Warped AdS₆times S² in Type IIB supergravity I: Local solutions

We investigate the existence of solutions with 16 residual supersymmetries to Type IIB supergravity on a space-time of the form $AdS_6 \times S^2$ warped over a two-dimensional Riemann surface $\Sigma$. The $SO(2,5) \times SO(3)$ isometry extends to invariance under the exceptional Lie superalgebra $F(4)$. In the present paper, we construct the general Ansatz compatible with these symmetries, derive the corresponding reduced BPS equations, and obtain their complete local solution in terms of two locally holomorphic functions ${\cal A}_\pm$ on $\Sigma$, subject to certain positivity and regularity conditions. Globally, $({\cal A}_+, {\cal A}_-)$ are allowed to be multiple-valued on $\Sigma$ and be holomorphic sections of a holomorphic bundle over $\Sigma$ with structure group contained in $SU(1,1) \times { C}$. Globally regular solutions are expected to provide the near-horizon geometry of $(p,q)$ 5-brane and 7-brane webs which are holographic duals to five-dimensional conformal field theories. A preliminary analysis of the positivity and regularity conditions will be presented here, leaving the construction of globally regular solutions to a subsequent paper.