Can the World's Greatest Detective catch a killer who leaves no trace? It's a tough decision to pit Batman against Death Note's Light Yagami.
DC Comics' Batman is often called one of the greatest detective characters in fiction, but would he have a chance to catch Kira? Light Yagami from Death Note is an equally brilliant killer with some supernatural abilities on his side, so it's hard to make a definitive decision. Comparing these two characters and some of their recognized achievements and abilities is the only way to find out.
Batman is a character with a long and colorful history, so he has some edge in the "feats" department. But an important part of this question is determining exactly which incarnation of Batman is opposed to Kira. For simplicity's sake, the best way to progress to other eras of Batman seems to be to use modern Batman from the comics, but it could be a little different.
Batman's talents include extensive physical and martial arts training, the immense wealth and resources at his disposal, and a genius-level intelligence empowered by the best schools. Batman's curiosity led him to learn as much as he could about a wide variety of fields, from forensics to magic and tactics. While this comparison will only look at Batman and Light, rather than allies and large teams, his ability to develop strong relationships with a wide variety of people is another critical skill of his.
Still, there are two factors that may be most important here: Batman's detective skills and his ability to instill fear in his enemies. Batman's ability to deduce and investigate leads him to unravel the secret identities of many of his villains, and he is extremely meticulous, finding clues that the police or other heroes might miss. Identifying patterns in a series of murders, even if they seem like natural causes, is exactly what he's best at. Batman's horror mastery is a unique tool in his arsenal that can work especially well against Kira, as fear has proven to be one of the most effective ways to provoke Kira to make mistakes.
For Light's part, he's a pretty ordinary high school-to-college student depending on when exactly he arrives in the Death Note story. Light is extremely intelligent, but it's hard to say whether he's smarter than Batman. Still, he escaped capture for years and outsmarted L, the greatest detective in his world. He also has no access to any private resources other than Death Note itself. However, Death Note has a number of abilities that make it extremely difficult to identify its user. It can kill without a trace; There's no way to identify a natural cause of a heart attack caused by the victim's name being written in the Death Note. Death Note can also kill in an almost infinite number of other ways, including latency, which makes it extremely difficult to ensure that all deaths for which Kira is responsible are discovered.
Limitations in Death Note include physical impossibilities, the need to know the victim's real name and face, and the need to actually write it down on paper. While the Eye of the Shinigami deal is unlikely for Light to accept, it could make it very dangerous for him to stand up to her in public. Although Kira primarily kills criminals, this infamous "not killing" isn't an argument that goes far with Batman. As with Batman, Light won't be assisted by other characters like Misa or Ryuk - something that's needed to escape that long.
In a universe where both exist, Batman and Kira would quickly fall into a similar game of cat-and-mouse, as in L. Batman would no doubt have noticed a string of criminals suddenly dying of heart attacks. (Especially if any member of his rogue gallery was targeted.) Batman's secret identity would protect him from any sudden retaliation. Light may have even looked at Batman before interacting with the Death Note, so he may be reluctant to set his sights on what he sees as yet another vigilante. This vigilante status might be one of Batman's biggest advantages in a match like this, as the Rent Task Force has to deal with government pressure and legality (not restricting the L too much). The type of surveillance Batman can implement with his money is definitely beyond what L and his task force can reach, which can help speed up the timeline.
In an ideal scenario for Light, he could pressure government officials to declare Batman a wanted man, threatening to kill him the next time. They would go after Batman, and even if they weren't able to catch him, it would cause major problems for his operation. But Batman has one main advantage that L doesn't have: He would be able to immediately recognize the supernatural nature of Kira's murders, as he was already familiar with things like magic, demons, and gods. As a result, Kira may have less time to experiment and learn about Death Note's limitations. As with L, if Batman's identity can reveal the masked hero's face and name would be Kira's main target, then it's game over. Enough characters in DC's continuity have been able to conclude that it's certainly possible—even likely—that Light could do with Batman's identity. Where this gets tricky is a certain bottleneck in Batman fandom: which is the real identity?
While some say the masked Batman is clearly second-person, others argue that his notion of revenge and his desire to stop crime is so inherent in who he is that Bruce Wayne is more of a “character” than Batman. Even if Light got Bruce Wayne's name and face…it might not actually matter. If Bruce really saw himself as Batman and thought so, wouldn't the name on the note be "Batman"? Of course, this doesn't help if other people associated with Batman and/or Bruce Wayne are also identified. Kira may eliminate many of Batman's friends on the way, but sooner or later Batman will find her and has the tools to rein in one so that typing is impossible.
If Batman destroyed the Death Note and Light lost his memories of being Kira, he could stay in Arkham for too long. This leaves one last interesting question: Can Light be rehabilitated if he survives one last showdown?
Hard to say, but Batman will undoubtedly try.