I have a Lefschetz thimble defined by the stable flow of the gradient a holomorphic function toward a critical point (as defined e.g. in Witten arXiv:1001.2933 and F.Pham "Vanishing homologies and the n variable saddlepoint method").
Is such thimble a holomorphic manifold? Maybe under more restrictive assumptions?
If by holomorphic manifold you mean that it happens to be a complex manifold then the answer is surely "not always", because in the case when the total space is $\mathbf{C}^3$ and the function is $(z_1,z_2,z_3)\mapsto z_1^2+z_2^2+z_3^2$ then the thimble living over the positive real axis will be $\mathbf{R}^3$ (which is odd-dimensional).
If you mean "is it a complex submanifold of the ambient space?" then the answer is "no". In general (at least in the nondegenerate finite-dimensional setting which I understand, where the critical points are Morse) the thimble is a Lagrangian submanifold diffeomorphic to a disc. As such it doesn't inherit a natural complex structure from the ambient space precisely because hitting a tangent vector to the thimble with $i$ sends it to the orthogonal complement of the thimble.
Maybe there's something about this Witten setting which is different and which I don't understand.
Given the Witten citation, you might be interested in the work of Andriy Haydys.
