[Paper Review] Conjugate varieties with distinct real cohomology algebras

François Charles|ArXiv.org|Jun 25, 2007

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper constructs smooth projective varieties over a number field whose complex conjugates—induced by different embeddings of the number field into ℂ—have non-isomorphic real cohomology algebras. Using Voisin’s methods on cohomology rings of abelian varieties with complex multiplication, the authors show that the real cohomology algebra detects the endomorphism ring structure, which varies under complex conjugation, thereby answering Grothendieck’s question about conjugate varieties having distinct real cohomology algebras.

ABSTRACT

Using constructions of Voisin, we exhibit a smooth projective variety defined over a number field k and two complex embeddings of k, such that the two complex manifolds induced by these embeddings have non isomorphic cohomology algebras with real coefficients. This contrasts with the fact that the cohomology algebras with l-adic coefficients are canonically isomorphic for any prime number l, and answers a question of Grothendieck.

Motivation & Objective

• To answer a question posed by Grothendieck regarding whether conjugate algebraic varieties can have non-isomorphic real cohomology algebras.
• To demonstrate that the real cohomology algebra of a complex variety is not invariant under complex conjugation, despite invariance under l-adic cohomology and Betti cohomology with ℚ_l coefficients.
• To construct explicit examples of smooth projective varieties defined over a number field whose conjugate embeddings into ℂ yield non-isomorphic real cohomology algebras.
• To show that the real cohomology algebra can detect arithmetic data such as endomorphism rings of abelian varieties, even when such data are not visible in rational or l-adic cohomology.
• To extend previous results on non-homeomorphic conjugate varieties by showing that the difference persists at the level of real cohomology algebras, not just homotopy or fundamental group.

Proposed method

• Construct a smooth projective variety $X$ as a blow-up of $A imes A imes bP^N$ along five disjoint subvarieties $Z_1, ar{Z}_2, Z_3, Z_4, Z_5$, where $A = E imes E'$ is a product of elliptic curves with complex multiplication.
• Use the endomorphism rings of $A$—generated by complex multiplication endomorphisms $f$ and $f'$—to encode arithmetic data into the cohomology algebra of $X$ via the blow-up process.
• Leverage Voisin’s technique of using the cohomology algebra to reconstruct the endomorphism ring of an abelian variety, particularly by isolating classes corresponding to $f$ and $f'$ in $H^1(X, bR)$.
• Define key cohomology classes: $h$ (pullback of a hyperplane class from $bP^N$), $[D_i]$ (classes of exceptional divisors), and use the blow-up formula to relate cohomology of $X$ to that of the ambient space.
• Apply a key lemma showing that $(h + heta [D_5])^{N+1} = 0$ if and only if $ heta = 0$, which allows distinguishing cohomology classes under conjugation.
• Use the conjugation action of $ ext{Aut}(bC)$ on the variety and its cohomology to show that the real cohomology algebra of $X^ au$ differs from that of $X$ when $ au$ swaps the imaginary quadratic fields associated with the CM elliptic curves.

Experimental results

Research questions

• RQ1 Can two conjugate smooth projective varieties over a number field have non-isomorphic real cohomology algebras?
• RQ2 Does the real cohomology algebra of a complex variety detect arithmetic invariants such as the endomorphism ring of an abelian variety?
• RQ3 Is the real cohomology algebra invariant under complex conjugation, despite being invariant under l-adic cohomology and Betti cohomology with ℚ_l coefficients?
• RQ4 Can the cohomology algebra of a blow-up detect the action of complex multiplication endomorphisms on the cohomology of an abelian variety?
• RQ5 Do conjugate varieties with isomorphic l-adic cohomology algebras necessarily have isomorphic real cohomology algebras?

Key findings

• The paper constructs a smooth projective variety $X$ defined over a number field such that its conjugate varieties $X^ au$ under different complex embeddings of the base field have non-isomorphic real cohomology algebras.
• The real cohomology algebra $H^*(X, bR)$ encodes the endomorphism ring of the abelian variety $A = E imes E'$, including the action of complex multiplication endomorphisms $f$ and $f'$, which are not preserved under complex conjugation.
• The key distinguishing feature is the presence of a line in $H^2(X, bR)$ spanned by $[D_5]$, whose interaction with the hyperplane class $h$ leads to a non-trivial relation $(h + heta [D_5])^{N+1} = 0$ only if $ heta = 0$, which fails under conjugation.
• The proof shows that the real cohomology algebra of $X^ au$ is not isomorphic to that of $X$ because the conjugation $ au$ swaps the imaginary quadratic fields $k$ and $k'$, altering the eigenvalues of the endomorphisms $f$ and $f'$, which are encoded in the cohomology.
• This construction provides a positive answer to Grothendieck’s question: there exist conjugate varieties with distinct real cohomology algebras, even though their l-adic cohomology algebras are canonically isomorphic.
• The result implies that the real cohomology algebra is not a functorial invariant under complex conjugation, unlike l-adic cohomology, and thus carries finer arithmetic information than previously thought.