Can the seeds of desires bowed in the past lives can now be eradicated from the mind completely? Desires are the seeds which have effects later or sooner. Can we destroy such seeds without experiencing their effect or Karmas?
First of all, a seed of desire, or a seed of anger, etc. are not karmic seeds (or karmic imprints). They do not ripen in the same way at all. A seed of desire will give rise to a mind of desire. Second, desires are not seeds. The seeds are non-associated compositional factors while an instance of desire is a mind.

Yes, the seeds of the afflictions can be abandoned so that they never give rise to an afflicted mind, such as a mind of anger, lust, etc.

There are two types of abandonment:


Partial abandonment, and
Complete abandonment.


Partial abandonment is also called 'non-analytical cessation' since such an abandonment happens when you engage in Samata and the Jhanas. When you reach a concentration of the form realm (such as a mind of calm abiding, etc.) afflictions of the desire realm are abandoned, but the seeds are still there. This is a partial abandonment, since the seeds are not yet abandoned.

The way to abandon the afflictions together with their seeds is by generating wisdom directly realizing emptiness. When the wisdom directly realizing emptiness acts as an antidote, it abandons afflictions together with their seeds, this is called analytical cessation and it is complete abandonment. According to Prasangika, the first moment of a wisdom directly realizing emptiness does not abandon all afflictions altogether (and their seeds), but merely the intellectually acquired afflictions and their seeds. It is only later, when the wisdom is conjoined with an accumulation of merit (or with the six perfections) that this wisdom is strong enough to abandon innate afflictions and their seeds.

Usually, according to so-called Hinayana tenets, the seeds are abandoned by way of generating wisdom directly realizing the sixteen aspects of the four noble truths.

In any case, such a wisdom is a mind of special insight and is thus a union of calm abiding and special insight. Calm abiding must be attained first.