[en] A description of light heavy-ion fusion, taking into account both entrace-channel characteristics and compound-nucleus properties, is derived within a unified theory of nuclear reactions. The dependence of the imaginary fusion potential on the level density of the compound nucleus is revealed. The ¹²C+¹²C, ¹²C+¹⁴N, ¹⁰B+¹⁶O and ¹⁶O+¹⁶O fusion cross sections are calculated for E_cm≤120 MeV and compared with experimental data. The excitation energy dependence of the level-density parameter of ²⁴Mg, ²⁶Al and ³²S is inferred below 5 MeV/A. A realistic nuclear level-density model, describing the experimental level-density parameters of highly excited nuclei, is shown to be consistent with both the global features and details of the fusion cross section. ¹²C+¹²C and ¹⁶O+¹⁶O fusion cross section oscillations are predicted at large excitation energies, reflecting the structure of the level density of the highly excited light compound nuclei. Differences of the ¹²C+¹⁴N and ¹⁰B+¹⁶O fusion reaction mechanisms are discussed in terms of specific entrance-channel characteristics. (orig.)