Algebraic cobordism of filtered vector bundles on varieties: Notes on a work of Lee and Pandharipande

TL;DR

This paper discusses the algebraic cobordism of filtered vector bundles on varieties, building on Lee and Pandharipande's work, and examines the compatibility of bundle operations with cobordism, raising questions on refined cobordisms.

Contribution

It extends the double point cobordism groups to filtered vector bundles and analyzes the compatibility of bundle operations within this framework.

Findings

Only dual operation is compatible with double point cobordisms in general.

Constructs double point cobordism groups for filtered vector bundles.

Raises questions on refined and higher algebraic cobordisms.

Abstract

The construction of double point cobordism groups of vector bundles on varieties in the work [Lee-P] (arXiv:1002.1500 [math.AG]) of Yuan-Pin Lee and Rahul Pandharipande gives immediately double point cobordism groups of filtered vector bundles on varieties. We note also that among the four basic operations -- direct sum, tensor product, dual, and Hom -- on vector bundles on varieties, only taking dual is compatible with double point cobordisms of vector bundles on varieties in general, by a demonstration on an example of vector bundles on Calabi-Yau 3-folds. A question on refined and/or higher algebraic cobordisms of vector bundles on varieties is posed in the end.