Topological characterisation of affine \( \mathbb{C} \)-schemes (Stein manifolds)

1. The statement of Cieliebak-Eliashberg about topology of Stein manifold.
2. Morse theory and strategy of the proof.
3. \( J \)-convex functions
4. Extension of complex structure.
5. Extension of \( J \)-convex function.

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This is the note of my presentation as final exam of the course Introduction au h-principe de Gromov given by Patrick Massot at Departement de Mathématiques d'Orsay. I presented how Cieliebak and Eliashberg cieliebak_stein_2013 applied h-principle to characterise the topology of Stein manifold. Further discussion can be found in the book eliashberg_stein_2012.

1 The statement of Cieliebak-Eliashberg about topology of Stein manifold.

Stein manifolds are affine schemes over \( \mathbb{C} \). There are many equivalent definitions of Stein manifolds, but we will use the following two:

They are complex manifolds that can be properly embedded into \( \mathbb{C}^N \) for some \( N\in \mathbb{Z}_{>0} \).
They are complex manifolds that admit an exhaustive, strictly plurisubharmonic (PSH) function \( \phi \). The exhaustive part means that \( \phi \) is a proper, real-value function bounded below, and the strictly plurisubharmonic part means that \( -dd^c \phi(v,Jv) > 0 \) for all tangent vector \( v \), where \( d^c\phi = d\phi\circ J \).

Strictly PSH condition locally reads: \( \frac{\partial^2 \phi}{\partial z^i \bar\partial z^j}\xi^i\bar\xi^j >0 \), this means that the restriction of \( \phi \) on each complex line is subharmonic, i.e. it satisfies the sub-mean property.

Since we will frequently change the complex structure on manifold, it is better to make it appear explicitly in the notation of plurisubharmonicity. So instead of saying a function \( \phi \) is (strictly) PSH with respect to the complex structure \( J \), we will say that \( \phi \) is \( J \)-convex.

Here are a few example of Stein manifolds:

The complex affine spaces \( \mathbb{C}^n \), a sub complex manifold of \( \mathbb{C}^N \).
Let \( X \) be a closed sub complex manifold of \( \mathbb{P}^N_{\mathbb{C}} \) and \( H \subset \mathbb{P}^N_{\mathbb{C}}\) be a complex hyperplan. Then \( X\setminus H \hookrightarrow \mathbb{P}^N_{\mathbb{C}}\setminus H = \mathbb{C}^N \) is a Stein manifold.

Our goal will be to answer the following question:

Question. Topologically, what are Stein manifolds.

In other words, we want to find a necessary and sufficient condition of a differential manifold \( V \) (without boundary) such that we can equip \( V \) with a complex structure \( J \) that makes it a Stein manifold. It is easy to see that the following two conditions are necessary:

\( V \) admits an almost complex structure.
\( V \) is an open manifold, i.e. there is no compact connected component of \( V \). This is an immediate consequence of Maximum modulus principle for holomorphic functions.

There is another less obvious necessary condition:

Let \( (V,J,\phi) \) be a Stein manifold. By a generic perturbation we can suppose that \( \phi \) is a Morse function (the Hessian of \( \phi \) is non-degenerate at the critical points of \( \phi \)). Then the index of any critical point \( p \) of \( \phi \) is less than \( n = \dim_\mathbb{C}V \).

The remark follows from a kind of Pigeonhole principle. Suppose that the index of \( p \) is strictly bigger than \( n \), then there exists a complex dimension in \( T_pV \) where the Hessian of \( \phi \) is definite negative, this means that there is a complex curve \( C \) passing by \( p \) on which the function \( \phi \) admits a local maximum at \( p \). This contradicts the pluriharmonicity.

It was proved by Cieliebak and Eliashberg that these conditions are sufficient.

Let \( V \) be a smooth manifold of real dimension \( 2n > 4 \) and \( J \) be an almost complex structure on \( V \) and \( \phi \) be an exhaustive Morse function without any critical points of index \( k > n \). Then there exists an integrable complex structure \( \tilde J \) such that \( (V,\tilde J) \) is a Stein manifold.

In fact, the authors proved that one can obtain \( \tilde J \) by a homotopy of \( J \) and the function \( \phi \) is \( \tilde J \)-convex. In the next part, we will not only modify the complex structure \( J \), but also modify the function \( \phi \) to a PSH function.

2 Morse theory and strategy of the proof.

3 \( J \)-convex functions

4 Extension of complex structure.

5 Extension of \( J \)-convex function.

Bibliography

[cieliebak_stein_2013] Cieliebak & Eliashberg, Stein structures: existence and flexibility, arXiv:1305.1619 [math], , (2013). link.
[eliashberg_stein_2012] Cieliebak & Eliashberg, From Stein to Weinstein and back : symplectic geometry of affine complex manifolds, Providence, R.I.: American Mathematical Society (2012).