  The $\sigma$ Orionis cluster is important for studying protoplanetary disk
evolution, as its intermediate age ($\sim$3-5 Myr) is comparable to the median
disk lifetime. We use ALMA to conduct a high-sensitivity survey of dust and gas
in 92 protoplanetary disks around $\sigma$ Orionis members with
$M_{\ast}\gtrsim0.1 M_{\odot}$. Our observations cover the 1.33 mm continuum
and several CO $J=2-1$ lines: out of 92 sources, we detect 37 in the mm
continuum and six in $^{12}$CO, three in $^{13}$CO, and none in C$^{18}$O.
Using the continuum emission to estimate dust mass, we find only 11 disks with
$M_{\rm dust}\gtrsim10 M_{\oplus}$, indicating that after only a few Myr of
evolution most disks lack sufficient dust to form giant planet cores. Stacking
the individually undetected continuum sources limits their average dust mass to
5$\times$ lower than that of the faintest detected disk, supporting theoretical
models that indicate rapid dissipation once disk clearing begins. Comparing the
protoplanetary disk population in $\sigma$ Orionis to those of other
star-forming regions supports the steady decline in average dust mass and the
steepening of the $M_{\rm dust}$-$M_{\ast}$ relation with age; studying these
evolutionary trends can inform the relative importance of different disk
processes during key eras of planet formation. External photoevaporation from
the central O9 star is influencing disk evolution throughout the region: dust
masses clearly decline with decreasing separation from the photoionizing
source, and the handful of CO detections exist at projected separations $>1.5$
pc. Collectively, our findings indicate that giant planet formation is
inherently rare and/or well underway by a few Myr of age.
