The Hilbert polynomial of the sheaf Ω^{T} of germs of T -symmetrical tensor differential forms on complete intersections

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

14F10 , 14M10 , 14M17

Abstract. For each Young tableau T the global T -symmetrical tensor forms on the manifold Y define a birational invariant. To calculat these invariants above all exact sequences of sheaves are required. In the case of smooth complete intersections Y such sequences are constructed as ”T -powers” of a familiar exact sequence. As an application we calculate the Hilbert polynomials χ(Y, Ω^{T} (t)) . Moreover, in the complex case explicit formulas for the dimensions dim H^{q}(ℙ_{ℂ}^{N}, Ω^{T} (t)) are given as a generalization of the well known Bott formulas concerning the classical alternating differential forms.

For each T the dimension dim H^{0}(Y, Ω^{T}) is a birational invariant of the manifold Y. In the case of some special Young tableaux T one gets for instance the plurigenera, the Hodge numbers h^{p,0} or the cotangential m-genus dim H^{0}(Y, S^{m}Ω^{1}) [5]. In general the invariants dim H^{0}(Y, Ω^{T}) are independent of the plurigenera. Unfortunately calculation of these invariants and of the higher cohomology groups nowadays is rather hard, except in some special cases [6], and requires more sophisticated instruments. Above all, exact sequences of sheaves are required. The main goal of this paper is to construct in the case of smooth complete intersections Y such sequences as ”T -powers” of a familiar exact sequence. An explicit formula for the Hilbert polynomial χ(Y, Ω^{T} (t)) is given as an application. The ground field k is assumed to be algebraically closed with char(k) = 0 or char(k) > number of boxes of T. Moreover, in the complex case the dimensions dim H^{q}(ℙ^{N}, Ω^{T} (t)) are calculated using the Bott theorem on homogeneous vector bundles [1], [4].

1. Young diagrams and Young tableaux The irreducible representations of the symmetric group S_{r} correspond to the conjugacy classes of S_{r}, i.e., to the partitions (l) : r = l_{1} + l_{2} + ⋯ + l_{d} with l_{i} ∈ ℤ and l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{d} > 0. The partition (l) can be described by a Young diagram with r boxes and with the row lengths l_{1}, l_{2}, …, l_{d} . The lengths of its columns are d_{j} = #{i ∈ ℤ : l_{i} ≥ j} (j = 1, 2, …, l_{1} ; l = l_{1} = length T ; d = d_{1} = depth T ; ∑ d_{j} = ∑ l_{i} = r) . Set l_{i} = 0 if i > d and d_{j} = 0 if j > l. The box inside the i-th row and the j-th column of the Young diagram has its own ”hook length” h_{i,j} = l_{i} − j + d_{j} − i + 1 and the degree of the irreducible representation is

ν_{(l)} = \frac{r!}{∏ h_{i,j}} = \frac{r!}{d!} · ∏_{i=1}^{d} \frac{i!}{(l_{i}+d−i)!} · ∏_{1≤i<j≤d}(\frac{l_{i}−l_{j}}{j−i} + 1) = r! · det((\frac{1}{Γ(l_{i}+1−i+j)}))_{i,j=1,2,... ,d} [2].

A Young tableau T to a given Young diagram with r boxes is an order relation of these boxes or simply a numbering of these boxes by the integers 1, 2, …, r . For a given Young tableau T an idempotent e_{T} in the group algebra k · S_{r} is introduced :

(1) e_{T} = \frac{ν_{(l)}}{r!} · ( ∑_{q∈Q_{T}} sgn(q) · q) · ( ∑_{p∈P_{T}} p)

with the subgroups P_{T} = {p ∈ S_{r} : p preserves each row of T } and Q_{T} = {q ∈ S_{r} : q preserves each column of T }. A Young tableau T is called a standard tableau if the sequence of box numbers inside any row or inside any column of T is monotonically increasing. The number of standard tableaux to a given Young diagram is equal to the degree ν_{(l)} of the corresponding irreducible representation of S_{r}. Let D(r) be the set of all standard tableaux with r boxes. Then the identity 1 ∈ k · S_{r} decomposes to the sum

(2) 1 = ∑_{T ∈D(r)} e_{T}

and these idempotents e_{T} are orthogonal to each other. Let Ω^{⊗r} = (Ω^{1})^{⊗r} be the sheaf of germs of regular tensor differential forms on the variety Y (in the case k = ℂ we can also take holomorphic forms instead). Then the symmetric group S_{r} and therefore the group algebra k · S_{r} act on Ω^{⊗r} by permutations of the spots inside the tensor product: p(a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{r}) = a_{p^{−1}(1)} ⊗ a_{p^{−1}(2)} ⊗ ⋯ ⊗ a_{p^{−1}(r)} ∀p ∈ S_{r}. From (2) we get

(3) Ω^{⊗r} = ⨁_{T ∈D(r)} Ω^{T} with Ω^{T} = e_{T} (Ω^{⊗r}) .

If the Young tableaux T and T̃ correspond to the same Young diagram ( i.e. to the same partition (l) ) then Ω^{T} and Ω_{T} are isomorphic as O_{Y} -moduls. If Y is a smooth n-dimensional variety then Ω^{T} is locally free. In case n < d = depth T one has Ω^{T} = 0 , otherwise rank (Ω^{T}) = ∏_{1≤i<j≤n}(\frac{l_{i}−l_{j}}{j−i} + 1) (l_{i} = 0 if i > d). Let Y ⊆ ℙ^{N} be a smooth n-dimensional projective variety whose canonical line bundle ω_{Y} is isomorphic to O_{Y} (n_{Y}) with n_{Y} ∈ ℤ. Then ( cf. [9]) :

(4) Ω^{T} ≅ Ω^{T′} ⊗ ω_{Y} ≅ Ω^{T′}(n_{Y}) if depth T = dim Y , length T > 1, where the Young tableau T′ arises from T by erasing the first column of T. Assume the Young tableau T has the lengths of columns d_{1}, d_{2}, …, d_{l} (> 0) with d_{1} = depth T ≤ dim Y. Let T^{∗} denote a Young tableau with the following lengths of columns: d_{j}^{∗} = n − d_{l+1−j} ∀j ∈ {1, 2, …, l} . Then obviously the equation (T^{∗})^{∗} = T holds if and only if depth T < n . By Serre duality [9] we have

(5) Hom(Ω^{T} (t), O_{Y}) ≅ Ω^{T^{∗}}(−t − l · n_{Y})

(6) H^{q}(Y, Ω^{T} (t)) ≈ H^{n−q}(Y, Ω^{T^{∗}}(−t + (1 − l)n_{Y})) .

2. The complex projective space ℙ_{ℂ}^{N} Now let k = ℂ and let T be a Young tableau with r boxes, the row lengths l_{i} , the column lengths d_{j}. Additionally we set μ_{i} = N + l_{i} − i , t_{i} = r + l_{i} − i (then

μ_{i} > μ_{i+1} , t_{i} > t_{i+1} ∀i ≥ 1), Δ(a_{1}, a_{2}, …, a_{m}) = ∏_{1≤i<j≤m}(a_{i} − a_{j}) (Vandermonde).

Theorem 1. Assume d = depth T ≤ N. Then the Hilbert polynomial χ(ℙ^{N}, Ω^{T} (t)) has the in pairs different zeros t_{1}, t_{2}, …, t_{N}. The fibration Ω^{T} (t) is acyclic if and only if t is one of the zeros t_{i}. For fixed N, T, t at most one of the cohomology groups H^{q}(ℙ^{N}, Ω^{T} (t)) (q = 0, 1, …) may be nontrivial (cf. [1] , [4]). Furthermore:

(7) χ(ℙ^{N}, Ω^{T} (t)) = \frac{Δ(t,t_{1},t_{2},... ,t_{N})}{Δ(N,N−1,... ,0)} = ( ∏_{i=1}^{N} i!)^{−1} · ∏_{1≤i<j≤N}(t_{i} − t_{j}) · ∏_{i=1}^{N}(t − t_{i})

(8) H^{q}(ℙ^{N}, Ω^{T} (t)) = 0 ∀t ∈ ℤ iff q ≠ 0, N, d_{1}, d_{2}, …, d_{l} (l = length T)

(9) H^{0}(ℙ^{N}, Ω^{T} (t)) = 0 iff t ≤ t_{1} dim H^{0}(ℙ^{N}, Ω^{T} (t)) = χ(ℙ^{N}, Ω^{T} (t)) if t ≥ t_{1}

(10) H^{N}(ℙ^{N}, Ω^{T} (t)) = 0 iff t ≥ t_{N} dim H^{N}(ℙ^{N}, Ω^{T} (t)) = (−1)^{N} · χ(ℙ^{N}, Ω^{T} (t)) if t ≤ t_{N}

(11) For each i ∈ {1, 2, …, l} with 0 < d_{i} < N holds t_{d_{i}+1} ≤ t_{d_{i}} − 2 and H^{d_{i}}(ℙ^{N}, Ω^{T} (t)) = 0 iff t ≤ t_{d_{i}+1} or t ≥ t_{d_{i}} dim H^{d_{i}}(ℙ^{N}, Ω^{T} (t)) = (−1)^{d_{i}} · χ(ℙ^{N}, Ω^{T} (t)) if t_{d_{i}+1} ≤ t ≤ t_{d_{i}} .

Remarks :

(12) In the special case d_{1} = r ≤ N , 0 = d_{2} = d_{3} = … one has Ω^{T} = Ω^{r} = ⋀_{r} Ω^{1} and the well known facts follow : χ(ℙ^{N}, Ω^{r}(t)) = \binom{t−1}{r} · \binom{t + N−r}{N−r} and dim H^{q}(ℙ^{N}, Ω^{r}(t)) = δ_{q,r} · δ_{t,0} for each q ∈ {1, 2, …, N − 1} .

(13) In the case l_{1} = r , 0 = l_{2} = l_{3} = … we have Ω^{T} = S^{r}Ω^{1} .

Therefore χ(ℙ^{N}, (S^{r}Ω^{1})(t)) = \frac{1}{N} · \binom{r + N−1}{N−1} · \binom{t + N−r}{N−1} · (t − 2r + 1) and dim H^{q}(ℙ^{N}, (S^{r}Ω^{1})(t)) = 0 ∀t ∀q with 1 < q < N (cf. (8)). The statements of the theorem remain true even in the case Ω^{T} = O_{ℙ^{N}}, i.e., l_{i} = 0 ∀i . Of course, that means χ(ℙ^{N}, O_{ℙ^{N}}(t)) = \binom{t + N}{N} .

(14) By Serre duality (5), (6) we get Hom(Ω^{T} (t), O_{ℙ^{N}}) ≅ Ω^{T^{∗}}(−t + l · (N + 1)), H^{q}(ℙ^{N}, Ω^{T} (t)) ≈ H^{N−q}(ℙ^{N}, Ω^{T^{∗}}(−t + (l − 1) · (N + 1))) and χ(ℙ^{N}, Ω^{T} (t)) = (−1)^{N} ·χ(ℙ^{N}, Ω^{T^{∗}}(−t+(l −1)·(N +1))), where T^{∗} denotes a Young tableau with the column lengths d_{j}^{∗} = N − d_{l+1−j} ∀j ∈ {1, 2, …, l}.

(15) The highest coefficient of the Hilbert polynomial χ(ℙ^{N}, Ω^{T} (t)) is (cf. [2]) :

( ∏_{i=1}^{N} i!)^{−1} · ∏_{1≤i<j≤N}(t_{i} − t_{j}) = \frac{Δ(t_{1},t_{2},... ,t_{N})}{Δ(N,N−1,... ,0)} = \frac{1}{N!} · ∏_{1≤i<j≤N}(\frac{l_{i}−l_{j}}{j−i} + 1) =

= \frac{1}{N!}· rank (Ω^{T}) = \frac{1}{N!} · ∏_{1≤i<j≤d}(\frac{l_{i}−l_{j}}{j−i} + 1) · ∏_{i=1}^{d} \begin{matrix}( & {N+l_{i}−i \atop l_{i}} & ) \\ ( & {d+l_{i}−i \atop l_{i}} & ) \end{matrix} = _{\frac{1}{∏_{i,j} h_{i,j}}} · ∏_{i=1}^{N} \frac{μ_{i}!}{i!}

Proof. ℙ_{ℂ}^{N} = 𝔊/𝔘 with 𝔊 = SL_{N}(ℂ) is a homogeneous space. Let λ_{1}, …, λ_{N+1} be the diagonal elements of the matrices of the Cartan subalgebra (∑ λ_{i} = 0). The irreducible representation Ad_{𝔊/𝔘} has the weights ω_{j} = λ_{j+1} − λ_{1} (j = 1, 2, …, N) and the highest weight ω_{1} = λ_{2} − λ_{1}. Let S_{(l)} = S_{(l_{1},l_{2},... ,l_{N})} be the Schur functor corresponding to the partition r = l_{1} + l_{2} + ⋯ + l_{N} (l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{N} ≥ 0). Then the representation S_{(l)}(Ad_{𝔊/𝔘}) is irreducible too. If φ denotes the 1-dimensional representation corresponding to the line bundle O_{ℙ^{N}}(1) then the representation S_{(l)}(Ad_{𝔊/𝔘}) ⊗ φ^{t} has the highest weight Λ = l_{1} · ω_{1} + ⋯ + l_{N} · ω_{N} + t · λ_{1} = = (t − l_{1} − ⋯ − l_{N}) · λ_{1} + l_{1} · λ_{2} + ⋯ + l_{N} · λ_{N+1} = (t − r) · λ_{1} + l_{1} · λ_{2} + ⋯ + l_{N} · λ_{N+1}. We set ϱ = N · λ_{1} + (N − 1) · λ_{2} + … + 2 · λ_{N−1} + λ_{N} . Then

(16) Λ + ϱ = (t − r + N) · λ_{1} + μ_{1} · λ_{2} + ⋯ + μ_{N} · λ_{N+1} , μ_{1} > μ_{2} > ⋯ > μ_{N}. Therefore one has the singular case iff ∃i ∈ {1, 2, …, N} with t − r + N = μ_{i} , that means, the fibration Ω^{T} (t) is acyclic iff t = r − N + μ_{i} = t_{i} . Thus the Hilbert polynomial χ(ℙ^{N}, Ω^{T} (t)) has the in pairs different zeros t = t_{i} . Under the assumption t − r + N > μ_{1} (i.e. t > t_{1}) the vector space H^{0}(ℙ^{N}, Ω^{T} (t)) is the corresponding irreducible representation module and by Weyl dimension formula one has dim H^{0}(ℙ^{N}, Ω^{T} (t)) = \frac{Δ(t−r+N,μ_{1},μ_{2},... ,μ_{N})}{Δ(N,N−1,... ,0)} = \frac{Δ(t,t_{1},t_{2},... ,t_{N})}{Δ(N,N−1,... ,0)}. From this equation we get (7) since χ(ℙ^{N}, Ω^{T} (t)) is a polynomial in t of degree N with χ(ℙ^{N}, Ω^{T} (t)) = = dim H^{0}(ℙ^{N}, Ω^{T} (t)) for t ≫ 0 . In order to prove (8)-(11) one has to check which Weyl chamber Λ + ϱ belongs to. First of all from (16) we have H^{0}(ℙ^{N}, Ω^{T} (t)) ≠ 0 iff μ_{1} < t − r + N, (i.e. iff t_{1} < t), H^{1}(ℙ^{N}, Ω^{T} (t)) ≠ 0 iff μ_{2} < t − r + N < μ_{1} (i.e. iff t_{2} < t < t_{1}), H^{2}(ℙ^{N}, Ω^{T} (t)) ≠ 0 iff t_{3} < t < t_{2} … and finally H^{N}(ℙ^{N}, Ω^{T} (t)) ≠ 0 iff t < t_{N}. If 0 < q < N and l_{q} = l_{q+1} then t_{q} = t_{q+1} + 1 and thus H^{q}(ℙ^{N}, Ω^{T} (t)) = 0 ∀t since the open interval (t_{q+1}, t_{q}) does not contain any integer. Expressing these results using the column lengths d_{1}, d_{2}, … of T one gets the statements (8)-(11). □

3. The T -power of an algebraic complex Let r be a positive integer and let R be a commutative ring with identity such that r! is a unit in R. Assume K : K_{0} ⟶^{d} K_{1} ⟶^{d} K_{2} ⟶^{d} ⋯ (d · d = 0) to be an algebraic complex of R-modules. Then the r-th tensor power P = K^{⊗r} of the complex K is defined as follows: P = K^{⊗r} : P_{0} ⟶^{δ} P_{1} ⟶^{δ} P_{2} ⟶^{δ} ⋯

with P_{s} = ⨁_{s_{1}+···+s_{r}=s} K_{s_{1}} ⊗ K_{s_{2}} ⊗ ⋯ ⊗ K_{s_{r}} and with δ(a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{r}) =

= ∑_{i=1}^{r}(−1)^{s_{1}+s_{2}+···+s_{i−1}} · a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{i−1} ⊗ d(a_{i}) ⊗ a_{i+1} ⊗ ⋯ ⊗ a_{r} if a_{j} ∈ K_{s_{j}} ∀j.

The symmetric group S_{r} acts on the tensor power by permutation of the spots. Now another action on P = K^{⊗r} is needed with the additional property p · δ = δ · p ∀p ∈ S_{r}. This action together with a fitting sign can be defined as follows :

(17) σ(p ; s_{1}, s_{2}, …, s_{r}) := ∑_{{i<j \atop p(i)>p(j)}} s_{i} · s_{j}

(18) p(a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{r}) := (−1)^{σ(p;s_{1},s_{2},... ,s_{r})} · a_{p^{−1}(1)} ⊗ a_{p^{−1}(2)} ⊗ ⋯ ⊗ a_{p^{−1}(r)} if a_{j} ∈ K_{s_{j}} ∀j ∈ {1, 2, …, r}.

From (2) one has P_{s} = ⨁_{T ∈D(r)} K_{s}^{T} , K^{⊗r} = ⨁_{T ∈D(r)} K^{T} , H^{∗}(K^{⊗r}) = ⨁_{T ∈D(r)} H^{∗}(K^{T})

with K_{s}^{T} = e_{T} (P_{s}) and K^{T} = e_{T} (K^{⊗r}) : K_{0}^{T} ⟶^{δ} K_{1}^{T} ⟶^{δ} K_{2}^{T} ⟶^{δ} · · · .

The complex K^{T} is called the T -power of the complex K. If the Young tableaux T and T̃ correspond to the same Young diagram then we have K^{T} ≅ K_{T} . Of course, the T -power of a complex K : ⋯ ⟶^{d} K_{2} ⟶^{d} K_{1} ⟶^{d} K_{0} (cp. (26)) can be defined in similar way . In this paper these constructions are used only under the additional assumption K_{2} = K_{3} = ⋯ = 0. That means if the product a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{r} ∈ P_{s} is different from zero then a_{i} ∈ K_{0} or a_{i} ∈ K_{1} ∀i. Moreover, there are exactly s spots i_{1}, i_{2}, …, i_{s} (1 ≤ i_{1} < i_{2} < ⋯ < i_{s} ≤ r) with a_{i_{1}}, a_{i_{2}}, …, a_{i_{s}} ∈ K_{1}. Finally in this case σ is equal to the number of inversions inside the sequence (p(i_{1}), p(i_{2}), …, p(i_{s})) (cf. (17)).

4. A free resolution of the sheaf Ω^{T} on ℙ^{N} Let T be a fixed Young tableau with r boxes. For ground fields k with char(k) = 0 or char(k) > r the Hilbert polynomial χ(ℙ^{N}, Ω^{T} (t)) will be computed using the above techniques.

Theorem 2. Let T be a Young tableau with r boxes. Assume d = depth T ≤ N and t_{i} = r + l_{i} − i ∀i (l_{i} = 0 if i > d). Then the following sequence is exact:

(19) 0 ⟶ Ω^{T} ⟶ ⨁_{b_{0}} O(−r) ⟶ ⨁_{b_{1}} O(1 − r) ⟶ … ⟶ ⨁_{b_{d}} O(d − r) ⟶ 0

with the integers

b_{s} = ( ∏_{i=1}^{N} i!)^{−1} · ∑_{1≤i_{1}<...<i_{s}≤d} Δ(t_{1}, t_{2}, …, t_{i_{1}} − 1, …, t_{i_{s}} − 1, …, t_{N}, t_{N+1}).

Corollary 1. For ground fields k with char(k) = 0 or char(k) > r we have

(20) χ(ℙ^{N}, Ω^{T} (t)) = ( ∏_{i=1}^{N} i!)^{−1} · ∏_{1≤i<j≤N}(t_{i} − t_{j}) · ∏_{i=1}^{N}(t − t_{i}) .

Proof. The exact sequence (19) and χ(ℙ^{N}, O(t)) = \binom{t + N}{N} are independent of char(k) . Therefore the corollary follows from the special case k = ℂ (cf. (7)). Of course it is possible to prove (20) directly using (19). □ From the Euler sequence one has the short exact sequence 0 ⟶ Ω^{1} ⟶ ⨁_{N+1} O(−1) ⟶ O ⟶ 0. The r-th tensor power of the complex

K : K_{0} ⟶ K_{1} ⟶ 0 with K_{0} = ⨁_{N+1} O(−1) , K_{1} = O , K_{2} = K_{3} = ⋯ = 0

gives the exact sequence

0 ⟶ Ω^{⊗r} ⟶ ⨁_{α_{0}} O(−r) ⟶ ⨁_{α_{1}} O(1 − r) ⟶ … ⟶ ⨁_{α_{r−1}} O(−1) ⟶ O ⟶ 0 ,

with α_{s} = \binom{r}{s} · (N + 1)^{r−s}. The T -power of the complex K is the exact sequence

0 ⟶ Ω^{T} ⟶ ⨁_{b_{0}} O(−r) ⟶ ⨁_{b_{1}} O(1 − r) ⟶ … ⟶ ⨁_{b_{r−1}} O(−1) ⟶ ⨁_{b_{r}} O ⟶ 0

with suitable numbers b_{s} and with ⨁_{b_{s}} O(s − r) = e_{T} (⨁_{α_{s}} O(s − r)). Because of e_{T} ∈ k · S_{r} the coefficients of e_{T} are constant. Therefore it is possible to compute the ranks b_{s} simply as dimensions of suitable k-vector spaces: Let V_{N} ⊂ V_{N+1} be k-vector spaces of dimension N resp. N +1 and let U = V_{N+1}/V_{N} be the factor space. The r-th tensor power of the complex K̃ : V_{N+1} ⟶ U ⟶ 0 is the exact sequence 0 ⟶ V_{N}^{⊗r} ⟶ V_{N+1}^{⊗r} ⟶ W_{1} ⟶ … ⟶ W_{r} ⟶ 0 with the vector spaces W_{s} = \begin{matrix}⨁ \\ ( & {r \atop s} & ) \end{matrix} V_{N+1}^{⊗(r−s)} ⊗ U^{⊗s} , dim W_{s} = α_{s}. The T -power of K̃ gives the

exact sequence 0 ⟶ V_{N}^{T} ⟶ V_{N+1}^{T} ⟶ W_{1}^{T} ⟶ W_{2}^{T} ⟶ … ⟶ W_{r}^{T} ⟶ 0 with W_{s}^{T} = e_{T} (W_{s}) and the required dimensions are dim W_{s}^{T} = b_{s}. V_{N+1}^{T} and V_{N}^{T} are the ordinary spaces of T -symmetrical tensors and by the Weyl dimension formula we have b_{0} = dim V_{N+1}^{T} = ( ∏_{i=1}^{N} i!)^{−1} · Δ(t_{1}, t_{2}, …, t_{N}, t_{N+1}) and

dim V_{N}^{T} = ( ∏_{i=1}^{N−1} i!)^{−1} · Δ(t_{1}, t_{2}, …, t_{N}) (d = depth T ≤ N) .

Theorem 2 follows from:

Lemma 1. For each s ∈ {0, 1, …, d} one has

(21) dim W_{s}^{T} = _{\frac{1}{∏_{i=1}^{N} i!}} · ∑_{1≤i_{1}<...<i_{s}≤d} Δ(t_{1}, t_{2}, …, t_{i_{1}} − 1, …, t_{i_{s}} − 1, …, t_{N}, t_{N+1}).

(22) If s > d = depth T then W_{s}^{T} = 0 and therefore b_{s} = 0.

Proof. Let a_{0}, a_{1}, …, a_{N} be a basis of V_{N+1}. The boxes of T are numbered by integers 1, 2, …, r and each product a_{i_{1}} ⊗ a_{i_{2}} ⊗ … ⊗ a_{i_{r}} (0 ≤ i_{ν} ≤ N) corresponds to a so called Young scheme, which arises from T by inscription of the index i_{ν} into the ν-th box of T . This Young scheme is called a standard scheme if the entries inside any row of T give a monotonously increasing sequence and the entries inside any column of T define a strictly monotonously increasing sequence. It is well known that there exist exactly dim V_{N+1}^{T} standard schemes and that the tensors e_{T} (a_{i_{1}} ⊗ a_{i_{2}} ⊗ … ⊗ a_{i_{r}}) with the corresponding products a_{i_{1}} ⊗ a_{i_{2}} ⊗ … ⊗ a_{i_{r}} form a basis of V_{N+1}^{T} . It is possible to describe the k-basis of W_{s}^{T} in a similar way: Let now a_{N+1} be a basis vector of U. Again any product ϱ = a_{j_{1}} ⊗ a_{j_{2}} ⊗ … ⊗ a_{j_{r}} ∈ W_{s} (0 ≤ j_{ν} ≤ N + 1) corresponds to a Young scheme, but this time exactly s boxes of T are covered by the index N + 1. Assume that there exist two of these s boxes both belonging to the same row of T . Let p ∈ P_{T} denote the transposition of these two boxes. Then from (1) we get e_{T} · p = e_{T} and as a consequence of the last remark in ch.3 one has p(a_{j_{1}} ⊗ a_{j_{2}} ⊗ … ⊗ a_{j_{r}}) = −a_{j_{1}} ⊗ a_{j_{2}} ⊗ … ⊗ a_{j_{r}}, that means e_{T} (a_{j_{1}} ⊗ a_{j_{2}} ⊗ … ⊗ a_{j_{r}}) = 0 and (22) follows. Now we consider the set of all Young schemes to the Young tableau T with the properties :

• The entries inside any row and similary inside any column of T give a monotonously increasing sequence.

• Exactly s boxes of T are covered by the index N + 1.

• Each row contains at most one box covered by N + 1.

• Erasing the s boxes covered by N + 1 one gets a standard scheme with r − s

boxes. Then the tensors e_{T} (a_{j_{1}} ⊗ a_{j_{2}} ⊗ … ⊗ a_{j_{r}}) with the corresponding products a_{j_{1}} ⊗ a_{j_{2}} ⊗ … ⊗ a_{j_{r}} form a basis of W_{s}^{T} and (21) follows. □

5. Complete intersections Let X, Y be smooth projective varieties such that Y is a complete intersection of X by algebraic hypersurfaces H_{i} : F_{i} = 0 ( n = dim Y , N = n + c = dim X , c = codim_{X} Y , deg H_{i} = deg F_{i} = m_{i} ∀i ∈ {1, 2, …, c}). Let T be a Young tableau with r boxes and with the column lengths d_{1}, d_{2}, …, d_{l} ( d_{1} ≥ d_{2} ≥ … ≥ d_{l} > 0 ) and let M(T) be the set of all integer matrices A = ((d_{ij})) with c + 1 rows, l = length T columns and with the properties:

(23) d_{1,j} = d_{j} ∀j ∈ {1, 2, …, l},

(24) d_{i,l} ≥ d_{i+1,l} ≥ 0 ∀i ∈ {1, 2, …, c} ,

(25) d_{i,j} ≥ d_{i+1,j} ≥ d_{i,j+1} ∀i ∈ {1, 2, …, c} ∀j ∈ {1, 2, …, l − 1} . The i-th row sum of such a matrix A will be denoted by ϱ_{i}(A) = ∑_{j=1}^{l} d_{i,j} , in particular we set ϱ(A) = ϱ_{c+1}(A) . Let μ = ∑_{j=1}^{c} d_{j} be the number of boxes inside the c front columns of T (d_{j} = 0 if j > l). Then r − μ ≤ ϱ(A) ≤ r ∀A ∈ M(T). Finally we set M_{s}(T) = {A ∈ M(T) : ϱ(A) = r − s} for each s ∈ {0, 1, …, μ}.

Theorem 3. The following sequence is exact :

0 ⟶ E_{T}^{μ} ⟶ E_{T}^{μ−1} ⟶ … ⟶ E_{T}^{2} ⟶ E_{T}^{1} ⟶ Ω_{X|Y}^{T} ⟶ Ω_{Y}^{T} ⟶ 0 with

E_{T}^{s} = ⨁_{A∈Ms(T)}Ω_{X|Y}^{T′(A)}(t(A)) , Ω_{X|Y}^{T′} = O_{Y} ⊗ Ω_{X}^{T′} , t(A) = ∑_{i=1}^{c}(ϱ_{i+1}(A) − ϱ_{i}(A)) · m_{i} ,

μ = ∑_{j=1}^{c} d_{j}. T′(A) denotes a Young tableau with ϱ(A) boxes and the column lengths

d_{c+1,1}, d_{c+1,2}, …, d_{c+1,l} , that means, T′(A) depends only on the last row of A.

In the case that ϱ(A) = 0 one has to substitute the sheaf Ω_{X|Y}^{T′(A)} by O_{Y}.

Proof. This exact sequence will be constructed as the T -power (cp. ch.3 ) of

(26) 0 ⟶ ⨁_{1≤i≤c}O_{Y} (−m_{i}) ⟶^{η} Ω_{X|Y}^{1} ⟶ Ω_{Y}^{1} ⟶ 0 (Ω_{X|Y}^{1} = O_{Y} ⊗ Ω_{X}^{1}).

The r-th tensor power of the complex 0 ⟶ K_{1} ⟶ K_{0} with K_{1} = ⨁_{1≤i≤c}O_{Y} (−m_{i}) , K_{0} = Ω_{X|Y}^{1} gives the exact sequence 0 ⟶ C^{r} ⟶ C^{r−1} ⟶ … ⟶ C^{1} ⟶ Ω_{X|Y}^{⊗r} ⟶ Ω_{Y}^{⊗r} ⟶ 0 with

C^{s} = ⨁_{i,j}B_{i,j}^{s} , where i = (i_{1}, i_{2}, …, i_{s}) (1 ≤ i_{1} < i_{2} < … < i_{s} ≤ r) runs through the

combinations of {1, 2, …, r} , s at a time, and j = (j_{1}, j_{2}, …, j_{s}) ∈ {1, 2, …, c}^{s},

and with B_{i,j}^{s} =Ω_{X|Y}^{1} ⊗⋯ ⊗O_{Y} (−m_{j_{1}})⊗⋯ ⊗O_{Y} (−m_{j_{s}})⊗⋯ ⊗Ω_{X|Y}^{1} .

1 …  i_{1}  …  i_{s}  … r

To each matrix A ∈ M_{s}(T) there corresponds such a pair i = i(A), j = j(A): At first for a given matrix A one has a chain of c+1 Young tableaux T_{1} = T, T_{2}, …, T_{c+1} embedded into the top-left of T. The column lengths of T_{k} are the elements of the k-th row of A : d_{k,1}, d_{k,2}, …, d_{k,l}. Because of (23) - (25) T_{k+1} is embedded into T_{k} and each row of T_{k} contains beside the boxes belonging to T_{k+1} at most one additional box. The r boxes of T are numbered by 1, 2, …, r, they correspond to the spots inside the tensor products. Therefore the boxes of each imbedded Young tableau T_{k} are ordered too. T_{c+1} contains exactly r − s boxes. The combinations i(A) and j(A) are defined as follows: i(A) = (i_{1}, i_{2}, …, i_{s}) consists of the numbers of those boxes of T which do not belong to T_{c+1} . The k-th component j_{k} of j(A) is equal to m if and only if the i_{k}-th box of T belongs to T_{m} and does not belong to T_{m+1} (k ∈ {1, 2, …, s}).

If one sets E_{T}^{s} = e_{T} (C^{s}) = ∑_{i,j} e_{T} (B_{i,j}^{s}) then C^{s} = ⨁_{T ∈D(r)}E_{T}^{s} and the T -power of (26) is the exact sequence 0 ⟶ E_{T}^{r} ⟶ E_{T}^{r−1} ⟶ … ⟶ E_{T}^{1} ⟶ Ω_{X|Y}^{T} ⟶ Ω_{Y}^{T} ⟶ 0 with sheaves E_{T}^{s} to be calculated . Analogously to the proof of lemma 1 one shows : e_{T} (B_{i,j}^{s}) = 0 if there exist a, b ∈ {1, 2, …, s} with a ≠ b, j_{a} = j_{b} such that the i_{a}-th box and the i_{b}-th box of T both belong to the same row of T. Therefore E_{T}^{s} = e_{T} (C^{s}) = 0 if s > μ = ∑_{j=1}^{c} d_{j} . On the other hand, if e_{T} (B_{i′,j′}^{s}) ≠ 0 then there exists a matrix A ∈ M_{s}(T) with

e_{T} (B_{i′,j′}^{s}) = e_{T} (B_{i(A),j(A)}^{s}) , i.e., E_{T}^{s} = e_{T} (C^{s}) = ∑_{A∈M_{s}(T)} e_{T} (B_{i(A),j(A)}^{s}) .

The set M_{s}(T) will be lexicographically ordered: A < Ã if ∃a ∈ {1, 2, …, c + 1} ∃b ∈ {1, 2, …, l} with d_{a,b} < d̃_{a,b} and d_{a,β} = d̃_{a,β} ∀β > b and d_{α,β} = d̃_{α,β} ∀β ∀α < a . Let A^{+} denote the successor of A inside M_{s}(T) and for each A ∈ M_{s}(T) let G(A) be the following subsheaf of E_{T}^{s} ⊆ C^{s}: G(A) := ∑_{{_{A}∈Ms(T) \atop A≤_{A}}} e_{T} (B_{i(_{A}),j(_{A})}^{s}) . Then theorem 3 follows from :

Lemma 2. For each A ∈ M_{s}(T) we have G(A) ≅ G(A^{+}) ⊕ Ω_{X|Y}^{T′(A)}(t(A)) with t(A) = ∑_{i=1}^{c}(ϱ_{i+1}(A) − ϱ_{i}(A)) · m_{i} . If A is the last element of M_{s}(T) set G(A^{+}) = 0.

Proof. Let π_{A}^{s} be the projection from C^{s} = ⨁_{i,j} B_{i,j}^{s} onto the component B_{i(A),j(A)}^{s}. Then from (1) we obtain

(27) π_{A}^{s}(e_{T} (B_{i(_{A}),j(_{A})}^{s})) = 0 for all matrices A, Ã ∈ M_{s}(T) with A < Ã. Remember that the Young tableau T′ = T′(A) is imbedded into T, hence P_{T′} and Q_{T′} are subgroups of P_{T} and Q_{T}, respectively. Let P_{T} = ⋃_{α}^{·}(P_{T′} · p_{α}) be a decomposition of P_{T} into rigth cosets of P_{T′} with p_{1} = 1. Then similarly to (27) we have

(28) π_{A}^{s}(p_{α}(B_{i(_{A}),j(_{A})}^{s})) = 0 for each α ≠ 1 and for all matrices A, Ã with A ≤ Ã. Let A = ((d_{α,β})) ∈ M_{s}(T) be a fixed matrix , T_{1}, T_{2}, …, T_{c+1} be the corresponding chain of Young tableaux, let R_{k} be the permutation group of those boxes of T_{k} which dont belong to T_{k+1} (k = 1, 2, …, c) and set Q_{k} = R_{k} ∩ Q_{T} ,

Q = Q_{1} × Q_{2} × … × Q_{c}. The order of Q is equal to |Q| = ∏_{α=1}^{c} ∏_{β=1}^{l}(d_{α,β} − d_{α+1,β})! .

Because of the last remark in ch.3 we have

(29) q(ω_{A}) = sgn(q) · ω_{A} ∀q ∈ Q and for each section ω_{A} of the sheaf B_{i(A),j(A)}^{s} . The direct product Q × Q_{T′} is a subgroup of Q_{T}. Let Q_{T} = ⋃_{β}^{·}(q_{β} · (Q × Q_{T′})) be a decomposition of Q_{T} into left cosets of Q × Q_{T′} with q_{1} = 1 . Then

(30) π_{A}^{s}(q_{β}(B_{i(_{A}),j(_{A})}^{s})) = 0 for each β ≠ 1 and for all matrices A, Ã with A ≤ Ã.

Moreover e_{T} = \frac{ν_{(l)}}{r!} · ( ∑_{q∈Q_{T}} sgn (q) · q) · ( ∑_{p∈P_{T}} p) =

= \frac{ν_{(l)}}{r!} · (∑_{β} sgn (q_{β}) · q_{β}) · ( ∑_{q∈Q} sgn (q) · q) · ( ∑_{q′∈Q_{T′}} sgn (q′) · q′) · ( ∑_{p′∈P_{T′}} p′) · (∑_{α} p_{α}) =

= \frac{ν_{(l)}}{r!} · \frac{(r−s)!}{ν_{(l′)}} · (∑_{β} sgn (q_{β}) · q_{β}) · ( ∑_{q∈Q} sgn (q) · q) · e_{T′} · (∑_{α} p_{α}) ,

where (l′) denotes the partition of r − s , corresponding to the Young tableau T_{c+1} = T′ = T′(A). For each section ω_{A} of the sheaf B_{i(A),j(A)}^{s} follows (cp. (29))

π_{A}^{s}(e_{T} (ω_{A})) = \frac{ν_{(l)}}{r!} · \frac{(r−s)!}{ν_{(l′)}} · π_{A}^{s}((∑_{β} sgn (q_{β}) · q_{β}) · ( ∑_{q∈Q} sgn (q) · q) · e_{T′}(ω_{A})) =

= \frac{ν_{(l)}}{r!} · \frac{(r−s)!}{ν_{(l′)}} · π_{A}^{s}((∑_{β} sgn (q_{β}) · q_{β}) · ∑_{q∈Q} e_{T′}(ω_{A})) =

= \frac{ν_{(l)}}{r!} · \frac{(r−s)!}{ν_{(l′)}} · |Q| · π_{A}^{s}((∑_{β} sgn (q_{β}) · q_{β}) · e_{T′}(ω_{A})) =

= \frac{ν_{(l)}}{r!} · \frac{(r−s)!}{ν_{(l′)}} · |Q| · e_{T′}(ω_{A}) , since neither after action by p_{α} ≠ 1 (cp. (28)) nor after action by q_{β} ≠ 1 (cp. (30)) a component belonging to B_{i(A),j(A)}^{s} could arise and because e_{T′}(ω) as well as ω is a section of B_{i(A),j(A)}^{s} . The sheaf Ω_{X|Y}^{T′(A)}(t(A)) can be imbedded into B_{i(A),j(A)}^{s} using the multiplication by the global sections ϑ_{κ} = x_{0}^{m_{κ}} · d\frac{F_{κ}}{x_{0}^{mκ}} ∈ Γ(Y, Ω_{X|Y}^{1} (m_{κ})) (κ = 1, 2, …, c) : The factor at the i_{k}-th place in the tensor product is ϑ_{j_{k}}, where i_{k} and j_{k} denote the k-th component of i(A) and j(A), respectively (k ∈ {1, 2, …, s}) . Let ω be a section of G(A). Then sections ω_{A} of B_{i(_{A}),j(_{A})}^{s} exist for all Ã ≥ A (Ã ∈ M_{s}(T)) with ω = ∑_{{_{A}∈Ms(T) \atop A≤_{A}}} e_{T} (ω_{A}) . From (27) we obtain π_{A}^{s}(e_{T} (ω_{A})) = 0 if

A < Ã. The homomorphism φ_{A} from G(A) onto Ω_{X|Y}^{T′(A)}(t(A)) is defined by φ_{A}(ω) = \frac{r!}{ν_{(l)}} · \frac{ν_{(l′)}}{|Q|·(r−s)!} · π_{A}^{s}(e_{T} (ω)) = \frac{r!}{ν_{(l)}} · \frac{ν_{(l′)}}{|Q|·(r−s)!} · π_{A}^{s}(e_{T} (ω_{A})) = e_{T′}(ω_{A}) . Because of (27) one has G(A^{+}) ⊆ Kerφ_{A}. In order to prove G(A^{+}) = Kerφ_{A} it suffices to consider, similarly to lemma 1, vector spaces and their dimensions , since the embedding η in (26) is defined as multiplication by the global sections ϑ_{κ} . Finally the homomorphism φ_{A} is a projection since for any section ω_{A} of the subsheaf Ω_{X|Y}^{T′(A)}(t(A)) ⊆ B_{i(A),j(A)}^{s} we have ω_{A} = e_{T′}(ω_{A}) and therefore φ_{A}(e_{T} (ω_{A})) = \frac{r!}{ν_{(l)}} · \frac{ν_{(l′)}}{|Q|·(r−s)!} · π_{A}^{s}(e_{T} (ω_{A})) = e_{T′}(ω_{A}) = ω_{A} , that means G(A) ≅ G(A^{+}) ⊕ Ω_{X|Y}^{T′(A)}(t(A)) , q.e.d. □

Using theorem 3 the Hilbert polynomial χ(Y, Ω^{T} (t)) can be calculated . Let σ_{i} denote the i-th elementary symmetrical polynomial of the degrees m_{1}, m_{2}, …, m_{c} of the complete intersection Y ⊂ X . For any polynomial f(t) ∈ ℚ[t] we set

f_{(m)}(t) := f(t) + ∑_{s=1}^{c}(−1)^{s} · ∑_{1≤i_{1}<...<i_{s}≤c} f(t − m_{i_{1}} − m_{i_{2}} − … − m_{i_{s}}) .

Then f_{(m)}(t) is a polynomial in t of degree deg f − c . The coefficient of t^{j} is a symmetrical polynomial of m_{1}, m_{2}, …, m_{c} the degree of which is lower or equal deg f − j. Moreover f_{(m)}(t) and therefore each coefficient is divisible by the product σ_{c} = m_{1} · m_{2} · . . . · m_{c} . Thus the coefficient of t^{j} vanishes in case j > deg f − c and otherwise depends only on the elementary symmetrical polynomials σ_{1}, σ_{2}, …, σ_{min{c,deg f−c−j}} and σ_{c}.

Theorem 4. For each Young tableau T with r boxes and each n-dimensional complete intersection Y ⊂ X of multidegree (m_{1}, m_{2}, …, m_{c}) one has :

χ(Y, Ω_{Y}^{T} (t)) = ∑_{A∈M(T)}(−1)^{r−ϱ(A)} · f_{(m)}^{T′(A)}(t + t(A)) with

t(A) = ∑_{i=1}^{c}(ϱ_{i+1}(A) − ϱ_{i}(A)) · m_{i} , f^{T′(A)}(t) = χ(X, Ω_{X}^{T′(A)}(t)) if ϱ(A) > 0 and

f^{T′(A)}(t) = χ(X, O_{X}(t)) if ϱ(A) = 0 ( cf. theorem 3 ). In the case X = ℙ^{N} the polynomials f^{T′(A)}(t) are given in (20).

Proof. Theorem 4 follows from theorem 3 and from the exact sequence :

0 ⟶ Ω_{X}^{T′}(−m_{1}−m_{2}−…−m_{c}) ⟶ ⨁_{1≤i≤c}Ω_{X}^{T′}(−m_{1}−m_{2}−…−m̂_{i}−…−m_{c}) ⟶ …

… ⟶ ⨁_{1≤i_{1}<i_{2}≤c}Ω_{X}^{T′}(−m_{i_{1}} − m_{i_{2}}) ⟶ ⨁_{1≤i≤c}Ω_{X}^{T′}(−m_{i}) ⟶ Ω_{X}^{T′} ⟶ Ω_{X|Y}^{T′} ⟶ 0. □

Remark : The coefficient of t^{j} in the Hilbert polynomial χ(Y, Ω^{T} (t)) is a symmetrical polynomial of m_{1}, m_{2}, …, m_{c} whose degree is lower or equal n + c − j. This symmetrical polynomial is divisible by the product σ_{c} = m_{1} · m_{2} · . . . · m_{c} and in the case j > n−c the quotient depends only on the elementary symmetrical polynomials σ_{1}, σ_{2}, …, σ_{n−j} . If dim Y < codim Y ( i.e. n < c ) then χ(Y, Ω^{T} (t)) depends only on σ_{1}, σ_{2}, …, σ_{n} and σ_{c} .

Example : Let Y and Z be smooth 3-dimensional complete intersections in X = ℙ^{8} of the multidegree m_{Y} = (45, 35, 34, 21, 18) and m_{Z} = (42, 42, 27, 25, 17) , respectively.

Then the elementary symmetrical polynomials σ_{1}, σ_{2}, σ_{3} and σ_{5} of these integers are equal and therefore we have χ(Y, Ω_{Y}^{T} (t)) = χ(Z, Ω_{Z}^{T}(t)) for each Young tableau T. In particular Y and Z have the same plurigenera P_{f} , since P_{1} = p_{g} = χ(Y, ω_{Y}) + 1 and P_{f} = χ(Y, ω_{Y}^{⊗f}) = χ(Y, Ω_{Y}^{T}) ∀f ≥ 2 where T denotes a rectangle with dim Y = 3 rows and f columns ( cp.(4) ) . On the other hand the birational invariant dim H^{0}(Y, Ω_{Y}^{T}) in general differs from dim H^{0}(Z, Ω_{Z}^{T}) [10] . Using the multidegrees (33, 21, 5, 5), (35, 15, 11, 3) and (66, 63, 29, 23, 6, 4), (69, 58, 36, 14, 11, 3) one gets a pair of complete intersection surfaces Y_{2}, Z_{2} ⊆ ℙ^{6} and a pair of 4-dimensional complete intersections Y_{4}, Z_{4} ⊆ ℙ^{10} , respectively, with similar properties. Note that in the complex case k = ℂ the complete intersection surfaces Y_{2} and Z_{2} are homeomorphic and that the 3-folds Y and Z are diffeomorphic [10] .

Acknowledgements: I would like to thank Aleksandr G. Elashvili (Tiflis) and Helmut A. Hamm (Münster) for the many helpful and stimulating discussions and the Deutsche Forschungsgemeinschaft for their generous support.

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