  We provide a construction of the multiplicative Borcherds lift for unitary
groups U(1,m), which takes weakly holomorphic elliptic modular forms and lifts
them to meromorphic automorphic forms having infinite product expansions and
taking their zeros and poles along Heegner divisors. The result is obtained
through the transfer of Borcherds' theory to unitary groups via a suitable
embedding of U(1,m) into O(2,2m).
  Further, we examine the values taken by the Borcherds products around the
cusps of the symmetric domain of the unitary group. Finally, as an application
of the lifting, we derive a modularity result for generating series with
Heegner divisors as coefficients, along the lines of Borcherds' generalization
of the Gross-Zagier-Kohnen theorem.
