By Luther Pfahler Eisenhart
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Additional info for An introduction to differential geometry with use of tensor calculus
29 shows that the space (Λp,q Tx X)∨prim is non-trivial. Hence, with m = min(p, q), there are at least m + 1 irreducible components in (Λp,q Tx X)∨ = Lr (Λp−r,q−r Tx X ∨ )prim . 0≤r≤m It suffices therefore to see that there are at most m + 1 irreducible U(n)components in the latter U(n)-module. It is well known that the irreducible U(n)-modules are in bijection with the eigenvectors of the action of the diagonal matrices. But these all act differently on the m + 1 covectors Lr (dzn−p+r+1 ∧ · · · ∧ dzn ∧ d¯ z1 ∧ · · · d¯ zq−r ).
5 that the cohomology of varieties with quotient singularities also admits a pure rational Hodge structure. 1 Basic Definitions We place the definition of a weight k Hodge structure (Def. 12) in a wider context. Let V be a finite dimensional real vector space and let VC = V ⊗ C be its complexification. 1. ) 34 2 Pure Hodge Structures The numbers hp,q (V ) := dim V p,q are Hodge numbers of the Hodge structure. The polynomial hp,q (V )up v q Phn (V ) = (II–1) p,q∈Z its associated Hodge number polynomial.
The assertions bout Lr then follow. −r−2s e) Let x ∈ V r and suppose that Λs+1 x = 0. )L y belongs to the kernel of Λs |V −r . Continuing the argument with x we inductively find an expression x = x0 + Lx1 + Lx2 + · · · + Ls xs , r−2j xj ∈ Vprim . This expression is unique: if x = 0 and j is the largest integer for which Lj xj = 0, then, by d) j ≥ r and if we apply Lj−r to both sides, d) also implies 0 = L2j−r xj and hence xj = 0 contrary to our assumption. This shows that we have a primitive direct sum decomposition of V µ .
An introduction to differential geometry with use of tensor calculus by Luther Pfahler Eisenhart