An Ohsawa-Takegoshi theorem on compact K\"ahler manifolds

In this article we prove a theorem of Ohsawa-Takegoshi type on compact K\"ahler manifolds. Our arguments follow the "standard" approach for this kind of extension results; however, there are many complications arising from the regularization process of quasi-psh functions on compact K\"ahler manifolds, and unfortunately we only obtain a particular case of the expected result. We remark that the additional hypothesis we are forced to make are natural, since they are in many situations; we hope to remove them in a near future.