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- Tags: - #geomtop/symplectic-topology - Refs: - #todo/add-references - Links: - To Review/2021-04-28_More_Weinstein_Notes - Weinstein

Stein Manifolds

Stein

Moral: rigid, complex-analytic.

Very rigid: uncountably many distinct biholomorphism Stein manifolds that are smooth \({\varepsilon}{\hbox{-}}\)small perturbations of \(B^n_{\mathbf{C}}\). 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 \({\mathbf{C}}^N\) (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 \(f({{\partial}}X)=f(X) \cap{{\partial}}Y\) and \(f(X) \pitchfork{{\partial}}Y\).

Examples:

• Any complex projective manifold \(X\subset {\mathbf{CP}}^N\),

  • I.e. a manifold that is a projective variety; locus of polynomial equations in \({\mathbf{P}}^n_{\mathbf{C}}\).
  • Any algebraic variety over \(k = {\mathbf{C}}\) 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 \(\phi:M\to {\mathbf{R}}\)
• Associate the 1-form \(d^{\mathbf{C}}\phi \coloneqq d\phi \circ J\).
• Associate the 2-form \(\omega_\phi \coloneqq-dd^{\mathbf{C}}\phi\).
• Suppose \(\phi\) is \(J{\hbox{-}}\)convex if the function \(g_\phi(v, w) \coloneqq\omega_\phi(v, Jw)\) defines a Riemannian metric
• Then \(\omega_\phi\) is a symplectic form compatible with \(J\), i.e. \(H_\phi \coloneqq g_\phi - i\omega_\phi\) 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 \({\partial}, \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu\)) is positive semidefinite. This is an equivalent condition.

The subspace of J-convex functions in \(C^\infty(M, {\mathbf{R}})\) 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 \(J{\hbox{-}}\)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 \(M_c\coloneqq f^{-1}(c)\) yield contact structures inducing orientations on \(M_x\) agreeing with the induced boundary orientation on \(f^{-1}(-\infty, c)\).

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