WEIL CLASSES AND ROSATI INVOLUTIONS ON COMPLEX ABELIAN VARIETIES

YU.G. ZARHIN* AND B.J.J. MOONEN

1. Abelian varieties, polarizations and divisor classes

Consider a complex torus X = 𝔱/Γ, where 𝔱 ≅ ℂ^{g} is a complex vector space of finite dimension g and Γ ≅ ℤ^{2g} a discrete lattice in 𝔱 of (maximal) rank 2g. We say that X is an abelian variety if it admits a polarization, by which we mean a positive-definite Hermitian form

H : 𝔱 × 𝔱 → ℂ

whose imaginary part

L = ℑ(H) : 𝔱 × 𝔱 → ℝ

takes integral values on Γ. Clearly, a sum of two polarizations is also a polarization. In particular, we can multiply a polarization by a positive integer to obtain another one.

There exists a polarization on X if and only if X (as a complex-analytic manifold) is algebraizable, which condition in turn is equivalent to the existence of a projective embedding X ↪ ℙ^{m} for some m. We refer to Mumford’s book [11] for the basic theory of abelian varieties.

Example 1.1 (elliptic curves). Every 1-dimensional complex torus X = ℂ/Γ admits a polarization. Namely, if Γ = ℤ · ω_{1} + ℤ · ω_{2} with ω_{1}/ω_{2} ∉ ℝ then for each positive integer n the Hermitian form

(z, w) ↦ n · \frac{zw̄}{| ℑ(ω_{1}ω̄_{2}) |}

defines a polarization on X. One can check that all polarizations on X are of this form.

It should be mentioned that, for g ≥ 2, “most” complex tori do not admit a polarization.

From now on, assume that X is an abelian variety and that H is a polarization. The form L = L_{H} := ℑ(H) is a non-degenerate skew-symmetric ℝ-bilinear form on 𝔱 such that

L(zx, zy) = L(x, y) ∀x, y ∈ 𝔱; z ∈ 𝕊 ,

where

𝕊 = {z ∈ ℂ^{∗} | zz̄ = 1} .

In other words, L is 𝕊-invariant. Conversely, if φ is an 𝕊-invariant skew-symmetric ℝ-bilinear form on 𝔱, then H_{φ}(x, y) = φ(ix, y) + i · φ(x, y) defines a Hermitian form on 𝔱 with ℑ(H_{φ}) = φ. If in addition φ(Γ, Γ) ⊆ ℤ then there are polarizations H_{1} and H_{2} such that

* Supported by the National Science Foundation.

φ = ℑ(H_{1}) − ℑ(H_{2}) .

Indeed, one may take H_{1} = nH + H_{φ} and H_{2} = nH for a sufficiently large positive integer n.

The natural map Γ ⊗ ℝ → 𝔱 is an isomorphism of real vector spaces, which we will take as an identification. Now consider the ℚ-vector space V = V_{X} := Γ ⊗ ℚ. One may view V as a ℚ-lattice in V_{ℝ} := V ⊗_{ℚ} ℝ = Γ ⊗ ℝ = 𝔱. We have

Γ ⊂ V ⊂ V_{ℝ} = 𝔱 .

We will consider 𝔱 = V_{ℝ} as a real vector space provided with a ℚ-lattice V and an action of 𝕊 which gives it the structure of a complex vector space.

Let

φ : V × V → ℚ

be a skew-symmetric ℚ-bilinear form on V. Let us extend φ by ℝ-linearity to the skew-symmetric ℝ-bilinear form

φ_{ℝ} : V × V → ℝ .

We call φ a divisor class if φ_{ℝ} is 𝕊-invariant. Clearly, the set D(X) of divisor classes is a ℚ-vector subspace of Hom_{ℚ}(∧_{ℚ}^{2}V, ℚ); in particular it is finite-dimensional. By what was said before, if φ is a divisor class then φ = ℑ(H_{φ}) for some Hermitian form H_{φ} on 𝔱 = V_{ℝ}. It readily follows that φ is a divisor class if and only if it can be presented as a ℚ-linear combination of imaginary parts of polarizations.

There is another way to describe the divisor classes, which involves the endomorphism algebra and Rosati involutions. First, notice that the endomorphism ring End(X) of the complex torus X is the ring of all ℂ-linear maps u : 𝔱 → 𝔱 such that u(Γ) ⊆ Γ. Here is an obvious alternative description: the ring End(X) consists of the homomorphisms v : Γ → Γ whose ℝ-linear extensions v_{ℝ} : 𝔱 → 𝔱 (identifying Γ ⊗ ℝ and 𝔱) commute with the action of 𝕊. This last description allows us to identify the endomorphism algebra End^{0}(X) := End(X) ⊗ ℚ with a ℚ-subalgebra of End_{ℚ}(V). It is known (see [11], p. 178, for example) that if X is an abelian variety, then End^{0}(X) is a semi-simple ℚ-algebra.

Associated to H, there is an involution v ↦ v′ on End^{0}(X), called the Rosati involution and characterized by

L(vx, y) = L(x, v′y) ∀v ∈ End^{0}(X) ; x, y ∈ V_{ℝ} .

Since H(x, y) = L(ix, y) + i · L(x, y), and since the v ∈ End^{0}(X) commute with the action of 𝕊, we also have

H(vx, y) = H(x, v′y) ∀v ∈ End^{0}(X) ; x, y ∈ V_{ℝ} .

This implies that the Rosati involution is a positive involution. In particular, the center of End^{0}(X) is a product of totally real number fields and of CM-fields. (Recall that a CM-field is a totally imaginary quadratic extension of a totally real number field).

For example, assume that End^{0}(X) is a number field. Then there are two possibilities:

• F = End^{0}(X) is a totally real number field. The only positive involution on F is the identity map. Therefore, the Rosati involution on End^{0}(X) is the identity map.

• F = End^{0}(X) is a totally imaginary quadratic extension of a totally real number field F_{0}. The only positive involution on F is the “complex conjugation” (the non-trivial automorphism of F over F_{0}). Therefore, the Rosati involution is the complex conjugation.

Now the description of the divisor classes in terms of a Rosati involution on the endomorphism algebra goes as follows. Start with one polarization form H, with imaginary part L and associated Rosati involution v ↦ v′ on End^{0}(X). Then one can show that the map which sends u ∈ End^{0}(X) to the ℚ-bilinear form

(x, y) ↦ L(ux, y)

on V induces a bijection

{u ∈ End^{0}(X) | u′ = u} ⟶^{∼} D(X) .

2. Hodge classes and Weil classes

Definition 2.1. Let n be a positive integer, and consider a skew-symmetric ℚmultilinear form ψ ∈ Hom_{ℚ}(∧_{ℚ}^{2n}V, ℚ). Extend ψ by ℝ-linearity to a form ψ_{ℝ} ∈ Hom_{ℝ}(∧_{ℝ}^{2n}V_{ℝ}, ℝ). We call ψ a Hodge class (in degree 2n) if ψ_{ℝ} is 𝕊-invariant.

Remark 2.2. Clearly, a skew-symmetric ℚ-bilinear form on V (with n = 1) is a Hodge class if and only if it is a divisor class. It is also clear that a linear combination of Hodge classes of the same degree 2n is a Hodge class of degree 2n and that the exterior product of Hodge classes ψ_{1}, …, ψ_{k} of degrees 2n_{1}, …, 2n_{k} is again a Hodge class, of degree 2(n_{1} + ⋯ + n_{k}).

Definition 2.3. A Hodge class is called decomposable if it can be written as a ℚ-linear combination of exterior products of divisor classes. Otherwise, it is called exceptional.

Easy linear algebra arguments imply that if dim(X) ≤ 3 then all Hodge classes are decomposable. For instance, assume dim(X) = 3. Then every form ψ ∈ Hom_{ℚ}(∧_{ℚ}^{4}V, ℚ) can be uniquely written as the exterior product of L and a skewsymmetric ℚ-bilinear form on V, since dim(V) = 6 and L : V × V → ℚ is nondegenerate. This implies that every Hodge class in degree 4 is of the form L ∧ φ where φ is a divisor class.

It was Mumford who first gave an example of an abelian fourfold X with an exceptional Hodge class of degree 4; see [13]. In his examples End^{0}(X) is a CMfield of degree 8 over ℚ (i.e., X is an abelian fourfold of CM-type) which contains an imaginary quadratic subfield k. The action of k on V must satisfy the following condition: it gives rise to the natural action of k ⊗_{ℚ} ℂ on V_{ℝ} = 𝔱 and this action makes 𝔱 a free k ⊗_{ℚ} ℂ = ℂ ⊕ ℂ-module of rank 2. This condition may be restated as follows: if k = ℚ(α) where α^{2} = −d ∈ ℤ_{<0}, and if H is a polarization on X, then the Hermitian form

(x, y) ↦ \frac{H(αx, y)}{\sqrt{−d}}

on 𝔱 has signature (2, 2). (This does not depend on the choice of \sqrt{−d} ∈ ℂ.)

In [22], A. Weil proposed a different approach to the construction of exceptional Hodge classes. His construction and its natural generalization proved to be useful in various aspects of arithmetic and geometry of abelian varieties; see for example [1], [2], [4], [6], [8], [15], [19], [18], [23]. Weil’s construction from [4], in its generalized form, works as follows.

Suppose E is a CM-field, and that there is a given ring homomorphism E ↪ End^{0}(X). Then V becomes a E-vector space of dimension m = 2g/[E : ℚ]. The trace map Tr_{E/ℚ} : E → ℚ gives rise to an isomorphism

Hom_{E}(∧_{E}^{m}V, E) ≅^{Tr_{E/ℚ}} Hom_{ℚ}(∧_{E}^{m}V, ℚ) ,

which yields an embedding

Hom_{E}(∧_{E}^{m}V, E) ≅^{Tr_{E/ℚ}} Hom_{ℚ}(∧_{E}^{m}V, ℚ) ⊂ Hom_{ℚ}(∧_{ℚ}^{m}V, ℚ) .

Let W = W_{E} ⊂ Hom_{ℚ}(∧_{ℚ}^{m}V, ℚ) be the image of this embedding; it is the set of all skew-symmetric m-linear forms ϕ on the ℚ-vector space V such that

ϕ(ex_{1}, x_{2}, …, x_{m}) = ϕ(x_{1}, ex_{2}, …, x_{m}) = ⋯ = ϕ(x_{1}, x_{2}, …, ex_{m})

for all e ∈ E and x_{1}, x_{2} ⋯ x_{m} ∈ V.

Remark 2.4. The [E:ℚ]-dimensional ℚ-vector space W_{E} has a natural structure of one-dimensional E-vector space, given by

(eϕ)(x_{1}, x_{2}, …, x_{m}) := ϕ(ex_{1}, x_{2}, …, x_{m}) .

Note also that

(e^{m}ϕ)(x_{1}, x_{2}, …, x_{m}) = ϕ(ex_{1}, ex_{2}, …, ex_{m}) .

Remarks 2.5. (i) Let r be a positive integer and let

ψ : V × ⋯ × V → ℚ

be an r-linear form on V which is not identically zero and such that

ψ(ex_{1}, x_{2}, …, x_{r}) = ψ(x_{1}, ex_{2}, …, x_{r}) = ⋯ = ψ(x_{1}, x_{2}, …, ex_{r})

for all e ∈ E and x_{1}, x_{2} ⋯ x_{r} ∈ V. If e ∈ E^{∗} is not an rth root of unity then there exist y_{1}, y_{2}, …, y_{r} ∈ V such that

ψ(ey_{1}, ey_{2}, …, ey_{r}) ≠ ψ(y_{1}, y_{2}, …, y_{m}).

To see this, first remark that

ψ(e^{r}x_{1}, x_{2}, …, x_{r}) = ψ(ex_{1}, ey_{2}, …, ex_{m})

for all x_{1}, x_{2} … x_{r} ∈ V. Now, choosing x_{1}, x_{2} … x_{r} ∈ V such that ψ(x_{1}, x_{2}, ⋯ x_{r}) ≠ 0, and putting

y_{1} = (e^{r} − 1)^{−1}x_{1}, y_{2} = x_{2}, …, y_{r} = x_{r} ,

we have

ψ(ey_{1}, ey_{2}, …, ey_{r}) − ψ(y_{1}, y_{2}, …, y_{r}) = ψ(e^{r}y_{1}, y_{2}, …, y_{r}) − ψ(y_{1}, y_{2}, …, y_{r}) =

= ψ((e^{r} − 1)y_{1}, y_{2}, …, y_{r}) = ψ(x_{1}, x_{2}, … x_{r}) ≠ 0.

(ii) Applying Remark 2.4, we obtain that if ϕ is a non-zero element of W_{E} and e ∈ E^{∗} is not a root of unity, then there exist x_{1}, x_{2} …, x_{m} ∈ V such that

ϕ(ex_{1}, ex_{2}, …, ex_{m}) ≠ ϕ(x_{1}, x_{2}, …, x_{m}).

We call the (non-zero) elements of W = W_{E} Weil classes, or Weil classes w.r.t. E. One easily checks that W_{ℝ} = W ⊗_{ℚ} ℝ is an 𝕊-stable subspace of

Hom_{ℚ}(∧_{ℚ}^{m}V, ℚ) ⊗_{ℚ} ℝ = Hom_{ℝ}(∧_{ℝ}^{m}V_{ℝ}, ℝ).

It is shown in [4] that W_{ℝ} consists of 𝕊-invariant elements (which, by definition, means that all elements of W are Hodge classes) if and only if the following condition holds: The action of E on V gives rise to the natural action of E ⊗_{ℚ} ℂ on V_{ℝ} = 𝔱 and under this action 𝔱 becomes a free E ⊗_{ℚ} ℂ-module. If this holds then, since 𝔱 has rank n = m/2 over E ⊗_{ℚ} ℂ, the number m must be even and n divides g.

It seems that most (but not all!) known examples of exceptional Hodge classes on abelian varieties, e.g. those discussed in [12], [13], [15], and [17], are Weil classes. For instance, if X is an abelian fourfold then each Hodge class can be presented as a linear combination of decomposable classes and Weil classes (if there are any), see [8], [10]. In Mumford’s example mentioned above, the exceptional Hodge classes are Weil classes w.r.t. the imaginary quadratic field k ⊂ E.

It is natural to ask when Weil’s construction leads to exceptional classes. (It is easy to see that if some non-zero element of W_{E} is exceptional then in fact all nonzero elements of W_{E} are exceptional.) Weil himself considered the case of (imaginary) quadratic fields E and proved that in this case Weil classes are exceptional for “generic” X (where the genericness includes the condition that E = End^{0}(X)). In a recent paper [9] the authors have given an explicit necessary and sufficient condition for (non-zero) Weil classes to be exceptional. This condition is formulated in terms of the endomorphism algebra of X and uses Albert’s classification of semi-simple finite-dimensional ℚ-algebras with a positive involution.

3. Main results

The motivation of the present paper is to give an explanation, in completely elementary terms, why in the examples of Mumford and Weil one gets exceptional Weil classes. Namely, we prove the following statement.

Theorem 3.1. Consider a complex abelian variety X with a given inclusion E ↪ End^{0}(X) of a CM-field E in its endomorphism algebra (sending 1 ∈ E to the identity on X). Assume that E is stable under all Rosati involutions on End^{0}(X) and that 𝔱 is a free E ⊗_{ℚ} ℂ-module. Then all non-zero elements of W = W_{E} are exceptional Hodge classes.

Remark 3.2. If Rosati involution leaves E stable then it acts on E as the complex conjugation.

Remark 3.3. If End^{0}(X) is a CM-field then there is only one Rosati involution, namely the complex conjugation. In particular, if End^{0}(X) contains an imaginary quadratic subfield k (e.g., End^{0}(X) = k) then the Rosati involution leaves k stable and, assuming that 𝔱 = V_{ℝ} is a free E ⊗_{ℚ} ℂ-module, we get the exceptionality of all non-zero Weil classes. This explains the examples of Mumford and Weil. In fact, in the situation considered by Weil we slightly improve the result, since we only need the equality End^{0}(X) = k.

Theorem 3.1 is an immediate corollary of the following statement.

Theorem 3.4. Let Pol_{E} be set of all polarizations H on X such that E is stable under the associated Rosati involution. Let

D_{E} := {ℑ(H) ∈ D(X) ⊂ Hom_{ℚ}(∧_{ℚ}^{2}V, ℚ) | H ∈ Pol_{E}}

be the set of their imaginary parts, viewed as divisor classes. Let r be an even positive integer and let ψ ∈ Hom_{ℚ}(∧_{ℚ}^{r}V, ℚ) be a non-zero skew-symmetric form with

ψ(ex_{1}, x_{2}, …, x_{r}) = ψ(x_{1}, ex_{2}, …, x_{r}) = ⋯ = ψ(x_{1}, x_{2}, …, ex_{r})

for all e ∈ E and all x_{1}, x_{2} …, x_{r} ∈ V. (This is the case, for example, if r = m := dim_{E}(V) and ψ ∈ W_{E} is a non-zero Weil class). Then ψ is not a ℚ-linear combination of exterior products of bilinear forms L_{H} (H ∈ Pol_{E}).

Remark 3.5. It is shown in [7], Lemma 9.2, that the set Pol_{E} is always non-empty.

Proof of Theorem 3.4. Choose a polarization H ∈ Pol_{E} and let

L_{H} = ℑ(H) : V × V → ℚ

be its imaginary part, viewed as a skew-symmetric ℚ-bilinear form on V. The assumption that E is stable under the Rosati involution u ↦ u′ attached to H means that u′ = ū is the complex conjugate of u and

L_{H}(ux, y) = L_{H}(x, ūy) ∀u ∈ E; x, y ∈ V .

Consider the multiplicative subgroup

𝕌_{E} = {u ∈ E^{∗} | uū = 1} .

Since E is a CM-field, 𝕌_{E} is infinite. Since E contains only finitely many roots of unity, 𝕌_{E} contains an element e which is not a root of unity. Since e ∈ 𝕌_{E}, we have

L_{H}(ex, ey) = L_{H}(x, ēey) = L_{H}(x, y) ∀x, y ∈ V .

This implies that if an alternating r-linear form ω on V can be presented as a linear combination of exterior products of bilinear forms L_{H} (H ∈ Pol_{E}) then

ω(ex_{1}, ex_{2}, … ex_{r}) = ω_{(}x_{1}, x_{2}, …, x_{r}) ∀x_{1}, x_{2}, …, x_{m} ∈ V .

Since e is not a root of unity, it follows from Remark 2.5(i) that there exist y_{1}, y_{2}, ⋯, y_{r} ∈ V such that

ψ(ey_{1}, ey_{2}, … ey_{r}) ≠ ψ(y_{1}, y_{2}, … y_{r}) .

4. Cohomological interpretation

Let us now explain the relevance of the above considerations in slightly different terminology. First, the lattice Γ is the first integral homology group H_{1}(X, ℤ) of X, and also we have natural isomorphisms V = H_{1}(X, ℚ) and V_{ℝ} = H_{1}(X, ℝ). The fact that X is a torus then implies that Hom_{ℚ}(∧_{ℚ}^{r}V, ℚ) ≅ H^{r}(X, ℚ) and Hom_{ℝ}(∧_{ℝ}^{r}V_{ℝ}, ℝ) ≅ H^{r}(X, ℝ). The exterior product of forms corresponds to the cupproduct of cohomology classes. The forms L = L_{H} ∈ Hom_{ℚ}(∧_{ℚ}^{2}V, ℚ) = H^{2}(X, ℚ) are rational multiples of the cohomology classes of hyperplane sections of X and each class of a hyperplane section coincide with L_{H} for some polarization H. The subspace

D(X) ⊂ Hom_{ℚ}(∧_{ℚ}^{2}V, ℚ) = H^{2}(X, ℚ)

is the set of rational multiples of cohomology classes of divisors on X. For even r the identification

Hom_{ℚ}(∧_{ℚ}^{r}V, ℚ) = H^{r}(X, ℚ)

identifies Hodge classes as defined here with conventional ones [3],[4],[11]. A celebrated conjecture of Hodge [20] asserts that all Hodge classes are algebraic (the converse is known to be true). For decomposable classes this is a consequence of Lefschetz’ theorem on (1, 1)-classes (see e.g. [5], Ch. 1, Sect. 2), and Weil suggested his construction as a source of possible counterexamples to the Hodge conjecture. See [17], [15], [16] and [21] for some rare examples where the algebraicity of exceptional Weil classes is proven.

Acknowledgements. The first named author (Y. Z.) would like to thank the organizers of the conference for fantastic warm hospitality.

References

[1] Y. André. Une remarque à propos de cycles de Hodge de type CM. Seminaire de Théorie des Nombres, Paris, 1989-90, Progress in Math. 102 (1992), 1–7.

[2] Y. André. Sur une théorie inconditionelle des motifs. Publ. Math. de l’I.H.E.S. 83 (1996), 5–49.

[3] P. Deligne. Théorie de Hodge II. Publ. Math. de l’I.H.E.S. 40 (1972), 5–57.

[4] P. Deligne. Hodge cycles on abelian varieties (notes by J.S. Milne). Lecture Notes in Math. 900 (1982), 9–100.

[5] P. Griffiths and J. Harris. Principles of Algebraic Geometry. John Wiley and Sons, New York, 1978.

[6] B. Gross. On periods of abelian integrals and a formula of Chowla-Selberg . Invent. Math. 45 (1978), 193–211.

[7] R.E. Kottwitz. Points on some Shimura varieties over finite fields. J. Amer. Math. Soc. 5 (1992), 373–444.

[8] B.J.J. Moonen and Yu.G. Zarhin. Hodge classes and Tate classes on simple abelian fourfolds. Duke Math. J. 77(1995), 553–581.

[9] B.J.J. Moonen and Yu.G. Zarhin. Weil classes on abelian varieties. J. reine angew. Math., to appear.

[10] B.J.J. Moonen and Yu.G. Zarhin. Hodge classes and Tate classes on abelian fourfolds, in preparation.

[11] D. Mumford. Abelian varieties (second edition). Oxford University Press, Oxford, 1974.

[12] V.K. Murty. Exceptional Hodge classes on certain abelian varieties. Math. Ann. 268 (1984), 197–206.

[13] H. Pohlman. Algebraic cycles on abelian varieties of complex multiplication type. Ann. of Math. 88 (1968), 161–180.

[14] K. Ribet. Hodge classes on certain types of abelian varieties. Amer. J. of Math. 105 (1983), 523–538.

[15] C. Schoen. Hodge classes on self-products of a variety with an automorphism. Compositio Math. 65 (1988), 3–32.

[16] C. Schoen. Addendum to: Hodge classes on self-products of a variety with an automorphism, to appear.

[17] T. Shioda. Algebraic cycles on abelian varieties of Fermat type. Math. Ann. 258 (1981), 65–80.

[18] A. Silverberg and Yu.G. Zarhin. Connectedness extensions for abelian varieties. Math. Z., to appear.

[19] A. Silverberg and Yu.G. Zarhin. Hodge groups of abelian varieties with purely multiplicative reduction. Izvestija RAN 56 (1996), 149–158; Izvestiya:Mathematics 60:2, 379–389.

[20] J. Tate. Algebraic cycles and poles of zeta functions. Arithmetical Algebraic Geometry, Harper and Row, New York (1965), 93-110.

[21] B. van Geemen. Theta functions and cycles on some abelian fourfolds. Math. Z. 221 (1996), 617–631.

[22] A. Weil. Abelian varieties and the Hodge ring. Collected papers, Vol. III, 421–429.

[23] Yu. G. Zarhin Torsion of abelian varieties, Weil classes and cyclotomic extensions. Math. Proc. Cambridge Phil. Soc., to appear.

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Institute for Mathematical Problems in Biology, Russian Academy of Sciences, Pushchino, Moscow Region, 142292, Russia

E-mail address: zarhin@math.psu.edu

Department of Mathematics, Princeton University, Princeton, NJ 08544, USA

E-mail address: moonen@math.Princeton.edu