That would point to a specific kind of art, instead of art in general.
So the intended meaning of the German sentence is "Italy is the home of art," and German uses the genitive feminine singular definite article der, where English would either use no article or use the plural "the arts."
But in the absence of any context, the German sentence could be referring to a particular art, and thus "Italy is the homeland of the art." is a possible and grammatically correct translation, which Duo currently rejects. I reported this problem. 28 June 2018.
I am inclined to think the sentence "Italien ist die Heimat der Kunst" is incorrect given that the translation "Italy is the home of the art" is not accepted (but rather '... home of the arts' is accepted. It would then appear that the base sentence, assuming the desired translation of 'der Kunst' is 'the arts,' should actually utilize the plural form of 'Kunst' which is 'Kuenste.'
"Italien ist die Heimat der Kuenste."
'Der Kuenste' being the genitive plural form, as opposed to the genitive singular form 'der Kunst.'
Art singular? That's the sentence Duo asked us to translate: Italien ist die Heimat der Kunst. Duo is rejecting a correct English translation: "Italy is the homeland of the art." Admittedly, in contrast, the alternative translation "of art" or "of the arts" is a well-known claim.
Of course, if you're referring to one specific art, you could say Italien ist die Heimat dieser Kunst.
Arrahn, Using the genitive plural der Künste" is an interesting and I think good suggestion. However, it appears that in Duo's sentence German uses der Kunst to mean "of art," not "of the art," and that's why Duo currently wants the English translation "of art" or "of the arts." (The statement is a well-known claim.)
I am very often not able to hear the loud speaker on my German-Englsh program. There I have only the possibility to skip but not more. And when it happened tree times or even more it is impossible to continue the lesson. It is really craza. Who has sometimes the same problem? Tell me, please.
I assume you meant ‘capital’ (‘Capitol’ is the name of various specific places, the best known of which is probably the seat of the US congress). ‘Home’ and ‘capital’ are not the same thing. When used metaphorically, they hint at similar things (a place where something flourishes and has a strong presence), but ‘home’ also strongly implies a place of origin, while ‘capital’ indicates superiority of some sort, a kind of first place. There is such a clear correspondence between ‘Heimat’–‘home’ and ‘Hauptstadt’–‘capital’ that translating one for the other would, in my opinion, be ill-advised.