Abbreviated Abstract: The near-infrared study of the Carina Nebula in this paper builds on the results of the Chandra Carina Complex Project (CCCP), that detected 14368 X-ray sources in the 1.4 square-degree survey region, an automatic source classification study that classified 10714 of these as very likely young stars in Carina, and an analysis of their clustering properties. We used HAWK-I at the ESO VLT to conduct a very deep near-IR survey with sub-arcsecond angular resolution, covering about 1280 square-arcminutes. The HAWK-I images reveal more than 600000 individual infrared sources, whereby objects as faint as J ~ 23, H ~ 22, and Ks ~ 21 are detected at S/N >= 3. While less than half of the Chandra X-ray sources have counterparts in the 2MASS catalog, the ~5 mag deeper HAWK-I data reveal infrared counterparts to 6636 (= 88.8%) of the 7472 Chandra X-ray sources in the HAWK-I field. We analyze near-infrared color-color and color-magnitude diagrams to derive information about the extinctions, infrared excesses, ages, and masses of the X-ray selected objects. The near-infrared properties agree well with the results of the automatic X-ray source classification. The shape of the K-band luminosity function of the X-ray selected Carina members agrees well with that derived for the Orion Nebula Cluster, suggesting that, down to the X-ray detection limit around 0.5-1 Msun, the shape of the IMF in Carina is consistent with that in Orion (and thus the field IMF). The fraction of stars with near-infrared excesses is rather small, <=10%, but shows considerable variations between individual parts of the complex. The distribution of extinctions for the diskless stars ranges from ~1.6 mag to ~6.2 mag (central 80th percentile), clearly showing a considerable range of differential extinction between individual stars in the complex.Comment: Accepted for the ApJS Special Issue on the Chandra Carina Complex Project (CCCP), scheduled for publication in May 2011. All 16 CCCP Special Issue papers are available at http://cochise.astro.psu.edu/Carina_public. A high-quality preprint is also available at http://www.usm.uni-muenchen.de/people/preibisch/publications.htm
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