Tian–Yau's conjectural identity between the equivariant alpha-invariant and normalized Lelong numbers on Fano manifolds, and the companion interior C^0 estimate for degenerate complex Monge–Ampère equations

In their 1987 study of Kähler-Einstein metrics on complex surfaces with positive first Chern class, Gang Tian and Shing-Tung Yau estimated the alpha-invariant $\alpha(M)$ — the standard analytic obstruction to Kähler-Einstein metrics on Fano manifolds — using branched coverings, potential estimates, and Lelong numbers of positive closed $(1,1)$-currents, and posed several questions they could not settle. Two components of that list were later resolved: Kähler-Einstein existence on smooth del Pezzo surfaces (including the delicate degree-one case) was settled by Tian in 1990 via a partial C^0 estimate, and the conjectured uniform boundedness of anticanonical volumes of Fano manifolds was settled by Kollár–Miyaoka–Mori (1992) and Birkar's boundedness theorems. What remains open is: (a) the conjectural identity $\alpha_G(M) = 1/L'_G(M)$, relating the $G$-invariant alpha-invariant to the reciprocal of a supremum of normalized Lelong numbers of $G$-invariant currents cohomologous to $c_1(M)$ — the 1987 paper proved only the weaker bound $\alpha_G(M) \ge 1/L_G(M)$; and (b) an interior estimate $|u(0)| \le C(\sup_{B_1}u + \int_{B_1}e^{-pu}dV)$ for plurisubharmonic solutions of a degenerate Monge–Ampère equation on the unit ball.

Background

A Fano manifold is a compact complex manifold $M$ whose first Chern class $c_1(M)$ can be represented by a positive $(1,1)$-form, equivalently whose anticanonical line bundle is ample; complex surfaces with $c_1>0$ are, in the Kähler case, the del Pezzo surfaces: the projective plane $\mathbb{CP}^2$ or $\mathbb{CP}^2$ blown up at $n \le 8$ points in general position. A Kähler-Einstein metric is a Hermitian metric whose Ricci form is proportional to its Kähler form; for Fano manifolds the proportionality constant must be positive, and deciding which Fano manifolds carry such a metric is the positive-curvature case of Calabi's problem. The problem reduces, via the continuity method, to a priori estimates for the complex Monge–Ampère equation, the fully nonlinear partial differential equation $\det(u_{i\bar j}) = F$ for a plurisubharmonic potential $u$ (a function that is subharmonic along every complex line; its complex Hessian $u_{i\bar j}$ has nonnegative eigenvalues).

To measure how singular Kähler potentials can become along the continuity method, Tian introduced the alpha-invariant: $\alpha(M)$ is the supremum of $\varepsilon>0$ such that $\int_M e^{-\varepsilon \varphi}dV$ is uniformly bounded for all $\varphi$ with $\omega + i\partial\bar\partial\varphi$ ranging over Kähler potentials in the class $c_1(M)$. Tian proved that $\alpha(M) > n/(n+1)$ (with $n=\dim M$) forces the Kähler-Einstein equation to be solvable, and this criterion remains one of the few practical tools, since the now-proved Yau–Tian–Donaldson conjecture characterizes existence by K-stability — an algebraic condition that is hard to verify directly (see the introduction of Macbeth's paper on higher alpha-invariants). The invariant admits an equivariant refinement $\alpha_G(M)$, where only potentials invariant under a compact subgroup $G$ of the automorphism group are considered; using group symmetry raises the invariant and enlarges the class of manifolds to which the criterion applies.

Two allied circles of ideas enter these questions. First, pluripotential theory: a positive closed $(1,1)$-current is a differential-form analogue of a positive measure whose local potentials are plurisubharmonic; its Lelong number $L(u,z) \ge 0$ measures the logarithmic concentration of the potential at a point $z$. Demailly–Kollár proved that $\alpha(M)$ coincides with the global log canonical threshold, an algebraic singularity invariant of effective Q-divisors Q-linearly equivalent to $-K_M$ (a proof streamlined in Demailly's appendix to the Cheltsov–Shramov survey of Fano threefolds, which also covers the equivariant equality $\alpha_G(M) = \mathrm{lct}(X,G)$), and their semicontinuity theorem for complex singularity exponents — the critical exponents $c$ for local integrability of $e^{-c\varphi}$ — launched the multiplier-ideal machinery underlying the modern theory. Second, degenerate Monge–Ampère estimates: Kołodziej's pluripotential $L^\infty$ theory controls the oscillation of solutions from integrability of the right-hand side, and a Harnack-type inequality used in the continuity method bounds the negative part of a potential in terms of its positive part.

Tian–Yau (1987) combined these ideas to produce Kähler-Einstein structures on $\mathbb{CP}^2\#\,n\overline{\mathbb{CP}^2}$ for $3 \le n \le 8$ (with symmetry assumptions in the hardest cases) and, alongside the results, explicitly posed the questions below: whether the bound they prove between $\alpha_G(M)$ and the Lelong numbers of invariant currents is sharp in the form of an exact identity; whether a certain interior estimate for degenerate Monge–Ampère equations holds; whether their alpha-lower bounds are optimal; and whether anticanonical volumes $c_1(M)^m$ are uniformly bounded over all Fano $m$-folds. The last two questions have affirmative answers in the later literature; the first two constitute the surviving open core.

Problem Statement

The problem is the surviving open core of the question list in Tian–Yau (1987).

(Primary.) Let $M$ be a compact complex manifold of complex dimension $m$ with positive first Chern class, and let $G \subseteq \mathrm{Aut}(M)$ be a compact subgroup of its holomorphic automorphism group. Fix a $G$-invariant Kähler metric $g$ on $M$ whose Kähler form represents $c_1(M)$, and let $L_g(u,z)$ denote the Lelong number of a $G$-invariant positive $d$-closed $(1,1)$-current $u$ cohomologous to $c_1(M)$ at a point $z \in M$, computed with respect to $g$. Let $\alpha_G(M)$ be Tian's $G$-invariant alpha-invariant: the supremum of the exponents $\varepsilon > 0$ such that $\int_M e^{-\varepsilon\varphi}\,dV$ is uniformly bounded as $\varphi$ ranges over the $G$-invariant Kähler potentials with $\omega + i\partial\bar\partial\varphi$ in the class $c_1(M)$. Let $L'_G(M)$ denote the supremum, over all such currents $u$ and over points $z \in M$ at which the analytic level set of the local potential of $u$ has dimension $p$ (an integer with $0 \le p \le m-1$), of the normalized Lelong number

$\frac{1}{m-p}\, L_g(u,z),$

where $L_g(u,z)$ is Tian–Yau's notation for the Lelong number of the current $u$ at $z$ with respect to $g$. Let $L_G(M)$ denote the same supremum computed without the normalization $1/(m-p)$, that is, $L_G(M) = \sup\{ L_g(u,z) \}$ over the same currents $u$ and points $z$. Tian–Yau proved the weaker bound $\alpha_G(M) \ge 1/L_G(M)$ and conjectured the exact identity

$\alpha_G(M) = 1/L'_G(M).$

Prove or disprove this identity. A disproof may exhibit a specific Fano manifold $M$ and compact subgroup $G$ for which the two sides differ.

(Secondary.) Does there exist an exponent $p > 0$ such that every plurisubharmonic solution $u$ of the degenerate Monge–Ampère equation $\det(u_{i\bar j}) = F$ on the unit ball, with $F$ a positive bounded function satisfying $0 < F \le \Lambda$ for a fixed constant $\Lambda > 0$, satisfies

$|u(0)| \le C\Big(\sup_{B_1} u + \int_{B_1} e^{-p u}\, dV\Big)$

for a constant $C$ independent of $u$? Prove such an estimate, or construct a counterexample family — that is, exhibit admissible right-hand sides $F$ and solutions $u$ for which no such $p$ and $C$ exist.

The problem consists of the two parts above, (Primary) and (Secondary).

The other two components of the 1987 list — optimal Kähler-Einstein existence on smooth del Pezzo surfaces including the degree-one case, and uniform boundedness of anticanonical volumes of Fano manifolds in each dimension — were resolved by later work and are not part of the asked problem.

References

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