# Kotschick D.'s Chern numbers and diffeomorphism types of projective PDF By Kotschick D.

In 1954, Hirzebruch requested which linear mixtures of Chern numbers are topological invariants of delicate advanced projective kinds. We provide a whole resolution to this query in small dimensions, and in addition turn out partial effects with out regulations at the size.

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Additional resources for Chern numbers and diffeomorphism types of projective varieties

Example text

Hence V2 ⊗ Vp−1 λ, μ ∈ F× q , Rp−3 ⊕ (Vp−1 ⊗ det). 5. Assume p > 2 and let r and s be f -tuples such that 0 ≤ rj , sj ≤ p − 1 for all j. (i) For 0 ≤ i ≤ f − 1 and all integers a, b we have: i dimFp HomΓ (Vs ⊗ detb , V2Fr ⊗ Vr ⊗ deta ) ≤ 1. i (ii) We have HomΓ (Vs ⊗ detb , V2Fr ⊗ Vr ⊗ deta ) = 0 if and only if sj = rj for all j = i and one of the following holds: (a) si = ri + 2 and b ≡ a (q − 1) (b) si = ri and b ≡ a + pi (q − 1) (c) si = ri − 2 and b ≡ a + 2pi (q − 1) (d) f = 1, p = 3, s0 = r0 = p − 1 and b ≡ a (q − 1).

Let τ be a subrepresentation of V2p−2−r . Assume that there exists an integer k ∈ {0, · · · , f } such that, if Vr(ε) ⊗ dete(ε) occurs in cosocΓ τ , then |ε| = k. Then the graded pieces of the cosocle ﬁltration of τ are given by: τi ∼ = Vr(ε) ⊗ dete(ε) . ε∈Σr (τ ) |ε|=k−i Proof. 7 together with ε ≺ δ ⇒ |ε| = |δ| − 1. 10. Let λ, λ ∈ I(x0 , · · · , xf −1 ) (see Chapter 3). We say λ and λ are compatible if, whenever λi (xi ) ∈ {p − 2 − xi − ±1, xi ± 1} and λi (xi ) ∈ {p − 2 − xi − ±1, xi ± 1} for the same i, then the signs of the ±1 are the same in λi (xi ) and λi (xi ).

Let δ be the extension corresponding to δ via Ext1P (χ, χ) ∼ = Hom(P, Fp ). By inducing to G we obtain an exact sequence: × 0 / IndG P χ / IndG P δ / IndG P χ / 0. By evaluating functions at identity, we see that this sequence splits if and only if δ = 0. This proves (i) as π IndG P χ. Let v1 , v2 be the basis of the underlying vector 8. EXTENSIONS OF PRINCIPAL SERIES 47 space of δ such that for all g ∈ P , we have gv1 = χ(g)v1 + χ(g)δ(g)v2 and gv2 = χ(g)v2 . Denote by U (resp. U s ) in this proof (and only in this proof) the unipotent subgroup of P (resp.