---
date: 2022-04-05 23:42
modification date: Sunday 10th April 2022 00:57:15
title: "Stein"
aliases: [Stein manifolds]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #geomtop/symplectic-topology 
- Refs:
	- #todo/add-references
- Links:
	- [To Review/2021-04-28_More_Weinstein_Notes](To%20Review/2021-04-28_More_Weinstein_Notes.md)
	- [Weinstein](Unsorted/Weinstein.md)

---

# Stein Manifolds


## Stein

Moral: rigid, complex-analytic.

> Very rigid: uncountably many distinct biholomorphism Stein manifolds that are smooth $\eps\dash$small perturbations of $B^n_\CC$. So we study them up to deformation of the manifold, i.e. homotopy of the space of structures.

**Definition (a)**:
$M^{2n}$ complex-analytic, *properly embedded* in some $\CC^N$ (biholomorphically, can take $N = 2n+1$) such that complex structure is inherited from ambient space.

Data: $M$ and $J$ an integrable complex structure.

> Note: *properly embedded* here seems to mean $f:X\to Y$ where $f(\bd X)=f(X) \intersect \bd Y$ and $f(X) \transverse \bd Y$.

**Examples**:

- Any complex projective manifold $X\subset \CP^N$,
  - I.e. a manifold that is a projective variety; locus of polynomial equations in $\PP^n_\CC$.
  - Any algebraic variety over $k = \CC$ is (essentially) birationally equivalent to such a manifold.

- Any connected non-compact Riemann surface (or closed with a puncture).

- Any smooth compact $2n$ dimensional manifold with $n>2$ and handles of index $\leq n$.
  - $n=2$ case works with modification
  - Every smooth $4$ manifold admits a bisection into two Stein 4-manifolds.

**Why useful**:

- Supposed to be an analog of affine varieties (as per Wikipedia, but should probably be quasi-projective).
- Every Stein manifold is Kahler (compatible complex + Riemannian + symplectic structures), large class interesting to AG
- Amenable to Hodge Theory
- Homotopy types of CW complexes (admits a homotopy equivalence, as do all manifolds)


**Definition (b)**:

Consider $(M^{2n}, J)$ where $M$ is a complex manifold and $J$ the structure of complex multiplication on $T_p M$.

- Pick a smooth functional $\phi:M\to \RR$
- Associate the 1-form $d^\CC \phi \definedas d\phi \circ J$.
- Associate the 2-form $\omega_\phi \definedas -dd^\CC \phi$.
- Suppose $\phi$ is $J\dash$convex if the function $g_\phi(v, w) \definedas \omega_\phi(v, Jw)$ defines a Riemannian metric
- Then $\omega_\phi$ is a symplectic form compatible with $J$, i.e. $H_\phi \definedas g_\phi - i\omega_\phi$ is a Hermitian metric
- Suppose $\phi$ is *exhausting*, i.e. preimages of compact sets are compact and $\phi$ is bounded from below (?)

> Note on exhausting J-convex functions: origins seem to be in analysis of multiple complex variables. In nicest cases, boils down to the "Levi matrix" (analog of Hessian for $\del, \bar \del$) is positive semidefinite. This is an equivalent condition.

> The subspace of J-convex functions in $C^\infty(M, \RR)$ is open and contractible, so well-approximated by Morse functions (and the bigger class of *generalized Morse functions*: nondegenerate, restricted critical points).


Theorem (Grauert, Bishop-Narasimhan)
: $M$ is Stein iff it fits this description.

So a Stein structure is a pair $(J, \phi)$, a complex structure and a $J\dash$convex exhausting Morse function.

Theorem
: If $n=2$, $M$ admits a Morse function $f$ such that away from critical points, taking complex tangencies at the preimages $M_c\definedas f\inv(c)$ yield contact structures inducing orientations on $M_x$ agreeing with the induced boundary orientation on $f\inv(-\infty, c)$.

> A type of filling? Etnyre seems to work on this kind of thing.