I'm trying to construct ladder operators on cohomology space, I searched for a similar procedure but I can't find anything. To be clearer, I consider the cohomology space of a compact Kähler manifold $M$: $H=\bigoplus_{k\in \mathbb{N}} H^k(M)$ where $H^k(M)= \bigoplus_{k=p+q}H^{p,q}(M)$ as a Fock space. Therefore, I claim that the differential $d= \partial + \bar{\partial}$ to be a mixed creator operator. I'm asking for the annihilation operator, which would be something like the interior product, but splits into holomorphic and antiholomorphic parts similar to the split of $d$. Any ideas?
EDIT
More specifically is there a contraction taking $(p, q)$-forms to $(p-1, q)$ and/or $(p, q-1)$ forms?
Here's a map that goes backwards.
If $M$ is equipped with a hermitian metric, then $\Omega^{p,q}(M)$ is an inner product space and $\bar{\partial} : \Omega^{p,q-1}(M) \to \Omega^{p,q}(M)$ has an adjoint $\bar{\partial}^* : \Omega^{p,q}(M) \to \Omega^{p,q-1}(M)$ satisfying $\langle\bar{\partial}\alpha, \beta\rangle = \langle\alpha, \bar{\partial}^*\beta\rangle$.
Likewise, $\partial : \Omega^{p-1,q}(M) \to \Omega^{p,q}(M)$ has an adjoint $\partial^* : \Omega^{p,q}(M) \to \Omega^{p-1,q}(M)$.
In terms of the Hodge star $\ast : \Omega^{p,q}(M) \to \Omega^{n-q,n-p}(M)$ induced by the hermitian metric, we have $\bar{\partial}^* = -\ast\partial\ast$ and $\partial^* = -\ast\bar{\partial}\ast$.