  We investigate the Dolbeault operator on a pair of pants, i.e., an elementary
cobordism between a circle and the disjoint union of two circles. This operator
induces a canonical selfadjoint Dirac operator $D_t$ on each regular level set
$C_t$ of a fixed Morse function defining this cobordism. We show that as we
approach the critical level set $C_0$ from above and from below these operators
converge in the gap topology to (different) selfadjoint operators $D_\pm$ that
we describe explicitly. We also relate the Atiyah-Patodi-Singer index of the
Dolbeault operator on the cobordism to the spectral flows of the operators
$D_t$ on the complement of $C_0$ and the Kashiwara-Wall index of a triplet of
finite dimensional lagrangian spaces canonically determined by $C_0$.
