Riemann–Roch variations

We construct a mirror-type correspondence that assigns variations (that is, local systems, $D$-modules or $l$-adic sheaves) to pairs $(V,C)$, where $V$ is a variety and $C$ is a complex of densely filtered vector bundles over $V$. We consider Calabi–Yau complete intersections in projective spaces. In the particular case when the complex is quasi-isomorphic to the tangent bundle on a generic Calabi–Yau complete intersection, this construction yields the variation that arises in the relative cohomology of the mirror-dual pencil. We call it the Riemann–Roch variation. The Riemann–Roch data of the divisorial sublattice in the $K$-group can be read off the Riemann–Roch local system since it encodes the information about the Euler characteristics of all $\mathscr O(i)$ sheaves (in an essentially non-commutative way).