The phases and properties of matter under global rotation have attracted much interest recently. In this paper we investigate the pairing phenomena in a system of fermions under the presence of rotation. We find that there is a generic suppression effect on pairing states with zero angular momentum. We demonstrate this effect with the chiral condensation and the color superconductivity in hot dense QCD matter as explicit examples. In the case of chiral condensation, a new phase diagram in the temperature-rotation parameter space is found, with a nontrivial critical point.Introduction.-The phases and properties of matter can become highly nontrivial under rotation, and have attracted a lot of interest recently. Such studies bear particular relevance for the strongly interacting matter of Quantum Chromodynamics (QCD). For example, astrophysical objects like neutron stars, made of dense QCD matter, can be rapidly spinning [1,2]. In relativistic heavy ion collision experiments, the typical collision events are off-central and the created QCD matter will carry a nonzero angular momentum on the order of 10 4 ∼ 10 5 with local angular velocity in the range 0.01 ∼ 0.1GeV [3][4][5][6][7][8]. There has also been impressive progress to study the rotating QCD matter using lattice gauge theory simulations [9].In rotating matter, many interesting transport phenomena could occur. For example, fluid rotation can induce anomalous transport processes in a system of chiral fermions, such as the chiral vortical effect [10][11][12] and chiral vortical wave [13]. These can lead to measurable experimental signals (see e.g. reviews in [14,15]). In such anomalous transport, it is found that the fluid rotation plays a very analogous role to a magnetic field. Indeed there appears to be interesting similarity between the chiral vortical effect and the chiral magnetic effect [10,16], as well as between the chiral vortical wave and the chiral magnetic wave [17,18].Apart from transport properties, it is of significant interest to explore the effects of rotation on the phase transitions of matter in both relativistic and non-relativistic cases. It is well known that a magnetic field can bring interesting effects on the thermodynamics and phase diagram of e.g. QCD matter [19][20][21][22][23][24], such as the magnetic catalysis and inverse catalysis (see reviews in [25,26]) on the chiral condensation. Given the close analogy between rotation and magnetic field, it is tempting to ask how the rotation could influence phase transitions. In this paper, we investigate the pairing phenomena in a system of fermions under rotation. We find a generic suppression effect on pairing states with zero angular momentum,