Ricci-Flat Metrics on Canonical Bundles

In this thesis, we studied a new two-parametric class of Kähler-Einstein surfaces $\mathcal{M}^{[\lambda_1,\lambda_2]}$ with explicit KE metrics with $SU(2)\times U(1)$ isometries, and have conical singularities. Topologically, every $\mathcal{M}^{[\lambda_1,\lambda_2]}$ is homeomorphic to $\mathbb{F}_2$, the second Hirzebruch surface, but are different as complex manifolds. We studied their differential geometry in detail regarding the behavior of the associated Riemannian curvature, geodesics, contact structure, and the nature of singularities. We used Calabi's ansatz to put explicit Ricci-flat metrics on $tot(K_\mathcal{M}^{[\lambda_1,\lambda_2]})$. These Ricci-flat metrics are $D3$-brane solutions of type IIB supergravity theories.