(Abridged) With the goal of investigating the nature of OVI absorbers at high redshifts, we study the effects of proximity to the background quasar. In a sample of sixteen quasars at z(QSO) between 2.14 and 2.87 observed at 6.6 km/s resolution with VLT/UVES, we detect 35 OVI absorption-line systems lying within 8000 km/s of z(QSO). The systems can be categorized into 9 strong and 26 weak OVI absorbers. The strong absorbers are defined by the presence of either broad, fully saturated OVI absorption or partial coverage of the continuum source, and have log N(OVI)>~15.0; these systems are intrinsic to the AGN. The weak (narrow) systems show no partial coverage or saturation, and are characterized by log N(OVI)<14.5 and have a median total velocity width of only 42 km/s. The incidence dN/dz of weak OVI systems within 2000 km/s of the quasar is 42+/-12. Between 2000 and 8000 km/s, dN/dz falls to 14+/-4, equal to the incidence of intervening OVI absorbers. Whereas the accompanying H I and C IV column densities are significantly lower (by a mean of ~1 dex) in the weak OVI absorbers within 2000 km/s of z(QSO) than in those at larger velocities, the OVI column densities display no dependence on proximity. Furthermore, significant offsets between the HI and OVI centroids in ~50% of the weak absorbers imply that (at least in these cases) the HI and OVI are not formed in the same phase of gas. In summary, we find no firm evidence that quasar radiation influences the OVI-bearing gas, suggesting the OVI is collisionally ionized rather than photoionized, possibly in the multi-phase halos of foreground galaxies. Non-equilibrium collisional ionization models are needed to explain the low temperatures in the absorbing gas, which are implied by narrow line widths (b<14 km/s) in over half of the observed OVI components.
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