  In this paper, we will show that for a smooth quasi-projective variety over
$\C,$ and a regular function $W:X\to \C,$ the periodic cyclic homology of the
DG category of matrix factorizations $MF(X,W)$ is identified (unde
Riemann-Hilbert correspondence) with vanishing cohomology
$H^{\bullet}(X^{an},\phi_W\C_X),$ with monodromy twisted by sign. Also,
Hochschild homology is identified respectively with the hypercohomology of
$(\Omega_X^{\bullet},dW\wedge).$
  One can show that the image of the Chern character is contained in the
subspace of Hodge classes. One can formulate the Hodge conjecture stating that
it is surjective ($\otimes\Q$) onto Hodge classes. For W=0 and $X$ smooth
projective this is precisely the classical Hodge conjecture.
