[Paper Review] Stability of tangent bundle on the moduli space of stable bundles on a curve

Jaya Nn Iyer|arXiv (Cornell University)|Nov 1, 2011

Geometry and complex manifolds17 references3 citations

TL;DR

This paper proves that the tangent bundle of the moduli space ${\mathcal{S}\mathcal{U}}_C(r,d)$ of stable vector bundles of rank $r \geq 3$ and fixed determinant with $\gcd(r,d)=1$ on a smooth projective curve $C$ of genus $g \geq 3$ is stable in the sense of Mumford-Takemoto. The proof uses the Hecke correspondence between moduli spaces and analyzes cohomological vanishing of sections on Grassmannian bundles to rule out destabilizing subsheaves, confirming a conjecture on stability of tangent bundles on Fano manifolds with Picard number one.

ABSTRACT

In this paper, we prove that the tangent bundle of the moduli space $\cSU_C(r,d)$ of stable bundles of rank $r>2$ and of fixed determinant of degree $d$ (such that $(r,d)=1$), on a smooth projective curve $C$ is always stable, in the sense of Mumford-Takemoto. This verifies a well-known conjecture, and is related to a conjectural existence of a Kähler-Einstein metric on Fano varieties with Picard number one.

Motivation & Objective

• To prove the stability of the tangent bundle on the moduli space ${\mathcal{S}\mathcal{U}}_C(r,d)$ for $r \geq 3$ and $\gcd(r,d)=1$.
• To confirm a conjecture that the tangent bundle is stable on Fano manifolds with Picard number one.
• To establish a link between the stability of tangent bundles and the existence of Kähler-Einstein metrics on Fano varieties.
• To extend previous results on tangent bundle stability from rank 2 to higher rank moduli spaces.

Proposed method

• Utilizes the Hecke correspondence between moduli spaces ${\mathcal{S}\mathcal{U}}_C(r,1)$ and ${\mathcal{S}\mathcal{U}}_C(r,1-h)$ for $0 < h < r$, which is realized as a Grassmannian bundle.
• Analyzes the structure of the cotangent sheaf and its destabilizing subsheaves via the direct sum of relative cotangent sheaves on the correspondence space.
• Applies cohomological techniques to show that certain Hodge cohomology groups twisted by powers of the ample line bundle vanish, particularly $H^0(G, q^{\prime*}L^{\prime - b'}) = 0$ for $b' > 0$.
• Uses Zariski trivialization and generic injectivity arguments to analyze the kernel and image sheaves in the exact sequence involving $\mathcal{K}$, $\mathcal{F}$, and $\mathcal{G}$.
• Employs the fact that rational connectedness of the moduli space implies $H^0(M, \Omega^t_M) = 0$ for $t > 0$, to rule out nontrivial global sections.
• Applies the determinant condition $\det(\mathcal{K}) = \mathcal{O}$ and derives a system of inequalities on the exponents of line bundle powers, leading to $a_1 = b_1 = 0$.

Experimental results

Research questions

• RQ1 Is the tangent bundle of the moduli space ${\mathcal{S}\mathcal{U}}_C(r,d)$ stable for $r \geq 3$ and $\gcd(r,d)=1$?
• RQ2 Can the Hecke correspondence be used to preserve or transfer stability properties of tangent bundles between moduli spaces?
• RQ3 Does the vanishing of certain cohomology groups on the Grassmannian bundle imply the non-existence of destabilizing subsheaves?
• RQ4 Can the determinant of the kernel sheaf $\mathcal{K}$ be used to derive contradictions under the assumption of destabilization?
• RQ5 Does the rational connectedness of the moduli space $\mathcal{S}\mathcal{U}_C(r,d)$ imply the vanishing of $H^0(M, \Omega^t_M)$ for $t > 0$?

Key findings

• The tangent bundle on ${\mathcal{S}\mathcal{U}}_C(r,d)$ is stable for all $r \geq 3$ and $\gcd(r,d)=1$ when $g(C) \geq 3$.
• The Hecke correspondence preserves the stability of the tangent bundle, allowing reduction to known cases via induction on rank.
• The vanishing of $H^0(G, q^{\prime*}L^{\prime - b'})$ for $b' > 0$ is established using the positivity of $q^{\prime*}L'$ on fibers, which rules out positive-degree destabilizing subsheaves.
• The determinant of the kernel sheaf $\mathcal{K}$ is trivial, i.e., $\det(\mathcal{K}) = \mathcal{O}$, which leads to a system of equations on line bundle exponents that forces $a_1 = b_1 = 0$.
• The existence of a non-zero section in $H^0(M, \Omega^t_M)$ for $t > 0$ is ruled out due to rational connectedness of $M$, which implies such cohomology groups vanish.
• The proof confirms that no destabilizing subsheaf of the cotangent bundle can exist, thus establishing the stability of the tangent bundle.