Cubic surfaces and cubic threefolds, jacobians and intermediate jacobians

Yuri G. Zarhin

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA and Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

zarhin@math.psu.edu

To my teacher Yuri Ivanovich Manin with admiration and gratitude

In this paper we study principally polarized abelian varieties that admit an automorphism of order 3. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do guarantee that those polarized varieties are not jacobians of curves. As an application, we get another proof of the (already known) fact that intermediate jacobians of certain cubic threefolds are not jacobians of curves.

1 Principally polarized abelian varieties that admit an automorphism of order 3

Let ζ_{3} = \frac{−1+\sqrt{−3}}{2} be a primitive (complex) cubic root of unity. It generates the multiplicative order 3 cyclic group μ_{3} of cubic roots of unity.

Let g > 1 be an integer and (X, λ) a principally polarized g-dimensional abelian variety over the field C of complex numbers, δ an automorphism of (X, λ) that satisfies the cyclotomic equation δ^{2}+δ+1 = 0 in End(X). In other words, δ is a periodic automorphism of order 3, whose set of fixed points is finite. This gives rise to the embeddings

Z[ζ_{3}] ↪ End(X), 1 ↦ 1_{X}, ζ_{3} ↦ δ,
Q(ζ_{3}) ↪ End^{0}(X), 1 ↦ 1_{X}, ζ_{3} ↦ δ.

By functoriality, Q(ζ_{3}) acts on the g-dimensional complex vector space Ω^{1}(X) of differentials of the first kind on X. This provides Ω^{1}(X) with a structure of Q(ζ_{3}) ⊗_{Q} C-module. Clearly,

Q(ζ_{3}) ⊗_{Q} C = C ⊕ C

where the summands correspond to the embeddings Q(ζ_{3}) → C that send ζ_{3} to ζ_{3} and ζ_{3}^{−1} respectively. So, Q(ζ_{3}) acts on Ω^{1}(X) with multiplicities a and b that correspond to the two embeddings of Q(ζ_{3}) into C. Clearly, a and b are non-negative integers with a + b = g.

Theorem 1.1. If g+2 < 3 | a−b | then (X, λ) is not the jacobian of a smooth projective irreducible genus g curve with canonical principal polarization.

Proof. Suppose that (X, λ) ≅ (J(C), Θ) where C is an irreducible smooth projective genus g curve, J(C) its jacobian with canonical principal polarization Θ. It follows from the Torelli theorem in Weil’s form [10, 11] that there exists an automorphism ϕ : C → C, which induces (by functoriality) either δ or −δ on J(C) = X. Replacing ϕ by ϕ^{4} and taking into account that δ^{3} is the identity automorphism of X = J(C), we may and will assume that ϕ induces δ. Clearly, ϕ^{3} is the identity automorphism of C, because it induces the identity map on J(C) and g > 1. The action of ϕ on C gives rise to the embedding

μ_{3} ↪ Aut(C), ζ_{3} ↦ ϕ.

Let P ∈ C be a fixed point of ϕ. Then ϕ induces the automorphism of the corresponding (one-dimensional) tangent space T_{P} (C), which is multiplication by a complex number c_{P}. Clearly, c_{P} is a cubic root of unity.

Lemma 1.2. Every fixed point P of ϕ is nondegenerate, i.e., c_{P} ≠ 1.

Proof (of Lemma 1.2). The result is well-known. However, I failed to find a proper reference.

Suppose that c_{P} = 1. Let O_{P} be the local ring at P and 𝔪_{P} its maximal ideal. We write ϕ_{∗} for the automorphism of O_{P} induced by ϕ. Clearly, ϕ_{∗}^{3} is the identity map. Since ϕ is not the identity map, there are no ϕ_{∗}-invariant local parameters at P. Clearly, ϕ_{∗}(𝔪_{P}) = 𝔪_{P} , ϕ_{∗}(𝔪_{P}^{2} ) = 𝔪_{P}^{2} . Since T_{P} (C) is the dual of 𝔪_{P} /𝔪_{P}^{2} and c_{p} = 1, we conclude that ϕ_{∗} induces the identity map on 𝔪_{P} /𝔪_{P}^{2} . This implies that if t ∈ 𝔪_{P} is a local parameter at t (i.e., its image t̄ in 𝔪_{P} /𝔪_{P}^{2} is not zero) then t′ := t + ϕ_{∗}(t) + ϕ_{∗}^{2}(t) is ϕ_{∗}-invariant and its image in 𝔪_{P} /𝔪_{P}^{2} equals 3t̄≠ 0. This implies that t′ ia a ϕ_{∗}-invariant local parameter at P. Contradiction.

Corollary 1.3. D := C/μ_{3} is a smooth projective irreducible curve. The map C → D has degree 3, its ramification points are exactly the images of fixed points of ϕ and all the ramification indices are 3.

Lemma 1.4. D is biregularly isomorphic to the projective line.

Proof (of Lemma 1.4). The map C → D induces, by Albanese functorialy, the surjective homomorphism of the corresponding jacobians J(C) → J(D) that kills all the divisors classes of the form (Q) − (ϕ(Q)) (Q ∈ C). This implies that it kills (1 − δ)J(C). On the other hand, 1 − δ : J(C) → J(C) is, obviously, an isogeny. This implies that the image of J(C) in J(D) is zero and the surjectiveness implies that J(D) = 0. This means that the genus of D is 0.

Corollary 1.5. The number h of fixed points of ϕ is g + 2.

Proof (of Corollary 1.5). Applying Hurwitz formula to C → D, we get

2g − 2 = 3 · (−2) + 2 · h.

Lemma 1.6. Let ϕ^{∗} : Ω^{1}(C) → Ω^{1}(C) be the automorphism of Ω^{1}(C) induced by ϕ and τ its trace. Then

τ = aζ_{3} + bζ_{3}^{−1}.

Proof (of Lemma 1.6). Pick a ϕ-invariant point P_{0} and consider the regular map

α : C → J(C), Q ↦ cl((Q) − (P_{0})).

It is well-known that α induces an isomorphism of complex vector spaces

α^{∗} : Ω^{1}(X) ≅ Ω^{1}(C).

Clearly,

ϕ^{∗} = α^{∗}δ^{∗}α^{∗}^{−1}

where δ^{∗} : Ω^{1}(J(C)) = Ω^{1}(J(C)) is the automorphism induced by δ. This implies that the traces of ϕ^{∗} and δ^{∗} do coincide. Now the very definition of a and b implies that the trace of ϕ^{∗} equals aζ_{3} + bζ_{3}^{−1}.

End of proof of Theorem 1.1. Let B be the set of fixed points of ϕ. We know that #(B) = g + 2. By the holomorphic Lefschetz fixed point formula [2, Th. 2], [6, Ch. 3, Sect. 4] (see also [9, Sect. 12.2 and 12.5]) applied to ϕ,

1 − τ̄ = ∑_{P ∈B} \frac{1}{1 − c_{P}}

where τ̄ is the complex-conjugate of τ. Recall that every c_{P} is a (primitive) cubic root of unity and therefore

| 1 − c_{P} |= \sqrt{3}, | \frac{1}{1 − c_{P}} |= \frac{1}{\sqrt{3}}

and

| 1 − τ̄ |≤ \frac{g + 2}{\sqrt{3}}.

Now

| 1 − τ̄ |^{2}= \frac{(a + b + 2)^{2} + 3(a − b)^{2}}{4} = \frac{(g + 2)^{2} + 3(a − b)^{2}}{4} .

This implies that

\frac{(g + 2)^{2}}{3} ≥ \frac{(g + 2)^{2} + 3(a − b)^{2}}{4} .

It follows that (g + 2)^{2} ≥ 9(a − b)^{2} and we are done.

2 Cubic threefolds

Let S : F (x_{0}, x_{1}, x_{2}, x_{3}) = 0 ⊂ P^{3} be a smooth projective cubic surface over C [7]. (In particular, F is an irreducible homogeneous cubic polynomial in x_{0}, x_{1}, x_{2}, x_{3} with complex coefficients.) Then the equation

y^{3} = F (x_{0}, x_{1}, x_{2}, x_{3})

defines a smooth projective threefold T ⊂ P^{4} provided with the natural action of μ_{3} that arises from multiplication of y by cubic roots of unity [1] (see also [3, 8]). We have the μ_{3}-invariant Hodge decomposition

H^{3}(T, C) = H^{3}(T, Z) ⊗ C = H^{1,2}(T) ⊕ H^{2,1}(T)

and the μ_{3}-invariant non-degenerate alternating intersection pairing

(,) : H^{3}(T, C) × H^{3}(T, C) → C.

In addition, both H^{1,2}(T) and H^{2,1}(T) are 5-dimensional isotropic subspaces and μ_{3} acts on H^{2,1}(T) with multiplicities (4, 1), i.e. ζ_{3} ∈ μ_{3} acts as diagonalizable linear operator in H^{2,1}(T) with eigenvalue ζ_{3} of multiplicity 4 and eigenvalue ζ_{3}^{−1} of multiplicity 1 ([3, Sect. 5], [1, Sect. 2.2 and Lemma 2.6]). (The proof is based on [5, Th. 8.3 on p. 488]; see also [4, pp. 338–339].)

Since both H^{1,2}(T) and H^{2,1}(T) are isotropic and the intersection pairing is non-degenerate, its restriction to H^{1,2}(T) × H^{2,1}(T) gives rise to the nondegenerate μ_{3}-invariant C-bilinear pairing

(,) : H^{1,2}(T) × H^{2,1}(T) → C.  (1)

It follows that μ_{3} acts on H^{1,2}(T) with multiplicities (1, 4). (This assertion also follows from the fact that H^{1,2}(T) is the complex-conjugate of H^{2,1}(T).) In particular, the action of μ_{3} on H^{1,2}(T) extends to the embedding

Z[μ_{3}] ↪ End_{C}(H^{1,2}(T)).  (2)

3 Intermediate jacobians

Let (J(T), θ_{T}) be the intermediate jacobian of the cubic threefold T [4, Sect. 3]; it is a principally polarized five-dimensional complex abelian variety. By functoriality, μ_{3} acts on J(T) and respects the principal polarization θ_{T}. As complex torus,

J(T) = H^{1,2}(T)/p(H^{3}(T, Z)),  (3)

where

p : H^{3}(T, C) = H^{3}(T, Z) ⊗ C = H^{1,2}(T) ⊕ H^{2,1}(T) → H^{1,2}(T)

is the projection map that kills H^{2,1}(T). The imaginary part of the Riemann form of the polarization coincides with the intersection pairing on H^{3}(T, Z) ≅ p(H^{3}(T, Z)).

It follows from (2) that the action of μ_{3} on J(T) extends to the embedding

Z[μ_{3}] ↪ End(J(T)).

Combining (1) and (3), we conclude that the μ_{3}-modules Ω^{1}(J(T)) = Hom_{C}(H^{1,2}(T), C) and H^{2,1}(T) are canonically isomorphic. Now the assertions of Sect. 2 about multiplicities imply that Z[ζ_{3}] acts on Ω^{1}(J(T)) with multiplicities (4, 1).

Since 3× | 4 − 1 |> 5 + 2, it follows from Theorem 1.1 that (J(T), θ_{T}) is not isomorphic to the canonically polarized jacobian of a curve. Of course, this assertion was proven by completely different methods in [4] for arbitrary smooth projective cubic threefolds.

References

1. D. Allcock, J.A. Carlson, D. Toledo, The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Algebraic Geometry 11 (2002), 659–724.

2. M. F. Atiyah, R. Bott, A Lefschetz fixed point formula for elliptic differential operators. Bull. Amer. Math. Soc. 72 (1966), 245–250.

3. J.A. Carlson, D. Toledo, Discriminant complements and kernels of monodromy representations. Duke Math. J. 97 (1999), 621–648.

4. C.H. Clemens, Ph. Griffiths, The intermediate jacobian of the cubic threefold. Ann. of Math. (2) 95 (1972), 281–356.

5. Ph. Griffiths, On the periods of certain rational integrals: I and II. Ann. of Math. (2) 90 (1969), 460–541.

6. Ph. Griffiths, J. Harris, Principles of algebraic geometry. John Wiley and Sons, New York, 1978.

7. Yu. I. Manin, Cubic forms, second edition. North Holland, 1986.

8. K. Matsumoto, T. Terasoma, Theta constants associated to cubic threefolds. J. Algebraic Geometry 12 (2003), 741–775.

9. J. Milnor, Dynamics in one complex variable. Vieweg, Braunschweig/Wiesbaden, 1999.

10. A. Weil, Zum Beweis des Torellischen Satzes. Gött. Nachr. 1957, no. 2, pp. 33–53; Œuvres, vol. III, [1957a].

11. A. Weil, Sur le théorème de Torelli. Séminaire Bourbaki 151 (Mai 1957).