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Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A Higgs bundle on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-modules $\phi: E\to E\otimes \Omega^1_X$ (the Higgs field) such that $$\phi\wedge \phi=0: E\to E\otimes \Omega^2_X.$$

If we replace the holomorphic vector bundle $E$ by a coherent sheaf $\mathcal{E}$, then we get a Higgs sheaf on $X$. See for example On Gieseker stability for Higgs sheaves.

Since the bounded derived category of coherent sheaves has been intensively studied in algebraic geometry, I wonder whether the bounded derived category of Higgs sheaves (or related concept) has ever been defined or studied in literatures.

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    $\begingroup$ This is possibly related to the full subcategory of $D^b(Coh(T^*X))$ consisting of objects whose support is finite/proper over $X$ $\endgroup$
    – Piotr Achinger
    Commented Nov 11, 2023 at 7:48
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    $\begingroup$ In a special case where $ X $ is a curve, $ \operatorname{Hom}(E, E \otimes \omega_X) \cong \operatorname{Ext}^1(E,E)^{\vee} $ by Serre duality. Therefore Higgs sheaves on $ X $ correspond to the cotangent bundle of the moduli stack of vector bundles on $ X $. $\endgroup$
    – Cranium Clamp
    Commented Nov 11, 2023 at 20:41

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