Voisin’s conjecture for zero-cycles on Calabi-Yau varieties and their mirrors. (English) Zbl 1437.14015
This paper studies a conjecture of C. Voisin [Chow rings, decomposition of the diagonal, and the topology of families. Princeton, NJ: Princeton University Press (2014; Zbl 1288.14001)], which is formulated as follows. For a smooth projective complex variety \(X\), let \(A^j(X)\) denote the Chow group of codimension \(j\) algebraic cycles on \(X\) modulo rational equivalence.
Conjecture. (Conjeture 4.37 in [C. Voisin, Chow rings, decomposition of the diagonal, and the topology of families. Princeton, NJ: Princeton University Press (2014; Zbl 1288.14001)]): Let \(X\) be a smooth projective complex variety of dimension \(n\) with \(j^{i,0}(X)=0\) for all \(0<j<n\). Then the following statements are equivalent:
(i) For any zero-cycle \(a, a^{\prime}\in A^n(X)\) of degree zero, \(a\times a^{\prime}=(-1)^n a^{\prime}\times a\in A^{2n}(X)\). Here \(a\times a^{\prime}\) denotes the cycle class \(p_1^*(a)\cdot p_2^*(a^{\prime})\in A^{2n}(X\times X)\) where \(p_i^*\) is the projection to the \(i\)-th compoments for \(i-1,2\),
(ii) the geometric genus \(p_g(X)\leq 1\).
The implication \(i)\Rightarrow (ii)\) is a theorem, while the other implication is the conjecture.
This notes presents a general critrion on the validity of the conjecture for some specific varieties.
The main result and its corollaries are formulated as follows.
Theorem. Let \(X\) be a smooth projective complex variety of dimension \(n\leq 5\) with \(h^{i,0}(X)=0\) for \(0 < i < n\), and \(p_g(X)=1\). Assume further that
(1) \(X\) is rationally dominated by a variety \(X^{\prime}\) of dimension \(n\) such that \(X^{\prime}\) has finite dimensional motive and that the Lefschetz standard conjecture \(B(X^{\prime})\) is true.
(2) \(X\) is \(\widetilde{N}_1^{\prime}\)-maximal: (for the definition, see Proposition 3.3).
(3) \(X\) is rationally dominated by a variety \(X^{\prime\prime}\) of dimension \(n\) and the Hodge conjecture is true for \(X^{\prime\prime}\times X^{\prime\prime}\).
Then the Voisin conjecture is true for \(X\), that is, any \(a\times a^{\prime}\in A^n_{\mathrm{hom}}(X)\) satisfies \(a\times a^{\prime}=(-1)^na^{\prime}\times a\in A^{2n}(X\times X)\).
Corollaries: (1) Let \(X\subset\mathbb{P}^5(\mathbb{C})\) be the sextic Fermat fourfold \(X_0^6+\cdots+X_5^6=0\). Then the Voisin conjecture is true for \(X\).
(2) Let \(X\subset\mathbb{P}^4(1^4,2)\) be the Calabi-Yau threefold defined as \(X_0^6+X_1^6+X_2^6+X_3^6+X_4^3=0\). Then The Voisin conjecture holds for \(X\).
Proof rests on results of C. Vial [Proc. Lond. Math. Soc. (3) 106, No. 2, 410–444 (2013; Zbl 1271.14010)] on refined Künneth decomposition, and refined Chow-Künneth decomposition (under the extra assupmtion of the existence of a finite-dimensional motive on \(X\)).
Conjecture. (Conjeture 4.37 in [C. Voisin, Chow rings, decomposition of the diagonal, and the topology of families. Princeton, NJ: Princeton University Press (2014; Zbl 1288.14001)]): Let \(X\) be a smooth projective complex variety of dimension \(n\) with \(j^{i,0}(X)=0\) for all \(0<j<n\). Then the following statements are equivalent:
(i) For any zero-cycle \(a, a^{\prime}\in A^n(X)\) of degree zero, \(a\times a^{\prime}=(-1)^n a^{\prime}\times a\in A^{2n}(X)\). Here \(a\times a^{\prime}\) denotes the cycle class \(p_1^*(a)\cdot p_2^*(a^{\prime})\in A^{2n}(X\times X)\) where \(p_i^*\) is the projection to the \(i\)-th compoments for \(i-1,2\),
(ii) the geometric genus \(p_g(X)\leq 1\).
The implication \(i)\Rightarrow (ii)\) is a theorem, while the other implication is the conjecture.
This notes presents a general critrion on the validity of the conjecture for some specific varieties.
The main result and its corollaries are formulated as follows.
Theorem. Let \(X\) be a smooth projective complex variety of dimension \(n\leq 5\) with \(h^{i,0}(X)=0\) for \(0 < i < n\), and \(p_g(X)=1\). Assume further that
(1) \(X\) is rationally dominated by a variety \(X^{\prime}\) of dimension \(n\) such that \(X^{\prime}\) has finite dimensional motive and that the Lefschetz standard conjecture \(B(X^{\prime})\) is true.
(2) \(X\) is \(\widetilde{N}_1^{\prime}\)-maximal: (for the definition, see Proposition 3.3).
(3) \(X\) is rationally dominated by a variety \(X^{\prime\prime}\) of dimension \(n\) and the Hodge conjecture is true for \(X^{\prime\prime}\times X^{\prime\prime}\).
Then the Voisin conjecture is true for \(X\), that is, any \(a\times a^{\prime}\in A^n_{\mathrm{hom}}(X)\) satisfies \(a\times a^{\prime}=(-1)^na^{\prime}\times a\in A^{2n}(X\times X)\).
Corollaries: (1) Let \(X\subset\mathbb{P}^5(\mathbb{C})\) be the sextic Fermat fourfold \(X_0^6+\cdots+X_5^6=0\). Then the Voisin conjecture is true for \(X\).
(2) Let \(X\subset\mathbb{P}^4(1^4,2)\) be the Calabi-Yau threefold defined as \(X_0^6+X_1^6+X_2^6+X_3^6+X_4^3=0\). Then The Voisin conjecture holds for \(X\).
Proof rests on results of C. Vial [Proc. Lond. Math. Soc. (3) 106, No. 2, 410–444 (2013; Zbl 1271.14010)] on refined Künneth decomposition, and refined Chow-Künneth decomposition (under the extra assupmtion of the existence of a finite-dimensional motive on \(X\)).
Reviewer: Noriko Yui (Kingston)
MSC:
| 14C15 | (Equivariant) Chow groups and rings; motives |
| 14C25 | Algebraic cycles |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
Keywords:
algebraic cycles;Chow groups; motives; Voisin’s conjecture; Calabi-Yau varieties of dimension at most 5References:
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