Basically a negative forward vol in the ATM vol term structure. For index options, it's probably rare. But for single name options, I've seen a bunch of examples on Bloomberg. Does this relationship admit some weak form of arbitrage?
Update: I think I have falsely claimed about the forward vol in this situation. Some people might call it "nagative forward vol", but now I don't think it makes much sense (might just be an analogy to interest rate). And as commented in one of the answers, the implied forward vol should be calculated as suggested on Wikipedia, assuming that the two periods are independent of each other. Otherwise, some correlation coefficient is needed in the calculation.
Now I read a bit more on this topic, this kind of phenomenon doesn't admit any arbitrage, but another situation would. If we factor in the time component and calculate the implied variance as $\sigma^2T$, then a "negative forward variance" would admit arbitrage under Black-Scholes dynamics.