plot explanation - Why didn't Sherlock Holmes investigate the events surrounding Irene Adler?

In the 2011 Sherlock Holmes: A Game of Shadows:

Moriarty and Irene Adler have an brief conversation in a restaurant. After the exchange, she passes out before exiting. Later in the film, Moriarty converses with Sherlock Holmes and tells him that Irene contracted "a deadly form of tuberculosis" and presents him with a bloodied handkerchief, seemingly as proof to the claim that she is dead.

Holmes accepts the information with little doubt and then the subject is dropped entirely for the rest of the movie. My question is: why would Sherlock Holmes take his enemy at his word? Why wouldn't he investigate these claims to make sure Moriarty wasn't bluffing? Considering his relationship with Irene Adler, I thought more time would have been spent developing this plot element.

Answer

The simplest answer is probably best. Holmes had already taken the measure of Moriarty and knew him to be cold-blooded, methodical, and ruthlessly efficient. Having traced his operations for a number of months, as well as the methods Professor Moriarty was willing to engage (the bomb in the crowded auction hall for instance, the assassination outside said auction hall) he knew that Moriarty would not have hesitated to have Ms. Adler removed from play, treating her as little more than a chess piece in his game to draw Holmes into the open.

Holmes was neither a sentimental man, nor a foolish one, and Professor Moriarty hoped to used Ms. Adler's demise as a means of putting Holmes off-balance. Do not mistake the stoicism displayed by Holmes as a sign of not caring. He cared deeply for Ms. Adler. But with Moriarty making the stakes as high as they were, Holmes could not allow his feelings to be clouded by emotion. Holmes had tried to warn Ms. Adler several times of the threat of Moriarty, and being a capable adventurer, she was certain she could handle herself. Likely, against any other threat, she would have been able to.

Holmes did not follow the lead, because there was no need to. Professor Moriarty did not bluff and was perfectly willing to kill anyone that stood between him and his goal. If her death upset Holmes, so much the better. Icing, nothing more.