Kudla's Modularity Conjecture and Formal Fourier-Jacobi Series
  We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogues of Fourier-Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla's conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties. 
Jan Hendrik Bruinier and Martin Westerholt-Raum
math.NT math.AG