On the infinitesimal Torelli problem of elliptic surfaces
The author studies the injectivity of the map \(\delta:H^ 1(X,\Theta)\to Hom_{{\mathbb{C}}}(H^{2,0}(X),H^{1,1}(X))\) given by the cup-product \(H^ 0(X,\Omega^ 2)\otimes H^ 1(X,\Theta)\to H^ 1(X,\Omega^ 1)\) deduced from the contraction \(\Theta \otimes \Omega^ 2\to \Omega^ 1\) for an elliptic surface \(\phi\) :\(X\to C\) with base curve C. He proves that: \(\delta\) is injective if \(h^{2,0}(X)>O\) and one of the following three conditions holds: (i) the functional invariant J(X) is not constant; (ii) J(X) is constant, distinct from O and 1, and either C is \({\mathbb{P}}_ 1({\mathbb{C}})\) or \((iii)\quad \chi(X,{\mathcal O}_ X))\geq 3.\) If \(\phi\) :\(X\to C\) is an elliptic bundle, then: (i) \(\delta\) is injective if \(g(C)=h^{1,0}(C)=1\), (ii) if \(b_ 1(X)\) is even and g(C)\(\geq 2\), \(\delta\) is not injective iff \(g(C)>2\) and C is hyperelliptic; (iii) if \(b_ 1(X)\) is odd and g(C)\(\geq 2\), \(\delta\) is never injective. - For Kodaira surfaces, the global Torelli does not hold, although the map \(\delta\) is injective. Editorial remark: \textit{A. Ikeda} constructed in [Adv. Math. 349, 125--161 (2019; Zbl 1414.14004)] an elliptic surface $Y$ and showed that the period map for the Hodge structure on $H^2(Y,\mathbb{Z})$ has one dimensional fibers and the general fiber has two irreducible components, which contradicts the main theorem of this work.
scientific article; zbMATH DE number 3843907
scientific article