Characterizations of Hermitian varieties by intersection numbers

In this paper, we give characterizations of the classical generalized quadrangles H(3, q ²) and H(4, q ²), embedded in PG(3, q ²) and PG(4, q ²), respectively. The intersection numbers with lines and planes characterize H(3, q ²), and H(4, q ²) is characterized by its intersection numbers with planes and solids. This result is then extended to characterize all Hermitian varieties in dimension at least 4 by their intersection numbers with planes and solids.