The $\Pi\sp 1\sb 2$-singleton conjecture

The real ${0^\char93 } = {\operatorname{Thy}}\left\langle {L,\varepsilon ,{\aleph _1},{\aleph _2}, \ldots } \right\rangle$ is a natural example of a nonconstructible definable real. Moreover ${0^\char93 }$ has a definition that is absolute: for some formula $\phi (x),{0^\char93 }$ is the unique real $R$ such that $L[R] \vDash \phi (R)$. Solovay conjectured that there is a real $R$ such that $0{ < _L}R{ < _L}{0^\char93 }$ and $R$ also has such an absolute definition. We prove his conjecture by constructing a $\Pi _2^1$-singleton $R$, $0{ < _L}R{ < _L}{0^\char93 }$. A variant of our construction produces a countable nonempty $\Pi _2^1$ set of reals not containing a $\Pi _2^1$-singleton. The latter result answers a question of Kechris.