[Paper Review] Dynamical construction of Kähler-Einstein metrics

Hajime Tsuji|arXiv (Cornell University)|Jun 25, 2006

Geometry and complex manifolds4 citations

TL;DR

This paper presents a dynamical construction of Kähler-Einstein metrics on smooth projective varieties with ample canonical bundle by taking the limit of Bergman kernels associated with higher pluricanonical systems. The key result shows that the limit of scaled Bergman kernels yields a Kähler-Einstein current, establishing a canonical link between Bergman metrics and Kähler-Einstein geometry via an iterative, dynamical process based on $L^2$-orthonormal bases and curvature estimates.

ABSTRACT

In this paper, I give a new construction of a Kähler-Einstein metrics on a smooth projective variety with ample canonical bundle. This result can be generalized to the construction of a singular Kähler-Einstein metric on a smooth projective variety of general type which gives an AZD of the canonical bundle. Also the variation of Bergman kernels and Kähler-Einstein volume form have been considered.

Motivation & Objective

• To establish a new dynamical construction of Kähler-Einstein metrics on smooth projective varieties with ample canonical bundle.
• To generalize this construction to singular Kähler-Einstein metrics on varieties of general type via analytic Zariski decompositions.
• To prove the existence of canonical singular hermitian metrics with semipositive curvature on direct images $f_*\mathcal{O}_X(mK_{X/S})$ for projective morphisms with general type fibers.
• To link the asymptotic behavior of Bergman kernels to Kähler-Einstein geometry through iterative metric constructions.

Proposed method

• Iteratively define a sequence of smooth hermitian metrics $h_m$ on $mK_X$ via the inverse of the Bergman kernel $K_m$ associated with $L^2$-orthonormal bases of $H^0(X, \mathcal{O}_X((m+1)K_X))$.
• Use the $L^2$ inner product weighted by the previous metric $h_m$ to define orthonormal bases for the next step, ensuring curvature control.
• Construct the limit metric $h_\infty = \liminf_{m\to\infty} \sqrt[m]{(m!)^n \cdot h_m}$ on $K_X$ to obtain a Kähler-Einstein current.
• Apply Fujita’s theorem and asymptotic volume estimates to relate the growth of $K_m$ to the canonical volume $\mu(X,K_X)$.
• Prove lower and upper bounds on $\limsup_{m\to\infty} \frac{1}{(m!)^{n/m}} (K_m)^{1/m}$ to show convergence to the Kähler-Einstein volume form.
• Extend the construction to relative settings via fiberwise Bergman kernels on general type fibers, proving semipositivity of curvature currents on direct image sheaves.

Experimental results

Research questions

• RQ1 Can Kähler-Einstein metrics on varieties with ample canonical bundle be constructed dynamically via limits of Bergman kernels?
• RQ2 How does the asymptotic behavior of Bergman kernels on pluricanonical systems relate to Kähler-Einstein geometry?
• RQ3 What is the curvature property of the direct image sheaf $f_*\mathcal{O}_X(mK_{X/S})$ for a family of general type varieties over a base?
• RQ4 Is the maximal Kähler-Einstein current on a general type variety unique?
• RQ5 Can the dynamical construction of Kähler-Einstein metrics be generalized to singular metrics on varieties of general type?

Key findings

• The limit metric $h_\infty = \liminf_{m\to\infty} \sqrt[m]{(m!)^n \cdot h_m}$ on $K_X$ defines a Kähler-Einstein current $\omega_\infty = \sqrt{-1} \partial\bar\partial \log h_\infty$.
• The asymptotic growth of $K_m$ satisfies $\limsup_{m\to\infty} \frac{1}{(m!)^{n/m}} \int_X (K_m)^{1/m} = (2\pi)^{-n} \int_X dV_E$, matching the Kähler-Einstein volume form.
• The upper bound $\limsup_{m\to\infty} \frac{1}{(m!)^{n/m}} (K_m)^{1/m} \leq \frac{1}{n!} \mu(X,K_X)$ holds, with equality in the limit.
• The maximal Kähler-Einstein current on a smooth projective variety of general type is unique.
• The direct image sheaf $f_*\mathcal{O}_X(mK_{X/S})$ admits a canonical singular hermitian metric $h_{E,m}$ with semipositive curvature current in the sense of Nakano.
• The construction extends to families: the limit metric on $K_{X/S}$ induces a semipositive curvature current on $f_*\mathcal{O}_X(mK_{X/S})$ for each $m$.