A remark on moduli spaces of 4-dimensional complete intersections

P. Brückmann Martin-Luther-Universität Halle-Wittenberg Fachbereich Mathematik und Informatik 06099 Halle (Saale), Germany

14J15, 14M10, 32G05, 32G13, 55N30

Abstract As a supplement to earlier results on two- and three-dimensional complete intersections the existence of homeomorphic four-dimensional complete intersections which lie in different dimensional components of the moduli space will be shown.

Narasimhan and Simha [9] proved that the isomorphism classes of complex analytic structures with ample canonical bundle on a topological space form a Hausdorff space. If V denotes a complete intersection then one knows by Sernesis result on infinitesimal deformations that the dimension m(V) of the moduli space component containing the isomorphism class of V is equal to dim H^{1}(V, T_{V}) [10], [11] . Libgober and Wood calculated this dimension [8] . An explicit formula is given in [2]. Finally Fuquan and Klaus proved, that two 4-dimensional complete intersections are homeomorphic, iff they have the same total degree, Pontrjagin numbers and Euler number [6]. In this paper we will show the following supplement to the results on surfaces and 3-dimensional complete intersections [2]:

Theorem 1 For each integer k > 1 there exist k homeomorphic complex 4-dimen-sional nonsingular complete intersections in the projective space ℙ_{ℂ}^{6k−2} isomorphism classes of which lie in different dimensional components of the moduli space.

Proof. Let V ⊆ ℙ^{N} be a smooth n-dimensional complete intersection of the multidegree d = (d_{1}, d_{2}, …, d_{r}) with n ≥ 2 and d_{j} ≥ 2 ∀j (r = N − n) . As usual let σ_{i} denote the i-th elementary symmetrical polynomial of the integers d_{1}, d_{2}, …, d_{r} and let s_{i} = ∑_{j=1}^{r} d_{j}^{i} be the sum of the i-th powers of these integers. For these σ_{i} and s_{i} one has the Newton relations. Assume that V is not a K3-surface or a quadratic hypersurface . Then we know from [2, Lemma 3]: m(V) = dim H^{1}(V, T_{V}) = dim H^{n−1}(V, Ω_{V}^{1} ⊗ ω_{V}) =

1 − (N + 1)^{2} + ∑_{i=1}^{r} \binom{N+d_{i}}{N} + ∑_{i=1}^{r} ∑_{j=1}^{r}(−1)^{j} ∑_{1≤k_{1}<k_{2}<...<k_{j}≤r} \binom{N+d_{i}−d_{k_{1}}−d_{k_{2}}−...−d_{k_{j}}}{N}

if we set \binom{m}{N} = 0 for m < N (m ∈ ℤ). Now we consider for instance the multidegrees e = (91, 80, 70, 44, 43, 32) and f = (88, 86, 64, 52, 35, 35) . It is easy to check that the total degree (= 2^{12}·5^{2}·7^{2}·11·13·43), the corresponding elementary symmetrical polynomials σ_{1}, σ_{2}, σ_{3}, σ_{4} and therefore the power sums s_{1}, s_{2}, s_{3}, s_{4} agree. The same is true for the composed multidegrees

d_{0,s}, d_{1,s−1}, …, d_{s,0} with d_{λ,μ} = (\underbrace{e, e, …, e}_{λ times}, \underbrace{f, f, …, f}_{μ times}) (λ + μ = s > 0).

Let V_{(λ,μ)} ⊆ ℙ^{6s+4} be a complex 4-dimensional smooth complete intersection of the multidegree d_{λ,μ}. The existence of this nonsingular complete intersection follows from the Bertini theorem. Then for each fixed s > 0 the complete intersections V_{(0,s)}, V_{(1,s−1)}, …, V_{(s,0)} have the same total degree deg V = 2^{12s}·5^{2s}·7^{2s}·11^{s}·13^{s}·43^{s}, the same Pontrjagin numbers p_{2} = (\binom{r′}{2} − r′s′_{2} + \frac{1}{2}(s′_{2}^{2} + s′_{4})) · deg V and p_{1}^{2} = (r′ − s′_{2})^{2} · deg V and the same Euler number e =( \binom{r′}{4} − \binom{r′}{3}s′_{1} + \frac{1}{2} \binom{r′}{2}(s′_{1}^{2} + s′_{2}) − \frac{1}{6}r′(s′_{1}^{3} + 3s′_{1}s′_{2} + 2s′_{3}) + \frac{1}{24}(s′_{1}^{4} + 6s′_{1}^{2}s′_{2} + 8s′_{1}s′_{3} + 3s′_{2}^{2} + 6s′_{4}) ) · deg V with r′ = 6s + 5 , s′_{i} = s · s_{i} . Therefore these complete intersections are homeomorphic. Note that they are diffeomorphic up to connected sum with a homotopy 8-sphere [6].

On the other hand the dimensions m(V_{(λ,μ)}) of their moduli space components form a strictly monotonously increasing sequence. This can be easily proved using the above formula because only a few binomial coefficients \binom{N+d_{i}−d_{k_{1}}−d_{k_{2}}−...−d_{k_{j}}}{N} are different from zero (N = 6s + 4). One has for s = λ + μ , 0 ≤ λ < s : m(V_{(λ+1,μ−1)}) − m(V_{(λ,μ)}) > \binom{N+91}{N} + \binom{N+80}{N} + \binom{N+70}{N} + \binom{N+44}{N} + \binom{N+43}{N}−

\binom{N+88}{N} − \binom{N+86}{N} − \binom{N+64}{N} − \binom{N+52}{N} − 2\binom{N+35}{N} − ((λ + 1)^{2} − λ^{2}) · \binom{N+91−32}{N}−

((λ + 1)(μ − 1) − λ · μ) · (2\binom{N+91−35}{N} + \binom{N+88−32}{N} + \binom{N+86−32}{N})−

2((μ − 1)^{2} − μ^{2}) · (\binom{N+88−35}{N} + \binom{N+86−35}{N}) − 2r^{2} · \binom{N+91−43}{N} − 2r\binom{r}{2} · \binom{N+91−32−32}{N}

with r = 6s , 2r^{2} < 2N^{2} < 4\binom{N+50}{2} , 2r\binom{r}{2} < N^{3} < 6\binom{N+30}{3}, (λ + 1)^{2} − λ^{2} = 2λ + 1 < 2s < \frac{1}{3}N < \frac{1}{3}(N + 60) , (λ + 1)(μ − 1) − λ · μ = μ − λ − 1 < s < \frac{1}{6}N < \frac{1}{6}(N + 57) and (μ − 1)^{2} − μ^{2} < 0 . Therefore m(V_{(λ+1,μ−1)}) − m(V_{(λ,μ)}) > \binom{N+91}{91} − \binom{N+88}{88} − \binom{N+86}{86} − \frac{1}{3}(N + 60)\binom{N+59}{59}−

\frac{2}{3}(N + 57)\binom{N+56}{56} − 4\binom{N+50}{2}\binom{N+48}{48} − 6\binom{N+30}{3}\binom{N+27}{27} =

\binom{N+91}{91}·( 1 − \begin{matrix}( & {91 \atop 3} & ) \\ ( & {N+91 \atop 3} & ) \end{matrix} − \begin{matrix}( & {91 \atop 5} & ) \\ ( & {N+91 \atop 5} & ) \end{matrix} − \frac{32·\binom{91}{32}}{3·\binom{N+91}{31}} − \frac{70·\binom{91}{35}}{3·\binom{N+91}{34}} − \frac{2·43·42·\binom{91}{43}}{\binom{N+91}{41}} − \frac{64·63·62·\binom{91}{64}}{\binom{N+91}{61}} ) .

The last expression is greater then 2.7 · 10^{20} for N = 21 , i.e., positive for all N ≥ 21 because it is monotonously increasing. On that account the sequence ( m(V_{(λ,s−λ)}) )_{λ=0,1,... ,s} is strictly monotonously increasing for s ≥ 3 , i.e., for N = 6s + 4 ≥ 22 . Finally in the cases s = 1, 2 the explicit calculation of the integers m(V_{(λ,μ)}) shows the monotonicity of the corresponding sequences. Setting k = s + 1 one has k four-dimensional complete intersections V_{(0,s)}, V_{(1,s−1)}, …, V_{(s,0)} in the projective space ℙ^{6s+4} = ℙ^{6k−2} with the desired properties. □

Remark : For each fixed s > 0 the sequence of the birational invariants ( dim H^{0}(V_{(λ,s−λ)}, Ω_{V}^{1} ⊗ ω_{V}) )_{λ=0,1,... ,s} is strictly monotonously increasing too, because the manifolds V_{(0,s)}, V_{(1,s−1)}, …, V_{(s,0)} have the same multiplicity of the canonical class n_{V} (> 0) with ω_{V} = O_{V} (n_{V}) and the same Hilbert polynomial χ(V, Ω_{V}^{1} (t)) [2, Lemma 2], [3]. As well known H^{q}(V, Ω_{V}^{1} (n_{V})) = 0 ∀q ∉ {0, 3} and therefore dim H^{0}(V, Ω_{V}^{1} ⊗ω_{V}) = χ(V, Ω_{V}^{1} ⊗ω_{V})+dim H^{3}(V, Ω_{V}^{1} ⊗ω_{V}) = χ(V, Ω_{V}^{1} (n_{V}))+m(V). Note that these manifolds have the same plurigenera [3].

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