Does the complement of sigma Kleene star exist?
Note that the complement of $A \subseteq \Sigma^*$ is defined as $\Sigma^* \setminus A$; so yes: the complement of $\Sigma^*$ itself is the empty set. 

To be completely accurate, you need to state with respect to which universe you want to take the complement. For instance, if you were to take the complement of $\Sigma^*$ w.r.t. $\Sigma'^*$ where $\Sigma' = \Sigma \mathbin{\dot\cup} \{a\})^*$, $a \not\in \Sigma$, it would not be empty but $\Sigma'^* \cdot \{a\} \cdot \Sigma'^*$ (the set of words with at least one $a$).