Modular invariance of vertex operator algebras satisfying C₂-cofiniteness

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We show that C₂-cofiniteness is enough to prove a modular invariance property of vertex operator algebras without assuming the semisimplicity of Zhu algebra. For example, if a VOA V=⊕ₘ₌₀∞Vₘ is C₂-cofinite, then the space spanned by generalized characters of V-modules is invariant under the action of SL₂(). In this case, the central charge and conformal weights are all rational numbers. Namely, a VOA satisfying C₂-cofiniteness is a rational conformal field theory in a sense. We also show that C₂-cofiniteness is equivalent to the condition that every weak module is an -graded weak module which is a direct sum of generalized eigenspaces of L(0).