A generalization of twistor lines for complex tori

Abstract

In this work we generalize the classical notion of a twistor line in the period domain of compact complex tori studied in [1]. We introduce two new types of lines, which are non-compact analytic curves in the period domain. We study the analytic properties of the compactifications of the curves, preservation of cohomology classes of type (1,1) along them and prove the twistor path connectivity of the period domain by the curves of one of the new types.

Twistor triangles in the period domain of complex tori

“…Choose I 1 ∈ S 1 ∩ S 3 and I 2 ∈ S 1 ∩ S 2 . As we know from [4] or [5], the G-action stabilizer G I ⊂ G of I ∈ Compl acts transitively on the set of twistor spheres containing I, so that we can find elements g 2 ∈ G I 2 such that S 2 = g 2 (S 1 ) and g 1 ∈ G I 1 such that S 3 = g −1 1 (S 1 ). Next, choose J 3 ∈ S 2 ∩ S 3 and set…”

“…It is easy to see, and this is explained in [5] that two non-proportional complex structures J 1 , J 2 belong to the same, uniquely defined, twistor sphere S if and only if J 1 J 2 + J 2 J 1 = 2αId for some α ∈ R such that |α| < 1 (such J 1 and J 2 generate the subalgebra H ⊂ End V R associated with S). This fact provides a natural generalization of the notion of a twistor sphere, namely, if J 1 J 2 + J 2 J 1 = 2αId for some general α ∈ R and J 1 = ±J 2 , then there is a canonically defined complex-analytic curve S(J 1 , J 2 ) in Compl containing ±J 1 , ±J 2 , it is the intersection of the subalgebra in End V R , generated by J 1 , J 2 with Compl ⊂ End V R .…”

“…If we do not specify whether a twistor line is compact or not, we can talk about it as a generalized twistor line. The geometry of such curves is studied in [5], where it is shown, in particular, that the (analytic or Zariski topology) closures of non-compact twistor lines in Gr(2n, V C ) ⊃ Compl are P 1 's.…”

“…For further discussion of twistor lines and the configurations of those we need the following lemma, which summarizes technical results proved in [4], see also [5].…”