In the literature, are there some researchs on non commutative analogy of Leray-Hirsch theorem in the context of non commutative Principal bundles?



Noncommutative Leray - Hirsch theorem in the context of noncommutative principal bundles
First of all, let me make clear that by no means would i like to pretend any expertise on the topic (in fact i feel i have a limited understanding of AG in general). However, the question reminded me that i recently came across a "quantum" generalization of Leray-Hirsch theorem (while i was actually searching for something quite different). 


So, maybe you would be interested to take a look at: 


* [Quantum Cohomology under Birational Maps and Transitions](https://arxiv.org/abs/1705.04799v1), arXiv:1705.04799 [math.AG]


and the references therein; especially: 


* [Invariance of quantum rings under ordinary flops II: A quantum Leray–Hirsch theorem](http://content.algebraicgeometry.nl/2016-5/2016-5-027.pdf), Algebraic Geometry 3 (5) (2016) 615–653,
* [Invariance of quantum rings under ordinary flops III: A quantum splitting principle](https://www.intlpress.com/site/pub/files/_fulltext/journals/cjm/2016/0004/0003/CJM-2016-0004-0003-a002.pdf), Cambridge Journal of Mathematics, Volume 4, Number 3, 333–401, 2016


Hope that is of some interest for your purposes. 


