  We obtain the complete local solutions with 16 supersymmetries to Type IIB
supergravity on a space-time of the form $AdS_{2}\times S^{6}$ warped over a
Riemann surface $\Sigma$ in terms of two locally holmorphic functions on
$\Sigma$. We construct the general Ansatz for the bosonic supergravity fields
and supersymmetry generators compatible with the $SO(2,1)\oplus SO(7)$ isometry
algebra of space-time, which extends to the corresponding real form of the
exceptional Lie superalgebra $F(4)$. We reduce the BPS equations to this
Ansatz, obtain their general local solutions, and show that these local
solutions solve the full Type IIB supergravity field equations and Bianchi
identities. We contrast the $AdS_{2}\times S^{6}$ solution with the closely
related $AdS_6\times S^2$ case and present our results for both in parallel.
Finally, we present a preliminary analysis of positivity and regularity
conditions for $AdS_{2}\times S^{6}$, but postpone the construction of globally
regular solutions to a subsequent paper.

High Energy Physics - Theory