The birth of Jesus is problematic. First, it was during the reign of Herod, who died in 4 BCE, and during the time when Quirinius was governor of Syria in 6 CE (the date of the census).
Second, no such census exists. The Romans were obsessive record keepers, and we’d have this in hand if it was a thing. There were many censuses taken by the Romans, but one commanded by Augustus of the whole empire never happened.
Third, the census required everyone to return to their ancestral homes. This is foolish: such a requirement would have thrown the empire into chaos, as people would have had to leave their lives and work for months on end, sometimes travelling thousands of miles. There’s no indication of what “ancestral” home this would have been (we can assume through the father’s line though, although there’s no good reason to assume this would give any accurate information). It was likely only included in the later gospel accounts to place Jesus in the lineage of David to fulfill the Old Testament prophecy. But since there’s no record of any of it, there’s no reason to assume the census was a thing anyway.
Fourth, nobody else in the ancient world seemed to notice a gigantic bright star shining day and night for weeks on end. Besides, stars are a terrible way to pinpoint a single location, as they don’t stay stationary in relation to our line of sight. Unless there was a very localized beacon directly above the location that literally no one else noticed (remember, there was a huge migration of people through the region at the time, if the census story was true), this probably didn’t happen.
Fifth, Herod the Great ordered the slaughtering of all the young boys in Bethlehem. Mass murder of this kind was fairly commonplace in the Bible, but not really typical of organized societies under Roman rule (unless ordered by the Romans). That said, the population of Bethlehem at the time is thought to have been about 1,000. The infanticide would have resulted in about 20 deaths, but there would have been somerecord of it somewhere. There isn’t. Even the much-touted Josephus — whose writings didn’t appear until nearly a century later, and so are only records of hearsay anyway — doesn’t record the incident. The first non-Biblical record of Herod’s infanticide appears in theof c.150 CE. This wouldn’t be considered an historical account by most practicing historians today, given that no records contemporary to the event have been found.
None of this necessarily refutes the existence of the historical figure of Jesus or negates his purported divinity in and of itself. But it does cast considerable doubt on the believability of the nativity story, considering the details only appear in two books (Matthew and Luke) in all of contemporary history (and those books were themselves not written until decades later), which contradict each other on a number of details, present demonstrable errors in chronology, history, and events, and have no corroborating source material anywhere else in the contemporary record.