# Existence of rigid objects in the derived category of a smooth projective variety

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 $$\dim X$$.

• Hint: elliptic curve. Jul 26, 2022 at 19:18
• The zero object is rigid. Jul 26, 2022 at 21:13
• @JasonStarr : good point! This is why I asked that, ideally, the Chern character would be non zero. Jul 26, 2022 at 21:19
• If you consider the "filtration" by good truncations, I believe that every rigid object on an elliptic curve is quasi-isomorphic to zero. Jul 27, 2022 at 0:12
• @JasonStarr : That may be. On the other hand, I know prove in the edit that any non zero object on an elliptic curve has non vanishing Ext^1. This seems however very specific to the case of curve ( because then $Coh(X)$ is hereditary) and I have added an extra hypothesis on $\dim X$ to improve my question. Jul 27, 2022 at 6:05

Let $$A$$ be an Abelian variety. For every scheme $$S$$ and every $$S$$-valued point $$x\in A(S)$$, denote by $$\mu_x$$ the associated translation automorphism of the $$S$$-scheme $$S\times A$$, i.e., $$\mu_x(y) = x+y$$.
Denote by $$\widehat{A}$$ together with the invertible sheaf $$\mathcal{P}$$ on $$\widehat{A}\times A$$ the relative $$\text{Pic}^0$$ of $$A$$, normalized so that $$\mathcal{P}|_{\widehat{A}\times\{0\}}$$ is the structure sheaf on $$\widehat{A}$$. Of course $$\widehat{A}\times A$$ is a commutative group scheme with its structure as the product of two commutative group schemes. Denote by $$G$$ the noncommutative group scheme structure on $$\widehat{A}\times A$$ defined by $$([\mathcal{L}],x)\bullet([\mathcal{M}],y) = ([\mathcal{L}\otimes \mu_x^*\mathcal{M}],x+y).$$ There is an "action" of $$G$$ on the bounded derived category of coherent sheaves on $$A$$ that associates to each $$([\mathcal{L}],x)$$ in $$G$$ and each bounded complex $$C^\bullet$$ of coherent sheaves on $$A$$ the associated bounded complex of coherent sheaves, $$\mathcal{L}\otimes \mu_x^*(C^\bullet).$$ (According to an article of Orlov, this action identifies $$G$$ with the identity component of the group of autoequivalences of the bounded derived category of coherent sheaves on $$A$$.)
If $$C^\bullet$$ is not quasi-isomorphic to the zero complex, i.e., if it is not an exact complex, there there exists an integer $$p$$ such that the cohomology sheaf $$h^p(C^\bullet)$$ is nonzero. In general, a "flat deformation" of the complex $$C^\bullet$$ does not necessarily give rise to a flat deformation of the coherent sheaf $$h^p(C^\bullet)$$, since base change is not left exact. However, for a connected, smooth group scheme $$G$$, for a $$G$$-equivariant family of deformations over $$G$$, the coherent sheaves $$h^p$$ are compatible with base change: this holds over a dense open of $$G$$ (since $$G$$ is reduced), and this dense open is $$G$$-invariant, thus it is all of $$G$$. Therefore, the deformations of $$C^\bullet$$ arising from the action of $$G$$ give rise to a deformation of $$h^p(C^\bullet)$$.
By hypothesis, the coherent sheaf $$h^p(C^\bullet)$$ on $$A$$ is nonzero. If the rank is positive, then the action of the normal subgroup $$\widehat{A}\times\{0\}$$ of $$G$$ on this sheaf is nontrivial by considering "det" of the coherent sheaf. If the rank of the sheaf is zero, i.e., if the support of the sheaf is a proper closed subscheme of $$A$$, then the action of the subgroup $$\{[\mathcal{O}_A]\}\times A$$ of $$G$$ on the sheaf is nontrivial since it "moves" this proper closed subscheme. Either way, the action of the group scheme $$G$$ on the sheaf is nontrivial.
Since the action of $$G$$ already produces nontrivial deformations of the sheaf $$h^p(C^\bullet)$$, it also produces nontrivial deformations of the complex $$C^\bullet$$. Thus, the only rigid complexes in the bounded derived category of $$A$$ are quasi-isomorphic to zero.