This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials H 0(X, SmΩX1 ⊗ script OX(k)) on subvarieties X ⊂ ℙN, 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, Qα,m(Xt) = dim Η0(X, Sm(ΩXt1 ⊗ αKXt)) along smooth families of projective varieties Xt are not invariant even for α arbitrarily large.
- Symmetric differentials
- Trisecant variety
- Vanishing and nonvanishing theorems
ASJC Scopus subject areas
- Geometry and Topology