$q$-Convex with Corners Function

• $q$-convex with corners functions are continuous functions on complex manifolds defined as the maximum of finitely many smooth $q$-convex functions, incorporating a localized corner structure.
• They enable strong cohomological control by ensuring vanishing of H^p groups in superlevel sets and guaranteeing extension properties for holomorphic line bundles.
• This framework generalizes classical convexity by relaxing smoothness and exhaustion conditions, providing counterexamples to the Andreotti–Grauert equivalence in complex analytic geometry.

A $q$-convex with corners function is a continuous function on a complex manifold that arises as a local maximum of finitely many smooth $q$-convex functions, generalizing classical convexity and enabling cohomological and extension results in complex geometry. The theory, developed notably in the work of Alaoui and others, reveals deep connections between local geometric properties and global holomorphic and cohomological structures, and provides counterexamples to long-standing equivalences in complex analytic geometry (Alaoui, 14 Jan 2026Alaoui, 2007).

1. Definition and Local Analytic Structure

Let $X$ be a complex manifold of dimension $n$. A $\mathcal C^\infty$ real-valued function $\phi$ on $X$ is called $q$-convex if at every point $x\in X$, its Levi form has at most $q-1$ non-positive eigenvalues on $T_xX$. Equivalently, $\phi$ features at least $n-(q-1)$ strictly positive directions in its complex Hessian.

A continuous function

$f\colon X\;\longrightarrow\;\mathbb R$

is termed $q$-convex with corners if for any $x\in X$ there exists a neighborhood $U\ni x$ and smooth $q$-convex functions $\phi_1, \dots, \phi_r$ on $U$ such that

$f|_U = \max\{\phi_1, \dots, \phi_r\}.$

The corner structure refers specifically to loci where two or more $\phi_i$ coincide and $f$ loses differentiability. The collection of all such functions is denoted by $F_q(X)$.

On every region where exactly one $\phi_i$ dominates, $f$ remains smooth and $q$-convex. However, $f$ itself may fail to be $C^2$ globally, and no single global Levi-discriminant applies; rather, convexity is tested piecewise through the envelope of local $q$-convex functions (Alaoui, 14 Jan 2026Alaoui, 2007).

2. Geometric and Cohomological Properties

A salient feature of $q$-convex with corners functions is their ability to control the geometry and topology of superlevel sets. For $f\in F_q(X)$, define superlevel domains $Y_c = \{x : f(x) > c\}$. For any coherent analytic sheaf $\mathcal F$ on $X$,

$$H^p(Y_c, \mathcal F) = 0 \quad \text{for } 1 \leq p \leq n-q-1,$$

and

$$H^0(X, \mathcal F) \rightarrow H^0(Y_c, \mathcal F)$$

is an isomorphism. These vanishing and restriction results, derived via classical Andreotti-Grauert methods and Mayer–Vietoris patching over the faces of the corners, establish the superlevel domains as locally cohomologically $q$-complete (Alaoui, 14 Jan 2026).

If $f$ is also an exhaustion from below (i.e., sublevel sets $\{x: f(x) < c\}$ are relatively compact for all $c$), then $X$ is called $q$-complete with corners. This case reproduces classical $q$-completeness results with weakened regularity requirements (Alaoui, 2007).

3. Examples, Non-Examples, and Model Domains

• Strictly Plurisubharmonic Case ($q=1$): Any strictly plurisubharmonic exhaustion is in $F_1(X)$, with finite maxima preserving the class.
• Higher $q$ in $\mathbb{C}^n$: Construct smooth functions with at most $q-1$ negative directions in the Levi form; maxima of such are $q$-convex with corners.
• Projective Models: In $\mathbb{P}^n$, partitioning coordinates and considering the complement of $q$-planes allows explicit construction of $q$-complete with corners domains via maximums of associated Peternell exhaustions:

$$\tilde{\phi}_i(z) = -\log\left( \sum_{j\in I_i} |z_j|^2 \right) + \log\left( \sum_{k=0}^n |z_k|^2 \right)$$

with

$f = \max_{0 \leq i \leq m} \phi_i$

yielding a $q$-convex with corners exhaustion (Alaoui, 2007).

• Failure in General Pseudoconvex Domains: Chen’s counterexample demonstrates that pseudoconvexity ($q=n$) does not guarantee line bundle extension, distinguishing $q$-convex with corners from general pseudoconvex domains (Alaoui, 14 Jan 2026).

4. Extension Theorem for Holomorphic Line Bundles

Alaoui's main result provides a Hartogs-type extension theorem for line bundles:

Theorem: Let $X$ be a complex manifold of dimension $n \geq 4$, suppose there exists $f\in F_q(X)$ with $1 \leq q \leq n-3$, and set $Y = \{ x \in X : f(x) > f(\xi_0) \}$ for fixed $\xi_0 \in X$. Then the restriction map

$$H^p(X,\mathcal{O}^*) \longrightarrow H^p(Y,\mathcal{O}^*)$$

is bijective for $p = 0,1,2$ and injective for $p = 3$. Consequently, every holomorphic line bundle over $Y$ extends uniquely up to isomorphism to $X$.

The proof uses local cohomology vanishing, Stein neighborhood arguments, the exponential sequence, and associated spectral sequences tied to local cohomology sheaves supported in the boundary $\{f = f(\xi_0)\}$, realizing the extension property without the necessity of global exhaustion or the previously imposed dimension bounds of $n \geq 3q$ (Alaoui, 14 Jan 2026).

5. $q$-Completeness With Corners: Comparison and Flexibility

The framework of $q$-convex with corners generalizes classical $q$-complete domains, embracing situations where a continuous function provides only local maxima of smooth $q$-convex functions, rather than a global smooth exhaustion. Earlier works (Fornaess–Sibony–Wold, Peternell) required stronger numerical hypotheses ($n \geq 3q$) and global exhaustivity, which are relaxed in the corners setting (Alaoui, 14 Jan 2026Alaoui, 2007).

This flexibility affords broader applicability in extension and vanishing theorems, enabling results in complex manifolds not admitting classical $q$-convex smooth exhaustions.

6. Counterexamples to Andreotti–Grauert Equivalence

Alaoui and collaborators provide explicit counterexamples to the conjectured equivalence between $q$-completeness and cohomological $q$-completeness. Notably, the complement of the Veronese surface in $\mathbb{P}^5$: $$A = \nu(\mathbb{P}^2) \subset \mathbb{P}^5, \quad X = \mathbb{P}^5 \setminus A$$ admits a continuous $3$-convex with corners exhaustion $f \in F_3(X)$, but topological properties force $H^5(X; \mathbb{Z}) = \mathbb{Z}/2\mathbb{Z} \neq 0$, precluding any smooth $3$-convex exhaustion. However, vanishing theorems for $f$ ensure $X$ is cohomologically $3$-complete. This demonstrates that weakening to “with corners” suffices for cohomological finiteness and vanishing below degree $n-q$, but does not guarantee the existence of genuinely smooth $q$-convex exhaustions, invalidating the full Andreotti-Grauert equivalence (Alaoui, 2007).

7. Further Directions and Geometric Implications

Superlevel sets $\{f > c\}$ of $q$-convex with corners functions are locally cohomologically $q$-complete: $$H^p(\{f>c\},\mathcal{F}) = 0 \quad \text{for } 1 \leq p \leq n-q-1,$$ and the maps $H^p(X,\mathcal{F}) \to H^p(\{f>c\},\mathcal{F})$ are isomorphisms for $p \leq n-q-1$, injections for $p = n-q$. These properties hold without exhaustion, provided sufficiently "non-critical" slices are taken via superlevel sets.

A plausible implication is the introduction of $F_q(X)$ as a highly flexible class, broadening the landscape of extension, vanishing, and finiteness theorems in complex analytic geometry and providing tools for constructing domains with prescribed geometric and topological attributes outside the reach of classical $q$-convexity results (Alaoui, 14 Jan 2026Alaoui, 2007).