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Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a projective variety? Ideally, I would also like the Chern character of this object to be non-zero.

In case $H^1(\mathcal{O}_X) =0$, any line bundle will do the job. On the other hand, if $H^1(\mathcal{O}_X) \neq 0$, the existence of the trace maps shows that the rank of such an object must be zero. I have some specific examples in mind (mostly structure sheaves of rigid subvarieties of some special varieties), but I would like to know if such objects exist in general on any smooth projective variety.

Edit: as Johan elliptically points out in the comment, the Grothendieck-Riemann-Roch Theorem shows that $\chi(F,F) =0$ for $F \in D^b(X)$, when $X$ is an elliptic curve. In particular, if $F$is a coherent sheaf, the non vanishing of $\mathrm{Hom}(F,F)$ implies necessarily that $\mathrm{Ext}^1(F,F) \neq 0$, as there are no higher Ext's. On the other hand, this line of reasoning doesn't yield anything for a general object, as there might be higher Ext's, even negative ones.

Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a projective variety of dimension bigger or equal to $2$ (see the edit below for comments on the dimensional hypothesis). Ideally, I would also like the Chern character of this object to be non-zero.

In case $H^1(\mathcal{O}_X) =0$, any line bundle will do the job. On the other hand, if $H^1(\mathcal{O}_X) \neq 0$, the existence of the trace maps shows that the rank of such an object must be zero. I have some specific examples in mind (mostly structure sheaves of rigid subvarieties of some special varieties), but I would like to know if such objects exist in general on any smooth projective variety.

Edit: as Johan elliptically points out in the comments, the Grothendieck-Riemann-Roch Theorem shows that $\chi(F,F) =0$ for $F \in D^b(X)$, when $X$ is an elliptic curve. In particular, if $F$ is a coherent sheaf, the non vanishing of $\mathrm{Hom}(F,F)$ implies necessarily that $\mathrm{Ext}^1(F,F) \neq 0$, as there are no higher Ext's. On the other hand, we know that that an object in the derived category of an elliptic curve is quasi-isomorphic to the direct sum of its shifted cohomology sheaves. From this, we can deduce that all objects have non vanishing $\mathrm{Ext}^1$.

This seems however a very specific phenomenon related to curve (as the category $Coh(X)$ is then hereditary and any object in the derived category is quasi-isomorphic to a sum of shifted coherent sheaves). This is why I will make an assumption on $\dim X$.

Let $X$ be a smooth projective variety (say over $\mathbb{C}$) with $\dim X \geq 3$ (see the edits for comments on this hypothesis). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a projective variety? Ideally, I would also like the Chern character of this object to be non-zero.

In case $H^1(\mathcal{O}_X) =0$, any line bundle will do the job. On the other hand, if $H^1(\mathcal{O}_X) \neq 0$, the existence of the trace maps shows that the rank of such an object must be zero. I have some specific examples in mind (mostly structure sheaves of rigid subvarieties of some special varieties), but I would like to know if such objects exist in general on any smooth projective variety.

Edit: as Johan elliptically points out in the comment, the Grothendieck-Riemann-Roch Theorem shows that $\chi(F,F) =0$ for any $F \in D^b(X)$, when $X$ is an elliptic curve. In particular, the non vanishing of $\mathrm{Hom}(F,F)$ implies necessarily that $\mathrm{Ext}^1(F,F) \neq 0$. On the other hand, this seems to be a phenomenon specific to curve ( because there can only be $\mathrm{Ext}^0$ and $\mathrm{Ext}^1$), so I will assume $\dim X \geq 3$.

Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a projective variety? Ideally, I would also like the Chern character of this object to be non-zero.

In case $H^1(\mathcal{O}_X) =0$, any line bundle will do the job. On the other hand, if $H^1(\mathcal{O}_X) \neq 0$, the existence of the trace maps shows that the rank of such an object must be zero. I have some specific examples in mind (mostly structure sheaves of rigid subvarieties of some special varieties), but I would like to know if such objects exist in general on any smooth projective variety.

Edit: as Johan elliptically points out in the comment, the Grothendieck-Riemann-Roch Theorem shows that $\chi(F,F) =0$ for $F \in D^b(X)$, when $X$ is an elliptic curve. In particular, if $F$is a coherent sheaf, the non vanishing of $\mathrm{Hom}(F,F)$ implies necessarily that $\mathrm{Ext}^1(F,F) \neq 0$, as there are no higher Ext's. On the other hand, this line of reasoning doesn't yield anything for a general object, as there might be higher Ext's, even negative ones.

Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a projective variety? Ideally, I would also like the Chern character of this object to be non-zero.

In case $H^1(\mathcal{O}_X) =0$, any line bundle will do the job. On the other hand, if $H^1(\mathcal{O}_X) \neq 0$, the existence of the trace maps shows that the rank of such an object must be zero. I have some specific examples in mind (mostly structure sheaves of rigid subvarieties of some special varieties), but I would like to know if such objects exist in general on any smooth projective variety.

Let $X$ be a smooth projective variety (say over $\mathbb{C}$) with $\dim X \geq 3$ (see the edits for comments on this hypothesis). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a projective variety? Ideally, I would also like the Chern character of this object to be non-zero.

In case $H^1(\mathcal{O}_X) =0$, any line bundle will do the job. On the other hand, if $H^1(\mathcal{O}_X) \neq 0$, the existence of the trace maps shows that the rank of such an object must be zero. I have some specific examples in mind (mostly structure sheaves of rigid subvarieties of some special varieties), but I would like to know if such objects exist in general on any smooth projective variety.

Edit: as Johan elliptically points out in the comment, the Grothendieck-Riemann-Roch Theorem shows that $\chi(F,F) =0$ for any $F \in D^b(X)$, when $X$ is an elliptic curve. In particular, the non vanishing of $\mathrm{Hom}(F,F)$ implies necessarily that $\mathrm{Ext}^1(F,F) \neq 0$. On the other hand, this seems to be a phenomenon specific to curve ( because there can only be $\mathrm{Ext}^0$ and $\mathrm{Ext}^1$), so I will assume $\dim X \geq 3$.