I am a physicist and first-time poster so I will do my best to make this question clear. Apologies in advance if it's trivial.
I am trying to compute the chi-y genus of the "Goldstein-Prokushkin manifold" formed from a certain torus fibration over a K3 surface. (http://arxiv.org/pdf/hep-th/0212307v2.pdf). In particular, the total space is non-Kahler with known Betti numbers and SU(3) structure group.
Formally, it looks like I will need to compute the dimensions of $H^p(X, \bigwedge^q E)$ where $E$ is the bundle over the total space X. The Hirzebruch-Riemann-Roch theorem relates this quantity to $\int_X ch(\bigwedge^q E)td(X)$ for each q.
However, in practice I have never computed these quantities and have struggled to find a sample computation. My question then is a.) if I need to use HRR to compute the chi-y genus, can someone provide some technical guidance or a reference to proceed? and b.) Is there perhaps an alternative way to compute these quantities using what's known about the manifold that is more transparent than HRR? (In the paper they construct a Gysin sequence relating cohomology classes of the total space to cohomology classes of the base K3 but I'm totally unfamiliar with Gysin sequences and moreover they were computing $H^p(X, \mathbb{R})$ rather than $H^p(X, \bigwedge^q E)$).
Thanks for your help!