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Rational Hodge isometries of hyper-Kähler varieties of
$K3^{[n]}$ type are algebraic
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 6 / June 2024
- Published online by Cambridge University Press:
- 07 May 2024, pp. 1261-1303
- Print publication:
- June 2024
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- Article
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Let
$X$ and 
$Y$ be compact hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of length 
$n$ subschemes of a 
$K3$ surface. A class in 
$H^{p,p}(X\times Y,{\mathbb {Q}})$ is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let 
$f:H^2(X,{\mathbb {Q}})\rightarrow H^2(Y,{\mathbb {Q}})$ be a rational Hodge isometry with respect to the Beauville–Bogomolov–Fujiki pairings. We prove that 
$f$ is induced by an analytic correspondence. We furthermore lift 
$f$ to an analytic correspondence 
$\tilde {f}: H^*(X,{\mathbb {Q}})[2n]\rightarrow H^*(Y,{\mathbb {Q}})[2n]$, which is a Hodge isometry with respect to the Mukai pairings and which preserves the gradings up to sign. When 
$X$ and 
$Y$ are projective, the correspondences 
$f$ and 
$\tilde {f}$ are algebraic.
