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## Degenerate twistor spaces for hyperkahler manifolds

Journal of Geometry and Physics. 2015. Vol. 91. P. 2-11.

Let M be a hyperkähler manifold, and η a closed, positive (1, 1)-form with rkη<dimM. We associate to η a family of complex structures on M, called a degenerate twistor family, and parametrized by a complex line. When η is a pullback of a Kähler form under a Lagrangian fibration L, all the fibers of degenerate twistor family also admit a Lagrangian fibration, with the fibers isomorphic to that of L. Degenerate twistor families can be obtained by taking limits of twistor families, as one of the Kähler forms in the hyperkähler triple goes to η.