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.
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F.B. partially supported by the NSF grant DMS-0404715. B.O. Partially supported by the NSF grant DMS-0707097.
Received: September 2006, Revision: May 2007, Accepted: June 2007
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Bogomolov, F., De Oliveira, B. Symmetric Tensors And Geometry of \({\mathbb{P}} ^N\) Subvarieties. GAFA Geom. funct. anal. 18, 637–656 (2008). https://doi.org/10.1007/s00039-008-0666-7
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DOI: https://doi.org/10.1007/s00039-008-0666-7