This memoir is devoted to the theory of vector-valued modular forms for orthogonal groups of signature \((2, n)\). The author's purpose is multi-layered: (1) to lay a foundation of the theory of vector-valued orthogonal modular forms; (2) to develop some aspects of the theory in more depth, such as geometry of the Siegel operators, filtrations associated to 1-dimensional cusps, decomposition of vector-valued Jacobi forms, square integrability etc; and (3) as applications to derive several types of vanishing theorems for vector-valued modular forms of small weight.

The author's vanishing theorems imply in particular vanishing of holomorphic tensors of degree less than \(n/2 − 1\) on orthogonal modular varieties, which is optimal as a general bound. The fundamental ingredients of the theory are the two Hodge bundles. The first is the Hodge line bundle which already appears in the theory of scalar-valued modular forms. The second Hodge bundle emerges in the vector-valued theory and plays a central role. It corresponds to the non-abelian part \(O(n, \mathbb{R})\) of the maximal compact subgroup of \(O(2, n)\). The main focus of this monograph is centered around the properties and the role of the second Hodge bundle in the theory of vector-valued orthogonal modular forms.