Tensor Differential Forms and Some Birational Invariants of Projective Manifolds

Abstract

Symmetry properties of tensors play an important role in physics. They correspond to the irreducible representations of the symmetric group, which can be described by young tableaux T. The global T-symmetrical tensor differential forms on the projective manifold Y define a birational invariant of Y. In the case of prime characteristic char(K)= p > 0 the pullback of the Frobenius provides an apportunity to define further discrete birational invariants of algebraic manifolds using the p^s-th powers (df)^p¨' instead of the differentials df. Using Sernesis result on infinitesimal deformations an explicit formula for the moduli space dimension of complete intersections is given. As an application among others a conjecture of Libgober and Wood will be confirmed concerning the existence of diffeomorphic three-dimensional complete interactions which lie in different dimensional components of the moduli space. Finally for arbitrary locally free sheaves F o Y the Chern classes of the T-power F^T are calculated as polynomials in Chern classes of F.