Let $X$ be a complex abelian variety. Then the dual abelian variety is $Pic^0(X)$. Is $Pic^0(X)$ is the set of degree zero line bundles on $X$? How do we define degree of a line bundle on a higher dimensional variety? Is $Pic^0(X)$ the set of line bundles on $X$ numerically equivalent to zero?
Another question I have is, if $P\in Pic^0(X)$, what can we say about $H^i(X,P)$? Is $H^0(X,P)=0$? It seems to me that the Euler characteristic is zero.