[Paper Review] Cone spherical metrics and stable vector bundles

Lingguang Li, Jijian Song+1 more|arXiv (Cornell University)|Aug 13, 2018

Geometric Analysis and Curvature Flows19 references3 citations

TL;DR

This paper establishes a correspondence between irreducible cone spherical metrics with integer multiple cone angles $2\pi\mathbb{Z}_{>1}$ and line subbundles of stable rank-two vector bundles on compact Riemann surfaces of genus $g > 1$. It proves a Lange-type existence theorem for stable extensions of line bundles and constructs a new class of irreducible spherical metrics via this algebraic framework.

ABSTRACT

Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. A cone spherical metric is called irreducible if each developing map of the metric does not have monodromy lying in ${ m U(1)}$. We establish on compact Riemann surfaces of positive genera a correspondence between irreducible cone spherical metrics with cone angles being integral multiples of $2π$ and line subbundles of rank two stable vector bundles. Then we are motivated by it to prove a theorem of Lange-type that there always exists a stable extension of $L^*$ by $L$, for $L$ being a line bundle of negative degree on each compact Riemann surface of genus greater than one. At last, as an application of these two results, we obtain a new class of irreducible spherical metrics with cone angles being integral multiples of $2π$ on each compact Riemann surface of genus greater than one

Motivation & Objective

• To establish a correspondence between irreducible cone spherical metrics with integer multiple cone angles and line subbundles of stable rank-two vector bundles on compact Riemann surfaces of genus $g > 1$.
• To prove a Lange-type theorem on the existence of stable extensions of $L^*$ by $L$ for line bundles $L$ of negative degree on such surfaces.
• To apply the correspondence and existence result to construct a new class of irreducible cone spherical metrics with integer multiple cone angles on every compact Riemann surface of genus greater than one.

Proposed method

• Utilizes developing maps of cone spherical metrics to relate their monodromy to unitary representations in $\mathrm{PSU}(2)$, characterizing irreducible metrics.
• Applies the theory of parabolic bundles and stable vector bundles, particularly focusing on line subbundles of stable rank-two bundles with trivial determinant.
• Employs algebraic geometry techniques, including sheaf extensions and local module decompositions, to analyze the structure of vector bundles and their subbundles.
• Uses the openness of stability in moduli spaces to show that generic extensions of sheaves yield stable bundles.
• Applies the Gauss-Bonnet formula and curvature constraints to ensure the metrics have constant curvature one with conical singularities.
• Constructs metrics via projective functions with unitary monodromy and ramification divisor equal to the divisor of cone points.

Experimental results

Research questions

• RQ1 Does there exist a correspondence between irreducible cone spherical metrics with cone angles in $2\pi\mathbb{Z}_{>1}$ and line subbundles of stable rank-two vector bundles on compact Riemann surfaces of genus $g > 1$?
• RQ2 Can one prove a Lange-type existence theorem for stable extensions of $L^*$ by $L$ when $L$ is a line bundle of negative degree on a genus $g > 1$ surface?
• RQ3 Does the correspondence between metrics and stable bundles yield a new construction of irreducible cone spherical metrics with integer multiple cone angles on every compact Riemann surface of genus greater than one?

Key findings

• There is a one-to-one correspondence between irreducible cone spherical metrics with cone angles in $2\pi\mathbb{Z}_{>1}$ and line subbundles of stable rank-two vector bundles with trivial determinant on compact Riemann surfaces of genus $g > 1$.
• For any line bundle $L$ of negative degree on a compact Riemann surface of genus $g > 1$, there exists a stable extension of $L^*$ by $L$, generalizing a classical result of Lange.
• A new class of irreducible cone spherical metrics with cone angles in $2\pi\mathbb{Z}_{>1}$ exists on every compact Riemann surface of genus greater than one.
• The construction relies on the existence of stable extensions and the algebraic structure of parabolic bundles, ensuring the metrics are irreducible via monodromy conditions.
• The method proves that generic extensions of sheaves in the moduli space yield stable bundles, leveraging the openness of stability in algebraic geometry.
• The results provide a complete algebraic framework for constructing irreducible spherical metrics with integer cone angles, resolving a long-standing existence question in special cases.