The explicit geometric constructions of bordism of Milnor hypersurface $H_{1,n}$ and $\mathbb{C} P^1\times \mathbb{C} P^{n-1}$

TL;DR

This paper constructs explicit complex bordisms between Milnor hypersurfaces and products of projective spaces, simplifying the bordism classification problem by reducing it to null-bordism of certain projective bundles.

Contribution

It provides new explicit geometric constructions of bordisms between key complex manifolds, advancing the understanding of their bordism classes.

Findings

Constructed explicit bordisms between $H_{1,n}$ and $ ext{CP}^1 imes ext{CP}^{n-1}$.

Reduced bordism classification to null-bordism of specific projective bundles.

Enhanced methods for analyzing complex bordism relations.

Abstract

In the present paper we construct two new explicit complex bordisms between any two projective bundles over $\mathbb{C} P^1$ of the same complex dimension, including the Milnor hypersurface $H_{1,n}$ and $\mathbb{C} P^1\times \mathbb{C} P^{n-1}$. These constructions reduce the bordism problem to the null-bordism of some projective bundle over $\mathbb{C} P^1$ with the non-standard stably complex structure.