[Paper Review] Irreducible cone spherical metrics and stable extensions of two line bundles

Lingguang Li, Jijian Song+1 more|arXiv (Cornell University)|Jan 23, 2020

Geometric Analysis and Curvature Flows46 references10 citations

TL;DR

This paper establishes a canonical surjective correspondence between stable extensions of two line bundles on a compact Riemann surface of genus $ g_X \geq 1 $ and irreducible cone spherical metrics with cone angles in $ 2\pi\mathbb{Z}_{>1} $. Using indigenous bundles and Hermitian-Einstein metrics, it proves that for $ g_X \geq 2 $, the space of effective divisors of even degree $ d > 12g_X - 7 $ represented by such metrics forms an arcwise connected Borel set of Hausdorff dimension at least $ 2(d + 3 - 3g_X) $, and for odd degree $ d > 2g_X - 2 $, almost all such divisors admit finitely many irreducible metrics.

ABSTRACT

A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in ${ m U(1)}$. By using the theory of indigenous bundles, we construct on a compact Riemann surface $X$ of genus $g_X \geq 1$ a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in $2 \pi \mathbb{Z}_{>1}$, which is generically injective in the algebro-geometric sense as $g_X \geq 2$. As an application, we prove the following two results about irreducible metrics: $\bullet$ as $g_X \geq 2$ and $d$ is even and greater than $12g_X - 7$, the effective divisors of degree $d$ which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension $\geq 2(d+3-3g_X)$ in ${ m Sym}^d(X)$; $\bullet$ as $g_X \geq 1$, for almost every effective divisor $D$ of degree odd and greater than $2g_X-2$ on $X$, there exist finitely many cone spherical metrics representing $D$.

Motivation & Objective

• To establish a correspondence between stable extensions of line bundles and irreducible cone spherical metrics on compact Riemann surfaces of genus $ g_X \geq 1 $.
• To characterize the moduli space of irreducible cone spherical metrics via stable extensions and the ramification divisor map.
• To resolve the existence and uniqueness problem for irreducible metrics representing effective divisors of both even and odd degrees using algebro-geometric techniques.
• To prove dimension and connectivity results for the set of effective divisors representable by irreducible metrics, particularly for even and odd degrees.
• To unify the treatment of cone spherical metrics across even and odd degree divisors, overcoming limitations of prior PDE-based methods that failed for even degrees due to bubbling.

Proposed method

• Construct a canonical surjective map from the moduli space of stable extensions of two line bundles to the moduli space of irreducible cone spherical metrics with cone angles in $ 2\pi\mathbb{Z}_{>1} $, using indigenous bundle theory.
• Define the ramification divisor map $ R(L_1, L_2) $, a real analytic map from the space of stable extensions to the complete linear system $ |L_1^{-1} \otimes L_2 \otimes K_X| $, which sends each extension to the divisor of the associated metric.
• Use the Hermitian-Einstein metric on the rank two vector bundle $ E $ to characterize the effective divisor represented by the corresponding irreducible metric.
• Employ algebraic geometry of vector bundles and cohomology to prove that the ramification divisor map is generically injective for $ g_X \geq 2 $, and surjective for odd degree divisors.
• Apply differential geometry on vector bundles to establish the properties of the ramification divisor map and prove its image contains all effective divisors representable by irreducible metrics.
• Use the structure of the moduli space of stable extensions and its embedding into projective space to analyze the image of the ramification map and derive dimension and connectivity results.

Experimental results

Research questions

• RQ1 What is the precise correspondence between stable extensions of line bundles and irreducible cone spherical metrics on compact Riemann surfaces?
• RQ2 For which effective divisors of even degree $ d > 12g_X - 7 $ do irreducible cone spherical metrics exist, and what is the topological structure of the set of such divisors?
• RQ3 For effective divisors of odd degree $ d > 2g_X - 2 $, how many irreducible cone spherical metrics represent a given divisor, and is this number uniformly bounded?
• RQ4 Can the ramification divisor map $ R(L_1, L_2) $ be used to characterize all effective divisors representable by irreducible metrics, and what are the geometric properties of its image?
• RQ5 What happens to the ramification divisor map in the limit as stable extensions degenerate to unstable ones, and can this lead to reducible metrics?

Key findings

• For $ g_X \geq 2 $, the set of effective divisors of even degree $ d > 12g_X - 7 $ that are representable by irreducible cone spherical metrics forms an arcwise connected Borel subset of Hausdorff dimension at least $ 2(d + 3 - 3g_X) $ in $ \operatorname{Sym}^d(X) $.
• For $ g_X \geq 1 $, and for almost every effective divisor $ D $ of odd degree $ d > 2g_X - 2 $, there exist finitely many cone spherical metrics representing $ D $, with the number of such metrics uniformly bounded in terms of $ d $ and $ g_X $.
• The ramification divisor map $ R(L_1, L_2) $ is surjective onto the complete linear system $ |L_1^{-1} \otimes L_2 \otimes K_X| $ when $ \deg L_2 - \deg L_1 $ is odd and positive, and its image contains exactly all effective divisors representable by irreducible metrics.
• The correspondence between stable extensions and irreducible metrics is generically injective for $ g_X \geq 2 $, with at most $ 2^{2g_X} $ preimages per metric.
• The ramification divisor map is a real analytic map from a Zariski open subset of the projective space $ \mathbb{P}(H^1(X, L_1 \otimes L_2^{-1})^s) $ to the complete linear system $ |L_1^{-1} \otimes L_2 \otimes K_X| $, and its image is exactly the set of effective divisors representable by irreducible metrics.
• The existence of irreducible metrics for even-degree divisors is established via algebro-geometric methods, overcoming the failure of PDE-based bubbling arguments in this case.