Translation:Tá sí ag tabhairt leabhair dó.
It's not the plural, but the genitive singular. This sentence literally translates to "She is at the giving of a book to him".
As far as I know you have always to use the genitive case after a verbal noun.
Sometimes I see the verb before the object, and sometimes, the other way around. What is the rule that I'm missing here?
why is it "dó"? I thought "him" uses "á" --according to the duolingo notes.
I think you're probably talking about the Tips & Notes for the Verbal Noun skill. Those notes describe a situation where a pronoun would be the object of the verbal noun, such as "they are cleaning him". As you can't have a pronoun as the object of the verbal noun, the sentence is reconstructed as "they are to his cleaning", where "to his" ends up as á. (note "to his", not "to him", and the case can also be made that it is "at his" rather than "to his").
That's not relevant in this case, where leabhair is the object of the verbal noun, and dó is just a prepositional pronoun indication the destination/recipient - dó being the combined form of the preposition do and the pronoun é.
go raibh maith agat! You've been very helpful. Just what I needed..
Would "Tá sí leabhar a tabhairt dó?" be acceptable? THese verbal nouns throw me for a loop every time
leabhar a thabhairt dó (note the séimhiú after a) means "to give him a book", and would be used in a sentence like ba mhaith léi leabhar a thabhairt dó - "she would like to give him a book".
You use the genitive form of a noun after the "ag (verbal noun)" construction, and the plural genitive is "leabhar", so " Tá sí ag tabhairt leabhar dó" is "She is giving books to him".
They object of a progressive verb (ag tabhairt) is in the genitive /tuiseal ginideach.
The genitive of leabhar is leabhair. Therefore ag tabhairt leabhair means "giving a book".