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There have been recently various results revolving around this question. Let me quote a few:

$\bullet$ For any line bundle $L$ on $X$, the graded algebra $\mathrm{Ext}^*(L,L)$ is always graded-commutative. More generally, for any autoequivalence $\Phi$ of $\mathrm{D}^b(X)$, the graded algebra $\mathrm{Ext}^*(\Phi(\mathcal{O}_X),\Phi(\mathcal{O}_X))$ is graded commutative. See for instance this short proof by Suarez-Alvarez.

$\bullet$ If $\Delta_X$ is the diagonal in $X \times X$, the the graded algebra $\mathrm{Ext}^*(\mathcal{O}_{\Delta_X},\mathcal{O}_{\Delta_X})$ is graded commutative. Hence, the Hochschild cohomology algebra on $X$ is graded commutative. This is eually proved in the paper by Suarez-Alvarez I mentionned above.

$\bullet$ If $\mathrm{rank}(E) \neq 0$, then the trace map shows that $\mathrm{Ext}^{*}(E,E)$ has the structure of a faithful $H^{*}(\mathcal{O}_X)$-algebra. This algebra structure is conjectured to be a derived invariant (in car $0$). This is proved in dimension $\leq 4$ (and is some other situations related to moduli theory). It will be disproved in car $p>0$ in a forthcoming paper of Addington and Bragg.

$\bullet$ Hochenegger and Krug proved that for any $E \in \mathrm{D}^b(X)$, if $\mathrm{Ext}^*(E,E) = k[t]/t^{n +1}$ with $\deg(t) \geq 2$, then the DG-algebra $\mathrm{RHom}(E,E)$ is automatically formal.

There have been recently various results revolving around this question. Let me quote a few:

$\bullet$ For any line bundle $L$ on $X$, the graded algebra $\mathrm{Ext}^*(L,L)$ is always graded-commutative. More generally, for any autoequivalence $\Phi$ of $\mathrm{D}^b(X)$, the graded algebra $\mathrm{Ext}^*(\Phi(\mathcal{O}_X),\Phi(\mathcal{O}_X))$ is graded commutative. See for instance this short proof by Suarez-Alvarez.

$\bullet$ If $\Delta_X$ is the diagonal in $X \times X$, then the graded algebra $\mathrm{Ext}^*(\mathcal{O}_{\Delta_X},\mathcal{O}_{\Delta_X})$ is graded commutative. Hence, the Hochschild cohomology algebra on $X$ is graded commutative. This is equally proved in the paper by Suarez-Alvarez I mentionned above.

$\bullet$ If $\mathrm{rank}(E) \neq 0$, then the trace map shows that $\mathrm{Ext}^{*}(E,E)$ has the structure of a faithful $H^{*}(\mathcal{O}_X)$-algebra. This algebra structure is conjectured to be a derived invariant (in car $0$). This is proved in dimension $\leq 4$ (and is some other situations related to moduli theory). It will be disproved in car $p>0$ in a forthcoming paper of Addington and Bragg.

$\bullet$ Hochenegger and Krug proved that for any $E \in \mathrm{D}^b(X)$, if $\mathrm{Ext}^*(E,E) = k[t]/t^{n +1}$ with $\deg(t) \geq 2$, then the DG-algebra $\mathrm{RHom}(E,E)$ is automatically formal.

There have been recently various results revolving around this question. Let me quote a few:

$\bullet$ The graded algebra $\mathrm{Ext}^*(E,E)$ is indeed always graded-commutative, see for instance this short proof by Suarez-Alvarez.

$\bullet$ If $\mathrm{rank}(E) \neq 0$, then the trace map shows that $\mathrm{Ext}^{*}(E,E)$ has the structure of a faithful $H^{*}(\mathcal{O}_X)$-algebra. This algebra structure is conjectured to be a derived invariant (in car $0$). This is proved in dimension $\leq 4$ (and is some other situations related to moduli theory). It will be disproved in car $p>0$ in a forthcoming paper of Addington and Bragg.

$\bullet$ Hochenegger and Krug proved that for any $E \in \mathrm{D}^b(X)$, if $\mathrm{Ext}^*(E,E) = k[t]/t^{n +1}$ with $\deg(t) \geq 2$, then the DG-algebra $\mathrm{RHom}(E,E)$ is automatically formal.

There have been recently various results revolving around this question. Let me quote a few:

$\bullet$ For any line bundle $L$ on $X$, the graded algebra $\mathrm{Ext}^*(L,L)$ is always graded-commutative. More generally, for any autoequivalence $\Phi$ of $\mathrm{D}^b(X)$, the graded algebra $\mathrm{Ext}^*(\Phi(\mathcal{O}_X),\Phi(\mathcal{O}_X))$ is graded commutative. See for instance this short proof by Suarez-Alvarez.

$\bullet$ If $\Delta_X$ is the diagonal in $X \times X$, the the graded algebra $\mathrm{Ext}^*(\mathcal{O}_{\Delta_X},\mathcal{O}_{\Delta_X})$ is graded commutative. Hence, the Hochschild cohomology algebra on $X$ is graded commutative. This is eually proved in the paper by Suarez-Alvarez I mentionned above.

$\bullet$ If $\mathrm{rank}(E) \neq 0$, then the trace map shows that $\mathrm{Ext}^{*}(E,E)$ has the structure of a faithful $H^{*}(\mathcal{O}_X)$-algebra. This algebra structure is conjectured to be a derived invariant (in car $0$). This is proved in dimension $\leq 4$ (and is some other situations related to moduli theory). It will be disproved in car $p>0$ in a forthcoming paper of Addington and Bragg.

$\bullet$ Hochenegger and Krug proved that for any $E \in \mathrm{D}^b(X)$, if $\mathrm{Ext}^*(E,E) = k[t]/t^{n +1}$ with $\deg(t) \geq 2$, then the DG-algebra $\mathrm{RHom}(E,E)$ is automatically formal.

There have been recently various results revolving around this question. Let me quote a few:

$\bullet$ The graded algebra $\mathrm{Ext}^*(E,E)$ is indeed always graded-commutative, see for instance this short proof by Suarez-Alvarez.

$\bullet$ If $\mathrm{rank}(E) \neq 0$, then the trace map shows that $\mathrm{Ext}^{*}(E,E)$ has the structure of a faithful $H^{*}(\mathcal{O}_X)$-algebra. This algebra structure is conjectured to be a derived invariant (in car $0$). This is proved in dimension $\leq 4$ (and is some other situations related to moduli theory). It will be disproved in car $p>0$ in a forthcoming paper of Addington and Bragg.

$\bullet$ Hochenegger and Krug proved that for any $E \in \mathrm{D}^b(X)$, if $\mathrm{Ext}^*(E) = k[t]/t^{n +1}$ with $\deg(t) \geq 2$, then the DG-algebra $\mathrm{RHom}(E,E)$ is automatically formal.

There have been recently various results revolving around this question. Let me quote a few:

$\bullet$ The graded algebra $\mathrm{Ext}^*(E,E)$ is indeed always graded-commutative, see for instance this short proof by Suarez-Alvarez.

$\bullet$ If $\mathrm{rank}(E) \neq 0$, then the trace map shows that $\mathrm{Ext}^{*}(E,E)$ has the structure of a faithful $H^{*}(\mathcal{O}_X)$-algebra. This algebra structure is conjectured to be a derived invariant (in car $0$). This is proved in dimension $\leq 4$ (and is some other situations related to moduli theory). It will be disproved in car $p>0$ in a forthcoming paper of Addington and Bragg.

$\bullet$ Hochenegger and Krug proved that for any $E \in \mathrm{D}^b(X)$, if $\mathrm{Ext}^*(E,E) = k[t]/t^{n +1}$ with $\deg(t) \geq 2$, then the DG-algebra $\mathrm{RHom}(E,E)$ is automatically formal.