Exact solutions to complex Type IIB supergravity for complex superalgebra F(4) and its real forms

We construct the general local solutions to complexified Type IIB supergravity which are invariant under the complexified Lie superalgebra $F(4)$. The geometry is a product of complexified maximally symmetric spaces $\mathcal M_{6 {\mathbb C}}$ and $\mathcal M_{2 {\mathbb C}}$ warped over a complexified surface $\Sigma_{\mathbb C}$. We classify the reality conditions that may be imposed consistently to obtain real form solutions within real forms of complex Type IIB supergravity. The latter comprise standard Type IIB, Type IIB$^\star$ and IIB$^\prime$, as well as theories with $3$, $5$, $7$ and $9$ time-like directions. Our classification of real solutions is consistent with and exhausts the real forms of $F(4)$, whose classification we confirm by elementary methods. The geometry of each real form solution is a product of real maximally symmetric spaces $\mathcal M_6$ and $\mathcal M_2$ warped over a Riemann surface $\Sigma$, with various signatures. The real solutions include, among others, known $AdS_6 \times S^2 \times \Sigma$ and $AdS_2 \times S^6 \times \Sigma$ solutions to standard Type IIB as well as new solutions of the form $dS_{1,5}\times S^2 \times \Sigma$ in Type IIB$^\star$. There are no real forms of the complex solutions with $\mathfrak{so} (7;{\mathbb R}) \oplus \mathfrak{so} (3;{\mathbb R})$ symmetry. We discuss the relevance of the complex solutions, and of analytic continuations from $dS_{1,5}\times S^2\times\Sigma$ to $S^6\times S^2\times\Sigma$ within complex Type IIB, in connection with holography for the polarized IKKT model.