Hill, Penrose, and Sparling have an example of a non-realizable CR structure, a 5-manifold $M^5$ that comes equipped with a "twisted version" of the Lewy operator for the quadric $Q^2$, $v = \frac{1}{2} z\bar{z}$.

In their example, which Sir Penrose briefly explains describe in a beautiful paper from the Poincare symposium from the early 80's, as their input data they take a 3-manifold equipped with a Lewy operator X such that one can find a smooth $g$ for which the PDE $$X(f) = g,$$ has no local solutions.

Penrose talks about exponentiating $g$ to obtain a CR line bundle over M^3 and with CR structure $X + g z \frac{\partial}{\partial \bar{z}}$ and $\frac{\partial}{\partial \bar{z}}$.

Maybe a complex analyst versed in sheaf theory may find it trivial, but my questions are:

1. why can $g$ be seen as a non-vanishing class in $H^{0,1}(M^3)$ (for the $\bar{\partial_b}$ cohomology)?
2. what does it mean to "exponentiate" such class to obtain a cpx. line bundle?

I guess ANY EXPLICIT EXAMPLES that'll bring forward the intuition in both questions are very welcome. And probably I don't quite understand what the operator $\bar{\partial}_b$ is.

Thanks,

Hill, Penrose, and Sparling have an example of a non-realizable CR structure, a 5-manifold $M^5$ that comes equipped with a "twisted version" of the Lewy operator for the quadric $Q^2$, $v = \frac{1}{2} z\bar{z}$.

In their example, which Sir Penrose briefly explains describe in a beautiful paper from the Poincare symposium from the early 80's, as their input data they take a 3-manifold equipped with a Lewy operator X such that one can find a smooth $g$ for which the PDE $$X(f) = g,$$ has no local solutions.

Penrose talks about exponentiating $g$ to obtain a CR line bundle over M^3 and with CR structure $X + g(z,\bar{z},w,\bar{w}) \lambda \frac{\partial}{\partial \lambda}$ and $\frac{\partial}{\partial \bar{\lambda}}$. Remark. $\lambda$ is a complex number, the fiber or ``vertical'' coordinates.

Maybe a complex analyst versed in sheaf theory may find it trivial, but my question is:

what does it mean to "exponentiate" such class to obtain a cpx. line bundle?

I guess ANY EXPLICIT EXAMPLES that'll bring forward the intuition in both questions are very welcome.

Thanks,

Hill, Penrose, and Sparling have an example of a non-realizable CR structure, a 5-manifold $M^5$ that comes equipped with a "twisted version" of the Lewy operator for the quadric $Q^2$, $v = \frac{1}{2} z\bar{z}$.

In their example, which Sir Penrose briefly explains describe in a beautiful paper from the Poincare symposium from the early 80's, as their input data they take a 3-manifold equipped with a Lewy operator X such that one can find a smooth $g$ for which the PDE $$X(f) = g,$$ has no local solutions.

Penrose talks about exponentiating $g$ to obtain a CR line bundle over M^3 and with CR structure $X + g z \frac{\partial}{\partial \bar{z}}$ and $\frac{\partial}{\partial \bar{z}}$.

Maybe a complex analyst versed in sheaf theory may find it trivial, but my questions are:

1. why can $g$ be seen as a non-vanishing class in $H^{0,1}(M^3)$ (for the $\bar{\partial_b}$ cohomology)?
2. what does it mean to "exponentiate" such class to obtain a cps. line bundle?

I guess ANY EXPLICIT EXAMPLES that'll bring forward the intuition in both questions are very welcome. And probably I don't quite understand what the operator $\bar{\partial}_b$ is.

Thanks,

Hill, Penrose, and Sparling have an example of a non-realizable CR structure, a 5-manifold $M^5$ that comes equipped with a "twisted version" of the Lewy operator for the quadric $Q^2$, $v = \frac{1}{2} z\bar{z}$.

In their example, which Sir Penrose briefly explains describe in a beautiful paper from the Poincare symposium from the early 80's, as their input data they take a 3-manifold equipped with a Lewy operator X such that one can find a smooth $g$ for which the PDE $$X(f) = g,$$ has no local solutions.

Penrose talks about exponentiating $g$ to obtain a CR line bundle over M^3 and with CR structure $X + g z \frac{\partial}{\partial \bar{z}}$ and $\frac{\partial}{\partial \bar{z}}$.

Maybe a complex analyst versed in sheaf theory may find it trivial, but my questions are:

1. why can $g$ be seen as a non-vanishing class in $H^{0,1}(M^3)$ (for the $\bar{\partial_b}$ cohomology)?
2. what does it mean to "exponentiate" such class to obtain a cpx. line bundle?

I guess ANY EXPLICIT EXAMPLES that'll bring forward the intuition in both questions are very welcome. And probably I don't quite understand what the operator $\bar{\partial}_b$ is.

Thanks,