Donaldson–Uhlenbeck–Yau theorem

Statement

Let $(X,\omega)$ be a compact connected complex $n$ -dimensional, $n\ge1$ , Kähler manifold and $E\to X$ a holomorphic vector bundle of positive rank. The Donaldson–Uhlenbeck–Yau theorem states that $E$ admits a Hermitian metric whose Chern connection is Hermitian Yang–Mills if and only if $E$ is slope-polystable with respect to $\omega$ . Here

\mu_\omega(E)=\frac{1}{\operatorname{rk}E} \int_X c_1(E)\wedge\frac{\omega^{n-1}}{(n-1)!},

Here $c_1(E)$ in the integral denotes a real representative of the first Chern class. For a positive-rank coherent subsheaf $F$ of the sheaf of holomorphic sections of $E$ , define $c_1(F)=c_1(\det F)$ , where $\det F=(\bigwedge^{\operatorname{rk}F}F)^{**}$ is its determinant line bundle; the double dual is taken over the sheaf $\mathcal O_X$ of holomorphic functions. Such $F$ is torsion-free, and its rank is its generic rank. Define $\mu_\omega(F)$ by the same integral divided by $\operatorname{rk}F$ . Stability requires $\mu_\omega(F)<\mu_\omega(E)$ for every coherent subsheaf $F\subset E$ with $0<\operatorname{rk}F<\operatorname{rk}E$ , and polystability means a direct sum of stable bundles having the same slope.

The two directions

The differential-geometric direction shows that a Hermitian Yang–Mills connection forces the slope inequalities and a splitting at equality; hence the associated holomorphic bundle is polystable. The analytic direction constructs a Hermitian–Einstein metric from stability by solving a nonlinear elliptic equation.

Stable and polystable forms

If $E$ is stable, its Hermitian–Einstein metric is unique up to multiplication by a positive scalar, and its Hermitian Yang–Mills connection is irreducible and unique up to unitary gauge. If $E$ is polystable, write

E=\bigoplus_j E_j

with the $E_j$ stable of common slope; the direct sum of their Hermitian–Einstein connections solves the equation on $E$ . Equal slope is essential, because the central constant in the Hermitian Yang–Mills equation is determined by the slope.

Scope and conventions

The theorem is also called the Kobayashi–Hitchin correspondence in this vector-bundle setting. The numerical constant in the equation $\sqrt{-1}\Lambda_\omega F_A=\lambda I$ depends on conventions for $\Lambda_\omega$ , $c_1(E)$ , and volume, but its proportionality to $\mu_\omega(E)$ does not.

Compactness, the Kähler form used to define slope, and holomorphicity are substantive hypotheses. A merely Hermitian base or an arbitrary smooth complex bundle is outside the theorem as stated; extensions require different stability notions and analytic hypotheses.

It suffices to test the stability inequality on saturated coherent subsheaves of positive rank and rank strictly below that of $E$ . Torsion subsheaves have rank zero and are not in the range of the displayed slope test.

References

S. K. Donaldson, “Anti Self-Dual Yang-Mills Connections over Complex Algebraic Surfaces and Stable Vector Bundles,” Proceedings of the London Mathematical Society s3-50 (1985), 1–26. DOI record. Relevant: the correspondence for algebraic surfaces.
Karen Uhlenbeck and Shing-Tung Yau, “On the Existence of Hermitian-Yang-Mills Connections in Stable Vector Bundles,” Communications on Pure and Applied Mathematics 39 (1986), S257–S293. DOI record. Relevant: existence for stable bundles over compact Kähler manifolds.