Adapted complex structures, also known as Grauert tubes, first arose in the early 1990s in connection with the homogeneous complex Monge-Ampère equation, as complex structures on the tangent or cotangent bundle of a Riemannian manifold M (or on certain subsets of these bundles). They also arise in Lie theory, geometric quantization and in the study of semiclassical pseudodifferential operators.

Adapted complex structures can be viewed as complex structures on the phase space of M that are compatible with the natural symmetries of phase space, and this is the approach I will take in my lectures. I will discuss various definitions of these structures and their fundamental properties, such as existence, uniqueness, regularity; also the connection with the Monge-Ampère equation and with curvature estimates for the metric of M.

Adapted complex structures, also known as Grauert tubes, first arose in the early 1990s in connection with the homogeneous complex Monge-Ampère equation, as complex structures on the tangent or cotangent bundle of a Riemannian manifold M (or on certain subsets of these bundles). They also arise in Lie theory, geometric quantization and in the study of semiclassical pseudodifferential operators.

Adapted complex structures can be viewed as complex structures on the phase space of M that are compatible with the natural symmetries of phase space, and this is the approach I will take in my lectures. I will discuss various definitions of these structures and their fundamental properties, such as existence, uniqueness, regularity; also the connection with the Monge-Ampère equation and with curvature estimates for the metric of M.

I will consider complex manifolds equipped with a plurisubharmonic exhaustion satisfying the top dimensional complex degenerate Monge-Ampère equation outside its minimal set. The minimal set is always small, and in the usual examples it is either a complex submanifold (Stoll's parabolic manifolds) or a maximal dimensional totally real submanifold (Grauert Tubes). The different nature of the minimal set largely determines properties of the ambient manifold, as rigidity properties and deformability features. This, in turn, has consequences in classification and/or characterization results. After a discussion of these aspects, I will outline a class of examples arising in the theory of almost homogeneous manifolds which naturally includes classical examples along with new ones with minimal set of "mixed nature".

I will describe joint work with Will Kirwin on using "imaginary time flows" to construct complex structures on cotangent bundles. First, I will discuss the geodesic flow. If one starts with the vertical polarization on a cotangent bundle and pushes forward by the "imaginary time geodesic flow", the result is a complex polarization, which turns out to be the adapted complex structure of Lempert-Szőke and Guillemin-Stenzel. If one instead uses a magnetic flow—describing the motion of a charged particle in the presence of a magnetic field—the resulting complex structure is new. I will describe several different ways of interpreting the imaginary-time flow and will discuss two concrete examples.