  For a vertex operator algebra $V$ with conformal vector $\omega$, we consider
a class of vertex operator subalgebras and their conformal vectors. They are
called semi-conformal vertex operator subalgebras and semi-conformal vectors of
$(V,\omega)$, respectively, and were used to study duality theory of vertex
operator algebras via coset constructions. Using these objects attached to
$(V,\omega)$, we shall understand the structure of the vertex operator algebra
$(V,\omega)$. At first, we define the set $\on{Sc}(V,\omega)$ of semi-conformal
vectors of $V$, then we prove that $\on{Sc}(V,\omega)$ is an affine algebraic
variety with a partial ordering and an involution map. Corresponding to each
semi-conformal vector, there is a unique maximal semi-conformal vertex operator
subalgebra containing it. The properties of these subalgebras are invariants of
vertex operator algebras. As an example, we describe the corresponding
varieties of semi-conformal vectors for Heisenberg vertex operator algebras. As
an application, we give two characterizations of Heisenberg vertex operator
algebras using the properties of these varieties.
