Mr. Donald - it looks like you and I posted about the exact same time, and said very similar things. I'd say that great minds think alike, or call it serendipitous, but I'm not sure that we'd agree on much else, considering you post under the sobriquet of Mr. Donald Trump or whatever it is.
For Fiscal, not that you need to read it, but this is pretty neat: the UK version of Westlaw allows this summary to be viewed without requiring a subscription.
https://uk.practicallaw.thomsonreut...ontextData=(sc.Default)&firstPage=true&bhcp=1
Short version:
0) Argentina sees itself as having jurisdiction, no question.
1) custody is "always assigned to the mother in cases where the child is under 5 years of age."
2) unmarried couples' division of property is not as in a "common law marriage" as in some places, but by contract (like we have in California), except as to the family home, and the "compensation rights for non-working party in charge of child education." This is apart from child support.
3) the above notwithstanding, you'll probably have to maintain her for some period of time, even if there's no alimony available to her. Proving infidelity on her party is pointless (even if you felt like you wanted to and could, which it doesn't sound like), as Argentina dispensed with fault-based marriage dissolutions a few years ago, and you're not even married.
4) child support in Argentina is absolutely brutal. You can pay until the kid is 25, you can be sued by the mother, the kid (when he reaches 18), and even whatever guy (or gal) the mother shacks up with next, provided they marry. They'll get everything they can from you, and it's based on a percentage of your income. She gets the benefit of the doubt burden of proof-wise (ie. it's on you to demonstrate that the kid doesn't have need, or that you can't/shouldn't continue to provide 35 or 40 or whatever percent of your salary, and even if she makes more or marries some new guy, good luck getting the judge to adjust it). The guide doesn't say this, but you can expect to be at a disadvantage as a relatively higher earning foreigner.
5) watch out for domestic violence accusations, as that could really harm you.
6) these are general rules and local rules may vary based on your location/court.