We now recall why the HYM equation implies $\mu$-semistability. Let $E^{\prime}\subset E$ be a holomorphic subbundle (or more generally, a proper coherent subsheaf). A basic feature of holomorphic geometry is that pointwise curvature decreases in subbundles: the Chern curvature $F_{E^{\prime},H}$ for the restricted Hermitian metric $H|_{E^{\prime}}$ satisfies

$$\sqrt{-1}F_{E^{\prime},H}\leq\sqrt{-1}F_{E,H}|_{E^{\prime}}.$$

Wedging both sides with $\omega_{X^{\vee}}^{n-1}$, taking the trace, and integrating over ${X^{\vee}}$, we get

$$\int_{X^{\vee}}c_{1}(E^{\prime})\wedge\omega_{X^{\vee}}^{n-1}\leq\mu(E)rk(E^{\prime}),$$

namely $\mu(E^{\prime})\leq\mu(E)$, which is $\mu$-semistability. Although this argument is very transparent, we wish to summarize its key features:

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  Even though connections and curvatures make sense in a more general setting, we need the integrability of Kähler geometry to obtain pointwise positivity.²²²² 22 It would be interesting if the physicists can explain this positivity from supersymmetry, which is closely related to the Kähler condition.

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  To derive $\mu$-stability, one integrates over $X^{\vee}$, which can be interpreted as the moduli space of constant maps into $X^{\vee}$. ²³²³ 23 Path integrals on the topological B-model typically localizes to the moduli space of constant maps. This suggests a worldsheet interpretation, which will be more apparent on the mirror side.

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  The input from complex geometry can be interpreted as a short exact sequence

  $$0\to E^{\prime}\to E\to E/E^{\prime}\to 0,$$

  which has a categorical meaning in $D^{b}Coh(X^{\vee})$ as a distinguished triangle.

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  The role of the Kähler form enters via cohomological integrals.

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  There is no need for the complex Monge-Ampère equation.

Most of these features are not specific to HYM connections, but similar arguments give rise to obstructions for a large class of PDEs involving holomorphic bundles, such as the deformed Hermitian Yang-Mills equation.