T -symmetrical tensor forms and Serre duality

P. Brückmann Martin-Luther-Universität Halle-Wittenberg Fachbereich Mathematik und Informatik

14F10 , 14F17 , 14M10 , 55N30 ,

Abstract. For any sheaf Ω^{T} of germs of T -symmetrical tensor forms on a projective manifold Y there exist a Young tableau T^{∗} and an integer p such that O_{Y} (p) ⊗ Ω^{T^{∗}} is dual to Ω^{T} . As an application a Lefschetz-type-theorem and some vanishing theorems for the cohomology groups of complete intersections are dualized.

By the Weyl construction it is well known that the sheaf Ω^{⊗r} = Ω^{1} ⊗ ⋯ ⊗ Ω^{1} of regular or holomorphic tensor forms on the manifold X splits up into the direct sum of subsheaves Ω^{T} , where T runs through the so called Young standard tableaux with r boxes. For each Young tableau T the finite dimensional vector space H^{0}(X, Ω^{T}) is a birational invariant of X. For special Young tableaux one gets the plurigenera P_{f}, the Hodge numbers h^{p,0} or the cotangential m-genus Q_{m} = dim H^{0}(X, S^{m}Ω^{1}) [3]. There are many examples showing the independence of the invariants H^{0}(X, Ω^{T}). Let e.g. Y and Z be complex 3-dimensional smooth complete intersections in ℙ_{ℂ}^{8} with multidegrees m_{Y} = (22, 16, 14, 5, 5) and m_{Z} = (20, 20, 11, 7, 4). Then the elementary-symmetrical polynomials σ_{1}, σ_{2}, σ_{3} and σ_{5} of these degrees are equal. Therefore Y and Z are diffeomorphic and they have the same Chern numbers, Hodge numbers and plurigenera. But chosing a Young tableau T^{⋆} with the row lengths l_{1} = 2, l_{2} = l_{3} = 1 one has dim H^{0}(Y, Ω_{Y}^{T^{⋆}}) < dim H^{0}(Z, Ω_{Z}^{T^{⋆}}) [4]. Taking for instance another Young tableau T′ with l_{1} = 2, l_{2} = 2, l_{3} = 1 we have inversely dim H^{0}(Y, Ω_{Y}^{T′}) > dim H^{0}(Z, Ω_{Z}^{T′}). Note that χ(Y, Ω_{Y}^{T}) = χ(Z, Ω_{Z}^{T}) for each Young Tableau T [6]. More examples are given in [7]. If X is a smooth n-dimensional manifold then Ω^{T} is locally free. In the case n < t = depth T holds Ω^{T} = 0, otherwise rank (Ω^{T}) = ∏_{1≤i<j≤n}(\frac{l_{i}−l_{j}}{j−i} + 1) with l_{i} = 0 if i > depth T and l_{i} ≥ l_{i+1} ≥ 0 ∀i [1]. If any two Young tableaux T and T′ correspond to the same partition r = l_{1} + l_{2} + ⋯ + l_{t} (l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{t} > 0) then Ω^{T} and Ω^{T′} are isomorphic as O_{X}-modules. Assume the ground field k to be algebraically closed with char k = 0 or char k > number of boxes of each Young tableau appearing in this paper.

1. The Serre duality

Theorem 1. Let X ⊆ ℙ^{N} be a smooth n-dimensional projective variety the canonical line bundle ω_{X} of which is isomorphic to O_{X}(n_{X}) (n_{X} ∈ ℤ). Then:

(1) If depth T = n then Ω^{T} ≅ Ω^{T′} ⊗ ω_{X} ≅ Ω^{T′}(n_{X}) , where the Young tableau T′ arises from T by erasing the first column of T.

(2) Assume depth T ≤ n. Let T^{∗} denote a Young tableau with the row lengths: l_{1}^{∗} = l_{1} − l_{n}, l_{2}^{∗} = l_{1} − l_{n−1}, …, l_{n−1}^{∗} = l_{1} − l_{2}, l_{n}^{∗} = 0 (l_{1} = l =lengthT). If the Young tableau T has the lengths of columns t_{1} ≥ t_{2} ≥ ⋯ ≥ t_{l} > 0 (t_{1} = depth T) then the column lengths of T^{∗} are t_{i}^{∗} = n − t_{l+1−i} . By Serre duality follows Hom (Ω^{T} (p), O_{X}) ≅ Ω^{T^{∗}}(−p − l · n_{X}) and therefore H^{q}(X, Ω^{T} (p)) ≈ H^{n−q}(X, Ω^{T^{∗}}(−p + (1 − l) · n_{X})) (p ∈ ℤ). In the case t_{1} = t_{2} = ⋯ = t_{l} = n one has to substitute the sheaf Ω^{T^{∗}} by O_{X}.

(3) H^{0}(X, Ω^{T} (λ · n_{X})) and H^{n}(X, Ω^{T} (μ · n_{X})) are birational invariants of X for all integers λ ≥ 0 and μ ≤ 1 − l, respectively.

Proof. Any section ϱ of Ω^{T} on an open subset U ⊆ X defines at each point x ∈ U a contravariant T -symmetrical tensor on the tangent space of x. For that reason we have to consider irreducible representations of GL_{n}(k). Let l_{1}, l_{2}, …, l_{n} be the row lengths of the Young tableau T and assume Ψ_{l_{1},l_{2},... ,l_{n}} denotes the corresponding irreducible representation (l_{i} = 0 if i > t_{1}). The first statement of the theorem follows from the equation Ψ_{l_{1}+1,l_{2}+1,... ,l_{n}+1} = Ψ_{l_{1},l_{2},... ,l_{n}} ⊗ det [1, §15.5.]. Let now λ_{1}, λ_{2}, …, λ_{n} be integers with λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n} and λ_{n} < 0. Then Ψ_{λ_{1},λ_{2},... ,λ_{n}} denotes the irreducible representation Ψ_{l_{1},l_{2},... ,l_{n}} ⊗ (det)^{λ_{n}} with l_{i} = λ_{i} − λ_{n} ∀i. The second statement follows from the fact that the dual of the representation Ψ_{λ_{1},λ_{2},... ,λ_{n}} is the representation Ψ_{−λ_{n},−λ_{n−1},... ,−λ_{1}} [1, §15.5.]. Finally Ω^{T} (λ · n_{X}) ≅ Ω^{T} ⊗ ω_{X}^{λ} and H^{n}(X, Ω^{T} (μ · n_{X})) ≈ H^{0}(X, Ω^{T^{∗}}((−μ + 1 − l) · n_{X})) imply (3). □

2. Complete intersections

Let X, Y ∈ ℙ^{N} be smooth projective varieties with d = dim Y < n = dim X such that Y is a complete intersection of X by hypersurfaces H_{1}, H_{2}, …, H_{n−d} of degree m_{1}, m_{2}, …, m_{n−d} respectively. Then holds ω_{Y} ≅ O_{Y} (∑_{i=1}^{n−d} m_{i}) ⊗ ω_{X} [2, II.8]. To apply the above theorem 1 assume ω_{X} ≅ O_{X}(n_{X}) with n_{X} ∈ ℤ. Then the dualisation of the Lefschetz type theorem 3 in [5] gives the following:

Theorem 2. Let σ denote the number of boxes inside the last n − d columns of the Young tableau T, i.e., σ = ∑_{i=1}^{n−d} t_{l+1−i} if n − d ≤ l and σ = r in case n − d ≥ l. Let T̃ be a Young tableau that arises from T by addition at the top of T a rectangle with n − d rows and l columns, i.e., T̃ has the lengths of columns t̃_{i} = t_{i} + n − d . Assume a, q ∈ ℤ to be fixed integers (q ≥ 0).

(1) If min{m_{1}, …, m_{n−d}} is high enough and q ≥ d · min{l, n − d} − σ then for each p ≥ a there exists a surjective homomorphism

H^{q}(Y, Ω_{Y}^{T} (p + (1 − l) · ∑_{i=1}^{n−d} m_{i})) ⟶ H^{q+n−d}(X, Ω_{T} ; X(p)).

(2) If min{m_{1}, …, m_{n−d}} is high enough and q > d · min{l, n − d} − σ then

H^{q}(Y, Ω_{Y}^{T} (p + (1 − l) · ∑_{i=1}^{n−d} m_{i})) ≅ H^{q+n−d}(X, Ω_{T} ; X(p)) ∀p ≥ a.

As an additional application some vanishing theorems are shown: Let X = ℙ^{N} and Y ⊆ ℙ^{N} be a smooth d-dimensional complete intersection as above. Then holds n_{Y} = ∑_{i=1}^{N−d} m_{i} − N − 1. Let λ be the number of boxes inside the N − d front columns of the Young tableau T , i.e., λ = ∑_{i=1}^{N−d} t_{i} with t_{i} = 0 if i > l.

The following vanishing theorems are known [5, theorem 4]:

(1) If t_{1} < q < d − λ then H^{q}(Y, Ω^{T} (p)) = 0 ∀p ∈ ℤ,

(2) If p < r − q and q < d − λ then H^{q}(Y, Ω^{T} (p)) = 0,

(3) If p < r + min{ l, m_{1} − 2, …, m_{N−d} − 2} and λ < d then H^{0}(Y, Ω^{T} (p)) = 0,

(4) If λ < d then H^{0}(Y, Ω^{T}) = 0. Now these vanishing theorems can be dualized:

Theorem 3. Let σ denote the number of boxes inside the last N − d columns of the above Young tableau T. Then holds

(1) If d · min{N − d, l} − σ < q < t_{l} then H^{q}(Y, Ω^{T} (p)) = 0 ∀p ∈ ℤ,

(2) If p > (1−l)(n_{Y} +d)+r−q and d·min{N−d, l}−σ < q then H^{q}(Y, Ω^{T} (p)) = 0.

(3) If p > (1 − l)(n_{Y} + d) + r − d − min{l^{∗}, m_{1} − 2, …, m_{N−d}} and d·min{N −d, l}−σ < d then H^{d}(Y, Ω^{T} (p)) = 0. Here l^{∗} denotes the number of columns of T with t_{i} < d, i.e., l^{∗} = ∑_{i=1}^{l} sgn (d − t_{i}).

(4) If l^{∗} > 0 and σ > d · min{l, N − d} − d then H^{d}(Y, Ω^{T} ((1 − l) · n_{Y})) = H^{d}(Y, Ω^{T} ⊗ ω_{Y}^{1−l}) = 0.

Note that the dualisation of theorem 7 in [5] makes it possible to compute under certain assumptions the dimensions dim H^{d}(Y, Ω^{T} (p)) by using the Riemann-Roch-theorem or the explicit formula given in [6].

Acknowledgements: I would like to thank Aleksandr G. Elashvili from the Academy of Science in Tiflis (Republic of Georgia) for the many helpful and stimulating discussions and the Deutsche Forschungsgemeinschaft for their generous support.

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