Isogenies between algebraic surfaces with geometric genus one (Q581493)

If X is a complex projective variety, X and its Albanese variety Alb(X) have isomorphic first cohomology groups and this isomorphism induces a functorial mapping \(X\to Alb(X)\). An analogous theory for second cohomology groups is developed for algebraic surfaces of geometric genus one over the complex numbers. Definition. Let X and Y be algebraic surfaces over \({\mathbb{C}}.\) (a) A \textit{cohomological isogeny} between X and Y is an isomorphism of rational Hodge structures \(T(X,{\mathbb{Q}})\to T(Y,{\mathbb{Q}})\) where for any algebraic surface Z, \(T(Z,{\mathbb{Z}})=H^ 2(Z,{\mathbb{Z}})/NS(Z)\) denotes the transcendental lattice and \(T(Z,{\mathbb{Q}})=T(Z,{\mathbb{Z}})\otimes {\mathbb{Q}}.\) (b) An \textit{isogeny} between X and Y is an irreducible algebraic cycle such that \([W]_{trans}: T(X,{\mathbb{Q}})\to T(Y,{\mathbb{Q}})\) is a cohomological isogeny. (c) An isogeny or a cohomological isogeny is said to be \textit{strict} if it maps the intersection form on T(X,\({\mathbb{Q}})\) to the intersection form on T(Y,\({\mathbb{Q}})\), and it is said to be \textit{integral} if it is compatible with an isomorphism of integral Hodge structures T(X,\({\mathbb{Z}})\to T(Y,{\mathbb{Z}}).\) The main result of the paper gives the ``classification'' of algebraic surfaces of genus one up to isogeny: Theorem. Let X be an algebraic surface with geometric genus one. Then the following hold true: There exists an algebraic K3 surface Y and a strict integral cohomological isogeny between X and Y. Such a Y is called an associated K3 surface of X. (b) If the minimal model of X is neither a K3 surface nor a logarithmic transform of an elliptic K3 surface, then any two associated K3 surfaces of X are isomorphic. - The proof of the theorem rests on the analysis of bilinear forms on \(T(X,{\mathbb{Z}}).\) Combining the above theorem with a result of \textit{Mukai} [``On the moduli space of bundles on K3 surfaces, I'', Proc. Symp. Vector Bundles (Tata Institute 1984)], the following corollary is obtained: Corollary. Let X be an algebraic surface with geometric genus one. Then the following hold true: (i) There is a unique strict integral isogeny class of associated K3 surfaces of X. (ii) If \(Y_ 1\) and \(Y_ 2\) are associated K3 surfaces of X and there exists a strict integral isogeny between X and \(Y_ 1\), then there exists a strict integral isogeny between X and \(Y_ 2\). In other words, if the isomorphism between second transcendental cohomology groups preserve both the intersection pairings and the integral Hodge structures, then most algebraic surfaces with geometric genus one have a unique associated K3 surface.