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Calabi-Yau algebra

Idea

The notion of Calabi-Yau algebra is an algebraic incarnation of the notion of Calabi-Yau manifold .

Definition

Definition

For $A$ a dg-algebra and $N$ a dg-bimodule over $A$, write

$$N^{!}:={\mathrm{RHom}}_{A\mathrm{Bimod}}(N,A\otimes A)$$N^! := RHom_{A Bimod}(N, A \otimes A)

for the dual $A$-bimodule, where $\mathrm{RHom}$ denotes the right derived hom-functor with respect to the model structure on dg-modules.

Definition

A homologically smooth dg-algebra $A$ is a Calabi-Yau algebra of dimension $d$ if there is a quasi-isomorphism of $A$-bimodules

$$f:A\stackrel{\simeq }{\to }{A}^{!}[d]$$f : A \stackrel{\simeq}{\to} A^![d]

such that

$$f\simeq f^{!}[d].$$f \simeq f^![d] \,.

This is (Ginzburg, def. 3.2.3).

Properties

Let $X$ be a smooth quasi-projective variety. Write $D^b(\mathrm{Coh}X)$ for the derived category of bounded chain complexes of coherent sheaves over $X$.

Definition

An object $ℰ\in D^b(\mathrm{Coh}X)$ is called a tilting generator if the Ext-functor satisfies

1. ${\mathrm{Ext}}^i(ℰ,ℰ)=0$ for all $i>0$;

2. ${\mathrm{Ext}}^{•}(ℰ,ℱ)=0$ implies $ℱ=0$;

3. the endomorphism algebra $\mathrm{End}(ℰ)=\mathrm{Hom}(ℰ,ℰ)$ has finite Hochschild dimension.

This appears as (Ginzburg, def. 7.1.1).

Remark

For $ℰ$ a tilting generator there is an equivalence of triangulated categories

$$D^b(\mathrm{Coh}X)\stackrel{\simeq }{\to }{D}^b(\mathrm{End}(ℰ)\mathrm{Mod})$$D^b(Coh X) \stackrel{\simeq}{\to} D^b(End(\mathcal{E})Mod)

to the derived category of modules over $\mathrm{End}(ℰ)$.

Proposition

For $X$ smooth connected variety which is projective over an affine variety, let $ℰ\mathrm{in}{D}^b(\mathrm{Coh}X)$ be a tilting generator, def. 3.

Then $\mathrm{End}ℰ$ is a Calabi-Yau algebra of dimension $d$ precisely if $X$ is a Calabi-Yau manifold of dimension $d$.

This appears as (Ginzburg, prop. 3.3.1).

Related concepts

• Calabi-Yau object

  • Calabi-Yau algebra, Calabi-Yau manifold

References

• Victor Ginzburg, Calabi-Yau algebras (arXiv:0612139)