[Paper Review] The Kähler-Ricci flow on Hirzebruch surfaces

Jian Song, Ben Weinkove|arXiv (Cornell University)|Mar 11, 2009

Geometry and complex manifolds13 references21 citations

TL;DR

This paper investigates the unnormalized Kähler-Ricci flow on Hirzebruch surfaces under invariant initial metrics, proving that the flow converges in the Gromov-Hausdorff sense to a limit space that is either a point, a $×1$, or an orbifold obtained by contracting an exceptional divisor. The results confirm the Feldman-Ilmanen-Knopf conjecture and extend to higher-dimensional analogues, showing metric collapse or contraction under specific curvature and initial class conditions.

ABSTRACT

We investigate the metric behavior of the Kahler-Ricci flow on the Hirzebruch surfaces, assuming the initial metric is invariant under a maximal compact subgroup of the automorphism group. We show that, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to $\mathbb{P}^1$ or contracts an exceptional divisor, confirming a conjecture of Feldman-Ilmanen-Knopf. We also show that similar behavior holds on higher-dimensional analogues of the Hirzebruch surfaces.

Motivation & Objective

• To understand the metric behavior of the unnormalized Kähler-Ricci flow on Hirzebruch surfaces under initial metrics invariant under a maximal compact subgroup of the automorphism group.
• To verify the conjecture by Feldman, Ilmanen, and Knopf on the Gromov-Hausdorff limit of the flow, which predicts collapse to a point, to $×1$, or to a divisor contraction.
• To extend the analysis to higher-dimensional analogues of Hirzebruch surfaces, showing similar limiting behavior under the same symmetry assumption.
• To provide evidence that the Kähler-Ricci flow may serve as an analytic tool for algebraic classification, particularly in birational geometry and the Yau-Tian-Donaldson conjecture.

Proposed method

• The authors analyze the unnormalized Kähler-Ricci flow $\partial\omega/\partial t = -\textrm{Ric}(\omega)$ on Hirzebruch surfaces $M_k$, assuming initial Kähler metrics invariant under a maximal compact subgroup of the automorphism group.
• They use local coordinates and a radial ansatz to derive asymptotic estimates for the evolving Kähler metric $g(t)$ near the exceptional divisor $D_0$, particularly as $t \to T$ where $T$ is the singular time.
• A key estimate $g_{i\overline{j}}(t) \leq a_t \chi_{i\overline{j}} + C e^{(k-n)\rho/n} \delta_{ij}$ is derived using Lemmas 4.4 and 4.5, controlling curvature blow-up near the divisor.
• The metric completion of $(M \setminus D_0, g_T)$ is studied, and it is shown that the singularity is integrable with finite diameter when $\beta = 2(n-k)/n < 2$, implying topological contraction.
• The Gromov-Hausdorff convergence is established by constructing a map $F: M \to \overline{M} \cong \mathbb{P}^n / \mathbb{Z}_k$ and showing that the distance between $(M, g(t))$ and the limit space $\overline{M}$ tends to zero as $t \to T$.
• The proof relies on the $C^\infty$ convergence of $g(t)$ on compact subsets of $M \setminus D_0$ and the decay of the metric component $a_t \to 0$ as $t \to T$.

Experimental results

Research questions

• RQ1 Does the unnormalized Kähler-Ricci flow on Hirzebruch surfaces converge in the Gromov-Hausdorff sense to a limit space that is a point, $\mathbb{P}^1$, or a quotient orbifold?
• RQ2 Under what conditions on the initial Kähler class and the parameter $k$ does the flow contract the exceptional divisor $D_0$?
• RQ3 Can the metric completion of the flow's singular limit be described as a finite-diameter metric space homeomorphic to $\mathbb{P}^n / \mathbb{Z}_k$?
• RQ4 Does the flow behavior on higher-dimensional analogues of Hirzebruch surfaces mirror that on the surface case under the same symmetry assumption?
• RQ5 How does the decay of the metric component $a_t$ as $t \to T$ influence the topology of the Gromov-Hausdorff limit?

Key findings

• The unnormalized Kähler-Ricci flow on Hirzebruch surfaces converges in the Gromov-Hausdorff sense to a limit space that is either a point, $\mathbb{P}^1$, or an orbifold $\mathbb{P}^n / \mathbb{Z}_k$ obtained by contracting an exceptional divisor.
• When $a_0(n+k) < b_0(n-k)$, the flow contracts the divisor $D_0$ as $t \to T$, and the metric completion of $(M \setminus D_0, g_T)$ has finite diameter and is homeomorphic to $\mathbb{P}^n / \mathbb{Z}_k$.
• The estimate $g_{i\overline{j}}(t) \leq a_t \chi_{i\overline{j}} + C e^{(k-n)\rho/n} \delta_{ij}$ ensures that the singularity is integrable with $\beta = 2(n-k)/n < 2$, which implies finite diameter and topological contraction.
• The Gromov-Hausdorff distance between $(M, g(t))$ and the limit space $\overline{M}$ tends to zero as $t \to T$, confirming convergence in the metric sense.
• The results extend to higher-dimensional analogues of Hirzebruch surfaces, showing similar collapse or contraction behavior under the same symmetry and initial metric assumptions.
• The flow’s behavior confirms the Feldman-Ilmanen-Knopf conjecture and provides analytic evidence for the Yau-Tian-Donaldson conjecture in birational geometry.