Revised Demailly's Affineness Criterion and Algebraization of Entire Grauert Tubes

Technical Summary: Revised Demailly's Affineness Criterion and Algebraization of Entire Grauert Tubes

Problem Statement
The paper addresses Burns' 1982 conjecture, which asks whether an entire Grauert tube $T^\infty M$ (the tangent bundle of a real-analytic Riemannian manifold $(M, g)$ equipped with a specific Stein structure) is an affine algebraic variety. While Stein manifolds are the analytic analogues of affine varieties, the former are defined by arbitrary holomorphic functions and are generally more flexible than the latter. The conjecture posits a deep link between the analytic condition of the Homogeneous Complex Monge–Ampère (HCMA) equation and the algebraic property of affineness.

Previous attempts to resolve this, notably by Aguilar and Burns, relied on Demailly's 1985 affineness criterion. This criterion requires the existence of a strictly plurisubharmonic exhaustion function $\phi$ and a function $\psi$ satisfying specific growth and Ricci curvature bounds. The obstruction identified in the literature is that the natural candidate for $\psi$, derived from the analytic continuation of the volume form of $M$, often possesses poles along a singular set $S$. Consequently, the required function $\psi$ takes the value $-\infty$ on $S$, rendering the original criterion inapplicable.

Methodology
The author employs a two-pronged approach: first, establishing a generalized version of Demailly's criterion that accommodates singularities, and second, applying this criterion to the specific geometry of entire Grauert tubes.

1. Revised Affineness Criterion: The paper introduces a modified version of Demailly's criterion (Theorem 1.2) for a Stein manifold $X$ with finite Monge–Ampère volume. Instead of requiring a global function $\psi$ satisfying the Ricci bound everywhere, the new criterion allows for a singular set $S = \{x \mid e^{-2\psi} \notin L^1_{loc}(x)\}$.

   • The methodology utilizes weighted $L^2$ estimates (Bombieri–Hörmander–Skoda type) to construct holomorphic functions $b_p$ that vanish on $S$ but are non-zero at specific points $p \in X \setminus S$.
   • By restricting the manifold to the open sets $X \setminus b_p^{-1}(0)$, the singularity set is "cut out," allowing the application of Demailly's original arguments to prove that these open subsets are biholomorphic to affine varieties.
   • The paper further analyzes the global structure of $X \setminus S$ under cohomological assumptions, showing it can be embedded as a Zariski open subset of a normal affine variety.

2. Application to Grauert Tubes: The author constructs the necessary geometric data for an entire Grauert tube $TM$:

   • Strictly Plurisubharmonic Exhaustion: A function $\phi = \log(1 + \cosh(\sqrt{2E}))$ is defined, where $E$ is the kinetic energy.
   • Complexified Metric: A non-degenerate meromorphic $(2,0)$-tensor $g_C$ is constructed by extending the base metric $g$ meromorphically along the leaves of the Monge–Ampère foliation (complexified geodesics).
   • Ricci Bound Function: The function $\psi = -\log \|\det(g_C)\|_\beta$ is defined, where $\beta = dd^c e^\phi$. The author proves that $\psi$ satisfies the required growth condition $\psi \leq A\phi + B$ and the Ricci curvature bound $Ric(\beta) + \frac{1}{2}dd^c\psi \geq 0$ in the sense of currents.
   • Singularity Identification: The set $S$ where the criterion fails is identified exactly as the pole set of $\det(g_C)$, termed the "tube singularity."

Key Results

• Theorem 1.2 (Revised Affineness Criterion): Establishes that if a Stein manifold $X$ has finite Monge–Ampère volume and admits a function $\psi$ satisfying the Ricci bound outside a set $S$, then $X \setminus S$ is covered by Zariski open subsets of affine varieties. If $S = \emptyset$ and cohomology is finite-dimensional, $X$ is affine.
• Theorem 1.3 (Algebraization of Entire Grauert Tubes): For any entire Grauert tube $TM$, the metric $g$ extends to a meromorphic tensor $g_C$. Let $S$ be the pole set of $\det(g_C)$. For any point $p \in TM \setminus S$, there exists a holomorphic function $b_p$ such that $TM \setminus b_p^{-1}(0)$ is biholomorphic to a smooth affine variety. Furthermore, $TM \setminus S$ is a Stein manifold covered by such affine open sets.
• Counter-examples to Global Affineness: The paper provides explicit examples (surfaces of revolution $M_d$ for $d \neq 0$) where the tube singularity $S$ is non-empty. In these cases, the transition matrix $\Phi'$ of parallel vector fields degenerates, causing $\det(g_C)$ to have poles. This demonstrates that the Burns conjecture does not hold in the strongest form (that $TM$ itself is affine) without further conditions on $S$.

Significance and Claims
The paper claims to provide a "partial answer" to Burns' conjecture. It establishes that an entire Grauert tube is affine outside a codimension-one subset (the tube singularity $S$). The significance lies in:

1. Generalizing Demailly's Criterion: The revised criterion (Theorem 1.2) is presented as a tool of independent interest, relaxing the regularity requirements on the Ricci potential $\psi$ to allow for singularities, thereby broadening the class of manifolds to which algebraization techniques can be applied.
2. Clarifying the Obstruction: The work identifies the tube singularity $S$ (the pole set of the complexified metric determinant) as the precise obstruction to the full affineness of the Grauert tube. It connects the failure of the Aguilar–Burns approach directly to the existence of these poles.
3. Connection to Good Complexification: The results suggest that if the tube singularity $S$ can be shown to be empty (or if the cohomology of $TM \setminus S$ satisfies specific finiteness conditions), the Burns conjecture would imply the Good Complexification Conjecture for manifolds admitting entire Grauert tubes.

The author remains modest regarding the full resolution of the Burns conjecture, noting that a complete proof requires a deeper understanding of the topology and structure of the singularity set $S$, for which no general control method currently exists. The paper concludes by posing questions regarding the cohomology of $TM \setminus S$ and the possibility of deforming the metric to eliminate $S$.