Abstract
This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials
$$H^{0}(X, S^{m}\Omega^{1}_{X} \bigotimes \mathcal {O}_{X}(k))$$
on subvarieties
$$X \subset {\mathbb{P}}_{ N}$$
, with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N − 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera,
$$Q _{\alpha, m}(X_{ t}) = {\rm dim} H^{0}(X, S^{m}(\omega^{1}_{X_t} \bigotimes \alpha K_{X_t} ))$$
along smooth families of projective varieties Xt are not invariant even for α arbitrarily large.
