H^n(X))$, where $sign(H^n(X))$ is the signature of a certain bilinear form defined on $H^n(X,\mz)$.

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