Slaveholders were under no legal obligation whatsoever to their enslaved children. They could force them to work, they had the same latitude to inflict violence on them as on any other enslaved person. They could sell their children as well. Frequently they did all of these things.

The ability of a slaveholder to do so had been built into the laws of North American slavery since the mid-17th century, when the Virginia Assembly codified the partus sequitur ventrem (Latin: that which comes forth follows the womb) principle that enslaved mothers gave birth to enslaved children. It's important to remember that the early colonists in what they called Virginia did not arrive with the intention or the legal framework for building a slave society where legal status and race were synonymous. That happened gradually over the course of the 17th century for a variety of reasons, and the construction of its legal architecture was contradictory, reactive, and somewhat improvisational. The 1662 law read:

WHEREAS some doubts have arrisen whether children got by any Englishman upon a negro woman should be slave or free, Be it therefore enacted and declared by this present grand assembly, that all children borne in this country shalbe held bond or free only according to the condition of the mother, And that if any christian shall committ fornication with a negro man or woman, hee or shee soe offending shall pay double the fines imposed by the former act.

Those "doubts" are likely a reference to freedom suits like that of Elizabeth Keys. In 1658 she brought a suit demanding her freedom to a Virginia court. Her mother had been enslaved but her father was a freeborn British subject, a gentleman and member of Virginia's House of Burgesses no less. Her father, Thomas Key, had acknowledged her, baptised her, and placed her under the guardianship of another Englishman at his death (in 17th century Virginia, illegitimate children were typically "indentured" to labor for a period of years before attaining truly free status). However, this guardian had sold Key's indenture to another, who forced her to labor beyond her term, in effect enslaving her (those who employed indentured servants were legally obligated to release them from their labors at the end of a term stipulated by the indenture).

Key knew her rights. As the daughter of a freeborn Englishman in a society where almost all forms of status, legal or otherwise, were patrilineal. And so she brought suit against the man who presumed to make her labor past her indenture: John Mottram. Incredibly, she won! But her victory here posed a potential problem for the colony's fortunes, and its ability to coerce enslaved workers into growing its tobacco. The Virginia Legislature soon acted to close off this route to emancipation (as they would do with baptism around the same time, upending the more antique notion of enslavement based on religious difference and further racializing the status). In the process, they upended centuries of English common law tradition. But veneration for tradition is often a matter of convenience, so I doubt they were too bothered.

The partus sequitur ventrem principle soon held sway in slave societies across the Americas, and as new ones were established in places like South Carolina and Georgia, they took that principle with them as a built in assumption. The implications of this decision are of course horrific. Not only did it free slaveholders from all obligation to their enslaved children, but it gave an economic incentive to commit sexual violence. In colonial and antebellum North America, there were no laws protecting enslaved women from sexual violence, and they would have been unenforceable if they were (another bit of legal maneuvering the colonial legislatures quickly developed: no black person, free or enslaved, could testify against a white person in a court of law). Sexual violence was thus not an "excess" of slavery, or the inevitable grim result of giving certain people far too much power over others. It was, in fact, a core component of slavery's daily operation and expansion.

So that's the legal obligation: non-existent.

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