On the topological decomposition of the hypersurfaces in projective toric manifolds

TL;DR

This paper investigates the topology of non-singular hypersurfaces in projective toric manifolds, providing a decomposition into connected sums involving spheres and analyzing how the hypersurface's topology relates to the ambient manifold's invariants.

Contribution

It introduces a topological decomposition of hypersurfaces in toric manifolds into connected sums with spheres, depending on the dimension's parity and degree conditions.

Findings

For odd dimensions, hypersurfaces decompose into connected sums with spheres and a manifold with specific Betti number properties.

For even dimensions with large degree, similar decompositions exist with relations involving Betti numbers and signatures.

The results connect hypersurface topology with ambient manifold invariants like Betti numbers and signatures.

Abstract

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 $H^n(X,\mz)$.