# $Ext$-algebra of stable vector bundles

Let $$X$$ be a smooth projective variety over $$\mathbb{C}$$ and $$E$$ a slope-stable vector bundle on $$X$$ with regard to some ample line bundle $$H$$.

Question: What can we say about the algebra structure of $$Ext^{\ast}(E,E)$$?

Since this is a fairly general question, let me be more precise.

Let us for simplicity assume that $$E$$ is a smooth point in the moduli spaces of stable sheaves $$M_H(v)$$ with $$v=v(E)$$ its Mukai vector. Are there some general results on the algebra structure $$Ext^{\ast}(E,E)$$? In particular, do we know how the (Yoneda) product looks like or what properties it has?

For curves and Fano surfaces for example the product is trivial, whereas for K3 surfaces the pairing $$Ext^1(E,E) \times Ext^1(E,E) \to Ext^2(E,E)\cong Hom(E,E) \cong \mathbb{C}$$ is perfect and skew-symmetric. I was wondering if there are any known structural results in higher dimensions?

For example, is the product (e.g. on Calabi-Yau varieties, as suggested by the K3 case) graded commutative? Since $$E$$ is assumed a smooth point in moduli, the obstructions to deform $$E$$ vanish and so the Maurer-Cartan equation gives that the product $$Ext^1(E,E) \times Ext^1(E,E) \to Ext^2(E,E)$$ is skew-symmetric

More generally, one can consider $$RHom(E,E)$$ as a differential-graded Lie algebra and as such one can associate to it a deformation functor which in this case controls the local deformation theory of $$E$$. This can be quite complicated, so for simplicity let us assume that this differential-graded Lie algebra is formal, i.e. quasi-isomorphic to its cohomology which is precisely $$Ext^{\ast}(E,E)$$. Are there on a smooth projective $$X$$ any a priori constraints on the algebra structure of $$Ext^{\ast}(E,E)$$?

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.

• Thank you very much for this great answer! May I quickly ask how you apply the result of Suarez-Alvarez to obtain that $Ext^{\ast}(E,E)$ is graded-commutative, i.e. what is the suspended monoidal category such that the endomorphism ring of the unit object equals $Ext^{\ast}(E,E)$? Jun 14, 2021 at 12:31
• @NicoBerger : you should have a look at section 2 of Suarez-Alvarez paper (and more precisely, section 2.1). Jun 14, 2021 at 20:20
• I am sorry, I still have troubles understanding this in general. If we take in section 2.1 $\mathcal{C}=Coh(X)$, then the unit object is $\mathcal{O}_X$ and one concludes that $Ext^{\ast}(\mathcal{O}_X,\mathcal{O}_X)$ is graded-commutative. So this also holds for objects in the orbit of $\mathcal{O}_X$ under $Aut(\mathrm{D}^b(X))$. But for a general slope-stable vector bundle $E$ what is the category $\mathcal{C}$ to which we apply the discussion of section 2.1? Jun 15, 2021 at 7:36
• @NicoBerger : you are right. I have had a wrong recollection of Suarez-Alvarez result. I used it some time ago to highlight some properties of Hochschild cohomology (where it applies, see section 2.5) and I wrongly remembered it was correct in a larger context. I will edit my answer. Jun 15, 2021 at 13:20