Inserted: 15 sep 2011
We discuss the deformation theory of special Lagrangian (SL) conifolds in complex space $C^m$. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. This category allows for the simultaneous presence of conical singularities and of non-compact, asymptotically conical, ends.
Our main theorem is the natural next step in the chain of results initiated by McLean and continued by the author and by Joyce. We survey all these results, providing a unified framework for studying the various cases and emphasizing analogies and differences. This paper also lays down the geometric foundations for our paper ``Special Lagrangian conifolds, II'' concerning gluing constructions for SL conifolds in $C^m$.