Twistor lines in the period domain of complex tori
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Twistor lines in the period domain of complex tori Nikolay Buskin1 · Elham Izadi2 Received: 28 June 2020 / Accepted: 28 August 2020 © Springer Nature B.V. 2020
Abstract As in the case of irreducible holomorphic symplectic manifolds, the period domain Compl of compact complex tori of even dimension 2n contains twistor lines. These are special 2spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a generic chain of twistor lines. We also prove a criterion of twistor path connectivity of loci in Compl where a fixed second cohomology class stays of Hodge type (1,1). Furthermore, we show that twistor lines are holomorphic submanifolds of Compl, of degree 2n in the Plücker embedding of Compl. Keywords Complex tori · Hyperkähler manifolds · Twistor lines · Twistor paths · Twistor path connectivity
1 Introduction Let M be a Riemannian manifold of real dimension 4m with metric g. Then M is called hyperkähler with respect to g (see [7, p. 548]) if there exist complex structures I , J and K on M, such that I , J , K are covariantly constant and are isometries of the tangent bundle T M with respect to g, satisfying the quaternionic relations I 2 = J 2 = K 2 = −1,
I J = −J I = K .
We call the ordered triple (I , J , K ) a hyperkähler structure on M compatible with g. A hyperkähler structure (I , J , K ) gives rise to a sphere S 2 of complex structures on M: S 2 = {a I + b J + cK |a 2 + b2 + c2 = 1}.
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Elham Izadi [email protected] Nikolay Buskin [email protected]
1
Department of Mechanics and Mathematics, Novosibirsk State University, 1 Pirogova st., Novosibirsk 630090, Russia
2
Department of Mathematics, University of California San Diego, 9500 Gilman Drive # 0112, La Jolla, CA 92093-0112, USA
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Geometriae Dedicata
We call the family M = {(M, λ)|λ ∈ S 2 } → S 2 a twistor family over the twistor sphere The family M can be endowed with a complex structure, so that it becomes a complex manifold and the fiber Mλ is biholomorphic to the complex manifold (M, λ), see [7, p. 554]. For every λ = a I + b J + cK ∈ S 2 , the closed alternating form g(λ·, ·) determines a Kähler class in H 1,1 ((M, λ), R). The known examples of compact hyperkähler manifolds are even-dimensional complex tori and irreducible holomorphic symplectic manifolds (IHS manifolds). We recall that an IHS manifold is a simply connected compact Kähler manifold M with H 0 (M, 2M ) generated by an everywhere non-degenerate holomorphic 2-form. Examples of IHS manifolds include K 3 surfaces and, more generally, Hilbert schemes of points on K 3 surfaces, generalized Kummer varieties. For IHS manifolds and complex tori there exist well-defined period domains, carrying the structure of a complex manifold. Every twistor family M determines an embedding of the base S 2 into the corresponding period domain as a 1-dimensional complex submanifold (for IHS manifolds this is k
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