  In this paper, we study the Khovanov homology of an alternating virtual link
$L$ and show that it is supported on $g+2$ diagonal lines, where $g$ equals the
virtual genus of $L$. Specifically, we show that $Kh^{i,j}(L)$ is supported on
the lines $j=2i-\sigma_{\xi}+2k-1$ for $0\leq k\leq g+1$ where
$\sigma_{\xi^*}(L)+2g= \sigma_{\xi}(L)$ are the signatures of $L$ for a
checkerboard coloring $\xi$ and its dual $\xi^*$. Of course, for classical
links, the two signatures are equal and this recovers Lee's $H$-thinness result
for $Kh^{*,*}(L)$. Our result applies more generally to give an upper bound for
the homological width of the Khovanov homology of any checkerboard virtual link
$L$. The bound is given in terms of the alternating genus of $L$, which can be
viewed as the virtual analogue of the Turaev genus. The proof rests on
associating, to any checkerboard colorable link $L$, an alternating virtual
link diagram with the same Khovanov homology as $L$.
  In the process, we study the behavior of the signature invariants under
vertical and horizontal mirror symmetry. We also compute the Khovanov homology
and Rasmussen invariants in numerous cases and apply them to show non-sliceness
and determine the slice genus for several virtual knots. Table 6 at the end of
the paper lists the signatures, Khovanov polynomial, and Rasmussen invariant
for alternating virtual knots up to six crossings.
