Pseudo-convex and pseudo-concave

Properties of domains in complex spaces, as well as of complex spaces and functions on them, analogous to convexity and concavity properties of domains and functions in the space $ \mathbf R ^ {n} $. A real-valued function $ \phi $ of class $ C ^ {2} $ on an open set $ U \subset \mathbf C ^ {n} $ is called $ p $- pseudo-convex (or $ p $- convex) if the Hermitian form

$$ H ( \phi ) = \sum _ { j,k } \frac{\partial ^ {2} \phi }{\partial z _ {j} \partial \overline{z}\; _ {k} } u _ {j} \overline{u}\; _ {k} $$

has, at each point of $ U $, at least $ n - p + 1 $ non-negative eigen values. If $ H ( \phi ) $ has at least $ n - p + 1 $ positive eigen values, then one says that $ \phi $ is strictly (or strongly) $ p $- pseudo-convex. In particular, a (strictly) $ 1 $- pseudo-convex function is a (strictly) plurisubharmonic function of class $ C ^ {2} $. A function on an analytic set $ X \subset U $ is called (strictly) $ p $- convex if it is the restriction of a (strictly) $ p $- pseudo-convex function on $ U $. Finally, a (strictly) $ p $- convex function on an arbitrary complex space $ X $ is a continuous function on $ X $ that is, in a neighbourhood of each point, a (strictly) convex function on the corresponding model (cf. Analytic space).

A complex space $ X $ is called $ ( p, q) $- convex-concave if there is a continuous function $ \phi : X \rightarrow \mathbf R $ and two numbers $ d _ {0} , c _ {0} $, $ - \infty \leq d _ {0} \leq c _ {0} \leq \infty $, such that for any $ c \geq c _ {0} $ and $ d \leq d _ {0} $ the set

$$ X _ {c,d} = \ \{ {x \in X } : {d < \phi ( x) < c } \} $$

is relatively compact in $ X $, while $ \phi $ is strictly $ p $- convex on $ X _ {\infty , c _ {0} } $ and strictly $ q $- concave on $ X _ {d _ {0} , \infty } $. If $ d _ {0} = - \infty $ or $ c _ {0} = \infty $, then $ X $ is called strongly $ p $- pseudo-convex or strongly $ q $- pseudo-concave, respectively. If $ d _ {0} = c _ {0} = - \infty $, then $ X $ is called $ p $- complete.

Examples.

1) An open set $ X $ with smooth boundary $ \partial X $ in a complex manifold $ M $ is called strongly $ p $- pseudo-convex (strongly $ p $- pseudo-concave) if every point $ x _ {0} \in \partial X $ has a neighbourhood $ U $ in which there is a strictly $ p $- pseudo-convex (strictly $ p $- pseudo-concave) function $ \phi $ such that $ X \cap U = \{ {x \in U } : {\phi ( x) < 0 } \} $( respectively, $ X \cap U = \{ {x \in U } : {\phi ( x) > 0 } \} $). Every strongly $ p $- pseudo-convex (strongly $ p $- pseudo-concave) relatively-compact open set is a strongly $ p $- convex (strongly $ p $- concave) manifold. If certain components of the boundary $ \partial X $ satisfy the $ p $- pseudo-convexity condition, while the remaining satisfy the $ q $- pseudo-concavity condition, then one obtains an example of a $ ( p , q ) $- convex-concave manifold.

2) Compact complex spaces are naturally considered to be $ 0 $- convex.

3) The class of $ 1 $- complete spaces coincides with the class of Stein spaces (cf. Stein space).

4) The class of strongly $ 1 $- convex spaces coincides with the class of spaces obtained from Stein spaces by proper modifications at a finite set of points.

5) Let $ X $ be a compact complex manifold of dimension $ n $, and let $ S $ be a closed submanifold of it all components of which have dimension $ q $. Then $ X \setminus S $ is strongly $ ( q + 1 ) $- concave, and if the normal bundle over $ S $ is positive, then $ X \setminus S $ is a strongly $ ( n - q ) $- convex space.

6) If $ S $ is a closed submanifold of codimension $ p $ in a Stein manifold $ X $, then $ X \setminus S $ is $ p $- complete.

7) A holomorphic vector bundle $ E $ of rank $ r $ over a manifold $ X $ is called $ p $- positive ( $ q $- negative) if there is a fibre-wise Hermitian metric $ h $ on $ E $ such that $ \chi ( \nu ) = - h ( \nu , \nu ) $ is a strictly $ ( p + r ) $- convex (respectively, $ - \chi $ is a strictly $ q $- convex) function on $ E $ outside the zero section (if $ p = q = 1 $ one obtains the concepts of a positive vector bundle and a negative vector bundle). If $ X $ is compact, then the space of a $ p $- positive vector bundle $ E $ is strongly $ ( p + r ) $- concave, while the space of a $ q $- negative vector bundle is strongly $ q $- convex. The space of a holomorphic vector bundle over a $ p $- complete space is always $ p $- complete.

For $ ( p , q ) $- convex-concave spaces one has proved theorems on the finite dimensionality and separability of certain cohomology spaces with values in coherent analytic sheaves (cf. Finiteness theorems in the theory of analytic spaces). Analogous finiteness theorems have been proved for strictly $ ( p , q ) $- convex-concave mappings (cf. [1], [2]). A space $ X $ is strongly $ 1 $- convex if and only if $ \mathop{\rm dim} H ^ {r} ( X , F ) < \infty $ for all $ r $ and any coherent analytic sheaf $ F $ on $ X $. If $ X $ is $ p $- complete, then $ H ^ {r} ( X , F ) = 0 $ for all $ r \geq p $ and any coherent analytic sheaf $ F $ on $ X $.

The homology groups of $ p $- convex and $ p $- complete spaces have the following properties. If $ X $ is an $ n $- dimensional reduced strongly $ p $- convex ( $ p $- complete) complex space, then $ \mathop{\rm dim} H _ {r} ( X , \mathbf C ) < \infty $( respectively, $ H _ {r} ( X , \mathbf C ) = 0 $) for $ r \geq n + p $. For strongly $ 1 $- convex spaces it is also known that $ H _ {r} ( X , \mathbf Z ) $ is a finitely-generated group for $ r \geq n + 1 $, while for $ p $- complete manifolds it is known that $ H _ {r} ( X , \mathbf Z ) = 0 $ for $ r \geq n + p $ and that $ H _ {n+} p- 1 ( X , \mathbf Z ) $ is free.

A complex space $ X $ is called pseudo-concave if there is a relatively-compact open set $ U $ in $ X $, intersecting each non-degenerate component of $ X $ and satisfying the following condition: Any point $ x _ {0} \in \partial U $ has a neighbourhood $ V $ in $ X $ such that for any $ x \in V $ sufficiently close to $ x _ {0} $,

$$ | f ( x) | \leq \sup _ {y \in V \cap U } | f ( y) | $$

for all holomorphic functions $ f $ in $ V $. If $ X $ is an $ n $- dimensional manifold, $ n \geq 2 $, then it is sufficient that $ U $ is a strongly $ ( n - 1 ) $- pseudo-concave set in $ X $. Any compact space is pseudo-concave. For pseudo-concave spaces $ X $ the following finiteness theorems have been proved: The space of holomorphic sections of any holomorphic vector bundle over $ X $ is finite-dimensional; if $ X $ is connected, then all holomorphic functions on $ X $ are constant; the field of meromorphic functions on $ X $ is an algebraic function field whose transcendence degree does not exceed $ \mathop{\rm dim} X $. The latter theorem has important applications to automorphic functions (cf. Automorphic function), based on the fact that the space $ D / \Gamma $, where $ \Gamma $ is a properly-discontinuous group of automorphisms of a bounded domain $ D \subseteq \mathbf C ^ {n} $, turns out to be pseudo-concave in many cases (one says in this case that $ \Gamma $ is a pseudo-concave group). E.g., arithmetic subgroups of automorphism groups of bounded symmetric domains are pseudo-concave.

References

[1]	J. Ermine, "Cohérence de certaines images directes à supports propres dans le cas d'un morphisme fortement -convexe" Ann. Scuola Norm. Sup. Pisa Cl. Sci. , 6 (1979) pp. 1–18
[2]	A.L. Onishchik, "Pseudoconvexity in the theory of complex spaces" J. Soviet Math. , 14 (1980) pp. 1363–1407 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 15 (1977) pp. 93–171

Comments

The Hermitian form $ H( \phi ) $ is usually called the Levi form of $ \phi $.

The fundamental domains for function theory in $ \mathbf C ^ {n} $ are the pseudo-convex ones, for they coincide with the domains of holomorphy (solution of the Levi problem). Especially well understood are the strongly pseudo-convex domains (the ones of the form $ \{ {z \in \mathbf C ^ {n} } : {\rho ( z) < 0 } \} $, where $ \rho $ is a strictly plurisubharmonic function). Good references are [a13]–[a15].

In many respects strongly $ p $- pseudo-convex domains have "better" properties than arbitrary $ p $- pseudo-convex domains. For simplicity (1-) pseudo-convex domains in $ \mathbf C ^ {n} $ will be discussed only. Thus, let $ D = \{ r( z) < 0 \} $ be a smoothly bounded domain with boundary $ M $ and $ p \in M $. Then $ D $ is called pseudo-convex at $ p $ if the Levi form $ H( r)( p) $ is positive semi-definite when restricted to the complex tangent space at $ p $, while $ D $ is called strongly pseudo-convex at $ p $ if the Levi form is positive definite when restricted to the complex tangent space at $ p $. A domain is called (strongly) pseudo-convex if it is (strongly) pseudo-convex at every boundary point. A pseudo-convex domain which is not strongly pseudo-convex is called a weakly pseudo-convex domain.

The weakly pseudo-convex domains in $ \mathbf C ^ {n} $ are the ones that can be exhausted by strongly pseudo-convex domains.

Some of the properties of strongly pseudo-convex domains that are not usually shared with — or do not have a proper analogue for — arbitrary weakly pseudo-convex domains, are:

a) One can solve the inhomogeneous Cauchy–Riemann equations with a gain: If $ f $ is a $ \overline \partial \; $- closed $ ( l , m+ 1 ) $- form and the coefficients of $ f $ belong to some Hölder space $ C ^ \alpha $ or Sobolev space $ \Lambda ^ \alpha $, then there is an $ ( l, m) $- form $ u $, with coefficients in $ C ^ \alpha $, respectively $ \Lambda ^ \alpha $, such that $ \overline \partial \; u = f $. More precisely, one has subelliptic estimates for the $ \overline \partial \; $- Neumann problem. This implies that the (unique) solution $ u $ with minimal $ L _ {2} $ norm of the inhomogeneous Cauchy–Riemann equations is well-behaved in the Sobolev norms as described above, cf. [a6]. As a consequence, the Bergman projection maps smooth functions to smooth functions.

b) Strongly pseudo-convex domains are locally biholomorphically equivalent to strongly convex domains.

c) A strongly pseudo-convex domain can be written as the interior of the intersection of a properly decreasing family of strongly pseudo-convex domains.

Apparently it makes a big difference whether the Levi form is positive definite or positive semi-definite. Analyzing this difference in conjunction with subelliptic estimates for the $ \overline \partial \; $- Neumann problem, the notion of domain of finite type was introduced by J.J. Kohn for domains in $ \mathbf C ^ {2} $, cf. [a9]. Closely related to Kohn's original definition and important in partial differential equations, see [a6], is a definition of finite type in terms of iterated commutators. Let $ T $ denote the complexified tangent bundle to $ M $, $ T ^ {1,0} $ the subbundle whose sections are $ ( 1, 0) $ vector fields, that is, the complex tangential vector fields, and let $ T ^ {0,1} $ be the complex conjugate bundle of $ T ^ {1,0} $. Now pick a real $ 1 $- form $ \eta $ which annihilates $ T ^ {1,0} \oplus T ^ {0,1} $. Let $ L $ be a local section of $ T ^ {1,0} $, $ L ( p) \neq 0 $. An iterated commutator of $ L $ and $ \overline{L}\; $ of length $ m $ is a local section $ X $ of $ T $ of the form

$$ \tag{a1 } X = [ \dots [ L _ {1} , L _ {2} ] \dots L _ {m+} 1 ] , $$

where $ L _ {j} \in \{ L , \overline{L}\; \} $. The type of $ L $ at $ p $ is the smallest integer $ q $ for which there is an iterated commutator $ X $ of $ L $ of length $ q + 1 $ such that $ \langle \eta , X \rangle ( p) \neq 0 $. Now one defines the (commutator) $ 1 $- type of $ M $ at $ p $:

$$ t _ {1} ( M, p) = \ \sup _ {L \in T ^ {1,0} } \{ \textrm{ type of } L \textrm{ at } p \} , $$

and $ M $ is of finite type if all points of $ M $ are of finite type. The commutator $ q $- type of a point, $ q = 1 \dots n- 1 $, was introduced by T. Bloom, cf. [a4]. However, it seems that only in the $ 2 $- dimensional case these notions are related to subelliptic estimates.

In the higher-dimensional case the following approach is useful. Again, let $ D $ be a smoothly bounded domain with boundary $ M $, defining function $ r $, and let $ p $ be a boundary point. (In the definitions one only needs a hypersurface $ M $.) Let $ z : ( \mathbf C , 0 ) \rightarrow ( \mathbf C ^ {n} , p ) $ be a germ of a holomorphic mapping sending 0 to $ p $. Let $ v( z) $ be the multiplicity of $ z $ at 0 and $ v( r \circ z ) $ the multiplicity of the composition $ r \circ z $ at 0. The point $ p $ is called a point of finite $ ( 1) $- type if there is a $ C $ such that

$$ \tag{a2 } \frac{v( r \circ z ) }{v( z) } \leq C $$

for all non-constant germs of holomorphic mappings $ z $ as above. The smallest $ C $ for which (a2) holds is called the type of $ p $ and is denoted by $ \Delta _ {1} ( M , p ) $. Roughly speaking, this means that $ M $ has finite order of contact with analytic curves at $ p $. The domain $ D $, or the hypersurface $ M $, is called of finite type if all its points are of finite type.

J. d'Angelo defined the notion of point of finite $ q $- type, $ q = 1 \dots n - 1 $, cf. [a1], [a3], [a4]. He proved that the set of points of finite type is an open subset of the hypersurface $ M $, cf. [a3], [a4].

There are many variants and refinements of these definitions. If one does not require pseudo-convexity in advance and $ q < n- 1 $, then the commutator $ q $- type has nothing to do with orders of contact. From now on finite type will be in the order-of-contact sense.

The following results were obtained. Kohn proved that if $ p $ is a boundary point of finite type of a smooth pseudo-convex domain in $ \mathbf C ^ {2} $, then there is a subelliptic estimate for the $ \overline \partial \; $- Neumann problem, cf. [a9]. P. Greiner proved the converse, cf. [a7]. D. Catlin extended these results to pseudo-convex domains in $ \mathbf C ^ {n} $, cf. [a2]. One has also obtained corresponding Hölder estimates for domains of finite type in $ \mathbf C ^ {2} $, cf. [a5]. Property c) is also shared by domains of finite type, cf. [a11]. There are domains of finite type that do not have property b), cf. [a10].

Finally, Catlin discovered "Catlin condition Pcondition P" , which is weaker than finite type and guarantees a compactness estimate for the $ \overline \partial \; $- Neumann problem. This implies estimates for Sobolev norms without gain, including the regularity of the Bergman projection and also property c): Condition $ P $ is equivalent to so-called weak B(remermann) regularity. This can be expressed as follows: Every continuous function on $ M $ is the boundary value of a plurisubharmonic function on $ D $, cf. [a11].

See also Bergman kernel function; Biholomorphic mapping.

A good reference for $ p $- convexity and $ q $- convexity is [a12].

References

[a1]	D. Catlin, "Boundary invariants of pseudo-convex domains" Ann. of Math. , 120 (1984) pp. 529–586
[a2]	D. Catlin, "Subelliptic estimates for the -Neumann problem on pseudo-convex domains" Ann. of Math. , 126 (1987) pp. 131–191
[a3]	J. d'Angelo, "Real hypersurfaces, orders of contact, and applications" Ann. of Math. , 115 (1982) pp. 615–637
[a4]	J. d'Angelo, "Finite type conditions for real hypersurfaces in $\mathbb{C}^n$" S.G. Krantz (ed.) , Complex Analysis , Lect. notes in math. , 1268 , Springer (1987) pp. 83–110
[a5]	C.L. Fefferman, J.J. Kohn, "Hölder estimates on domains of complex dimension two and on three dimensional CR manifolds" Adv. in Math. , 69 (1988) pp. 223–303
[a6]	G.B. Folland, J.J. Kohn, "The Neumann problem for the Cauchy–Riemann complex" , Annals Math. Studies , 75 , Princeton Univ. Press (1972)
[a7]	P. Greiner, "On subelliptic estimates of the $\overline{\partial}$-Neumann problem in $\mathbb{C}^2$" J. Diff. Geometry , 9 (1974) pp. 239–250
[a8]	L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985) pp. Chapt. 22
[a9]	J.J. Kohn, "Boundary behavior of $\overline{\partial}$ on weakly pseudo-convex domains" J. Diff. Geometry , 6 (1972) pp. 523–542
[a10]	J.J. Kohn, L. Nirenberg, "A pseudo-convex domain not admitting a holomorphic support function" Math. Ann. , 201 (1973) pp. 265–268
[a11]	N. Sibony, "Une classe de domaines pseudo-convexes" Duke Math. J. , 55 (1987) pp. 299–319
[a12]	G.M. Henkin, J. Leiterer, "Andreotti–Grauert theory by integral formulas" , Akademie Verlag (1988)
[a13]	G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1983)
[a14]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)
[a15]	R.M. Range, "Holomorphic functions and integral representations in several complex variables" , Springer (1986)