Open surfaces of small volume
  We construct a surface with log terminal singularities and ample canonical class that has $K_X^2=1/48 983$ and a log canonical pair $(X,B)$ with a nonempty reduced divisor $B$ and ample $K_X+B$ that has $(K_X+B)^2 = 1/462$. Both examples significantly improve known records. 
Valery Alexeev and Wenfei Liu
math.AG