Kuznetsov’s Fano threefold conjecture via Hochschild-Serre algebra

Let $X$ be Fano variety whose semi-orthogonal decomposition for bounded derived category is given by

where $E_{1},\ldots,E_{n}$ is an exceptional collection of vector bundles over $X$ and $\mathcal{K}u(X)$ is the right orthogonal complement of the collection, called Kuznetsov component. It has been widely believed that the Kuznetsov component encodes the essential birational geometric information of the Fano varieties. Thus extracting geometric information from Kuznetsov components is an important step to understand geometry of Fano varieties. There are numerous way to extract information from Kuznetsov components, which we briefly recall as follows.

In this section, we recall the terminology of $dg$-category of matrix factorization. We follow the context in [BFK14]. We refer the reader to [Kel06] for the basic of $dg$ categories. Denote by $\operatorname{Hqe(dg-cat)}$ the localized $\operatorname{dg-cat}$ with respect to the quasi-equivalences of $dg$ categories. Let $(X,G,L,\omega)$ be a quadruple where $X$ is a quasi-projective variety with $G$ action, where $G$ is a reductive algebraic group, $L$ is a $G$-equivariant line bundle and $\omega$ is a $G$-invariant section of $L$. Our main example is $(\mathbb{A}^{n+1},\mathbb{C}^{\ast},\mathcal{O}(d),\omega)$. The $\mathbb{C}^{\ast}$ action on $\mathbb{A}^{n+1}$ is given by $\lambda\cdot(x_{0},x_{1},\cdots,x_{n})=(\lambda^{a_{0}}\cdot x_{0},\lambda^{a_{1}}\cdot x_{1},\cdots,\lambda^{a_{n}}\cdot x_{n})$, $a_{0},a_{1},\cdots,a_{n}$ are integers such that $\operatorname{gcd}(a_{0},a_{1},\cdots,a_{n})=1$. $\mathcal{O}(d)$ is the trivial line bundle twisted with the character $\mathbb{X}_{d}:\mathbb{C}^{\ast}\rightarrow\mathbb{C}^{\ast},\quad\lambda\mapsto\lambda^{d}$. $\omega$ is a $\mathbb{C}^{\ast}$-invariant section of $\mathcal{O}(d)$. Namely $\omega$ is a degree $d$ polynomial, the weight of variable $x_{i}$ is $a_{i}$.

We have $dg$ category $\operatorname{Fact}(X,G,L,\omega)$, whose objects are a quadruple $(\mathcal{E}_{-1},\mathcal{E}_{0},\Phi_{-1},\Phi_{0})$, where $\mathcal{E}_{-1}$ and $\mathcal{E}_{0}$ are $G$-equivariant quasi-coherent sheaves, $\Phi_{-1}:\mathcal{E}_{0}\rightarrow\mathcal{E}_{-1}\otimes L$ and $\Phi_{0}:\mathcal{E}_{-1}\rightarrow\mathcal{E}_{0}$ are morphism of $G$-equivariant sheaves such that

The space of morphisms in $\operatorname{Fact}(X,G,L,\omega)$ are the internal Hom of $G$-equivariant sheaves while extending the pairs of morphisms to certain $\mathbb{Z}$-graded complexes. We point out the reference [BFK14] for interested readers. There is a category $\operatorname{Acyclic}(\operatorname{Fact}(X,G,L,\omega))$ which imitates acyclic complexes in the category of complexes of sheaves. The absolute derived category $D^{abs}[\operatorname{Fact}(X,G,L,\omega)]$ is the homotopy category of $dg$ quotient $\frac{\operatorname{Fact}(X,G,L,\omega)}{\operatorname{Acyclic}(\operatorname{Fact}(X,G,L,\omega))}\in\operatorname{Hqe(dg-cat)}$. Let $\operatorname{Inj}(X,G,L,\omega)\subset\operatorname{Fact}(X,G,L,\omega)$ be the dg sub-category whose components are $G$-equivariant injective quasi-coherent sheaves. We write [$\mathcal{A}$] as the homotopic category of any dg category $\mathcal{A}$.

Let $\operatorname{Inj_{coh}}(X,G,L,\omega)\subset\operatorname{Inj}(X,G,L,\omega)$ be a $dg$ sub-category whose objects are quasi-isomorphic to objects with coherent components in category $\operatorname{Fact}(X,G,L,\omega)$.

Define shifting functor

With cone construction, the homotopic category $[\operatorname{Inj_{coh}}(X,G,L,\omega)]$ is a triangulated category which is equivalent to the category of graded matrix factorization in [Orl09] for $(\mathbb{A}^{n+1},\mathbb{C}^{\ast},\mathcal{O}(d),\omega)$.

Denote by

the twisting functor that maps

to

Clearly, we have equality of functors $\{d\}:=\{1\}^{d}=[2]$.

Let $X\subset\mathbb{P}(a_{1},a_{2},\cdots,a_{n})$ be a smooth hypersurface of degree $d\leq n$ defined by $\omega$. Let

Roughly, $\mathcal{K}u(X)$ is identified with the essential subcategory of $B$-branes of $X$ in Physics. If $X$ a is Calabi-Yau variety, $Ku(X)=D^{b}(X)$. On $\mathrm{LG}$ side, the category $[\operatorname{Inj_{coh}}(\mathbb{A}^{n+1},\mathbb{C}^{\ast},\mathcal{O}(d),\omega)]$ is identified with the category of $B$-branes of Landau-Ginzburg model. Physically, B-branes of $X$ and $LG$ model are naturally equivalent, which was proved by Orlov [Orl09] mathematically. Namely, we have equivalence

Consider the natural enhancement $\operatorname{Inj_{coh}}(X)$, and let $\mathcal{K}u_{dg}(X)$ be a $dg$ subcategory that enhance $\mathcal{K}u(X)$. Orlov’s $\sigma$ /LG correspondence can be lifted to be the equivalence of dg categories.

According to [BFK14], the natural functors can be reinterpreted as kernels of Fourier-Mukai transforms, and the natural transformations between these functors are morphism of kernels. We write $\Delta(m)$ as the kernel of functor $-\otimes\mathcal{O}_{\mathbb{A}^{n+1}}(m)$.

Next we recall a key theorem in [BFK14, Theorem 1.2]. For $g\in\mathbb{C}^{\ast}$, we write $W_{g}$ as the conormal sheaf of fixed locus $(\mathbb{A}^{n+1})^{g}$ and $k_{g}$ the character of $\operatorname{det}(W_{g})$. We write $H^{\bullet}(d\omega_{g})$ as the Koszul cohomology of the Jacobian ideal of $\omega_{g}:=\omega|_{(\mathbb{A}^{n+1})^{g}}$. Let $\gamma=e^{\frac{2\pi i}{d}}$, and $\mu_{d}=\langle 1,\gamma,\gamma^{2},\cdots,\gamma^{d-1}\rangle$.

We refer the reader to [BFK14] for details of computation. We describe the multiplication under the isomorphism of Theorem 2.4. To make this self contain, we introduce some notions used in the proof.

Let $Z$ be a quasi-projective variety with $G$ action, $G$ is an algebraic group. Let $H$ be a closed subgroup of $G$. We have an action of $H$ on $G\times Z$ defined by

The fppf quotient of $G\times Z$ of $H$ is a scheme , which is denoted as $G\times^{H}Z$, see [BFK14, Lemma 2.16]. Consider morphisms

First, the pull back functor $l^{\ast}$ define an equivalence of equivariant quasi-coherent sheaves. Namely

is an equivalence [Tho87, Lemma 1.3]. We write $\alpha_{\ast}:\operatorname{Qcoh}_{G}G\times^{H}Z\rightarrow\operatorname{Qcoh}_{G}Z$ as the push forward functor of $\alpha$, and $\alpha^{\ast}:\operatorname{Qcoh}_{G}Z\rightarrow\operatorname{Qcoh}_{G}G\times^{H}Z$ as the pull back functor of $\alpha$.

We still write $\operatorname{Ind}^{G}_{H}$ and $\operatorname{Res}^{G}_{H}$ as derived functors of derived categories of equivariant sheaves. $\operatorname{Ind}^{G}_{H}$ is right adjoint functor of $\operatorname{Res}^{G}_{H}$. In our case, $Z=\mathbb{A}^{n+1}\times\mathbb{A}^{n+1}$, $G=\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}=\{(g_{1},g_{2})\in\mathbb{C}^{\ast}\times\mathbb{C}^{\ast}|g^{d}_{1}=g^{d}_{2}\}$, and $H=\mathbb{C}^{\ast}\subset G$ via diagonal embedding. The $G$ action on $Z$ is given by

By definition,

The multiplication

maps $(a,b)$ to $ab$ is the composition

To avoid cluttering the notation, we temporarily assume $m_{1}=m_{2}=t_{1}=t_{2}=0$. The sequence (2) is equivalent to

Here $a,b$ are regarded as $\mathbb{C}^{\ast}$ equivariant morphisms in $[\operatorname{Inj_{coh}}(\mathbb{A}^{n+1},\mathbb{C}^{\ast},0)]$ via diagonal embedding $\mathbb{C}^{\ast}\hookrightarrow\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}$. The morphism $b$ here is $\mathbb{L}\Delta^{\ast}\operatorname{Res}^{\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}}_{\mathbb{C}^{\ast}}\Delta^{\ast}(b):\mathbb{L}\Delta^{\ast}\operatorname{Res}^{\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}}_{\mathbb{C}^{\ast}}\operatorname{Ind}_{\mathbb{C}^{\ast}}^{\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}}\mathcal{O}_{\mathbb{A}^{n+1}}\rightarrow\mathbb{L}\Delta^{\ast}\operatorname{Res}^{\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}}_{\mathbb{C}^{\ast}}\operatorname{Ind}_{\mathbb{C}^{\ast}}^{\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}}\Delta_{\ast}\mathcal{O}_{\mathbb{A}^{n+1}}$.

Next, let $\Gamma_{g}$ be the graph $x\mapsto(g\cdot x,x)$. The $\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}$ action on $\bigoplus_{g\in\mu_{d}}\mathcal{O}_{\Gamma_{g}}$ is defined by $(g_{1},g_{2})\cdot(gx,\cdot x_{1})=(g_{1}g\cdot x,g_{2}\cdot x)$. By [BFK14, Lemma 5.31], $\operatorname{Ind}_{\mathbb{C}^{\ast}}^{\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}}\Delta_{\ast}\mathcal{O}_{\mathbb{A}^{n+1}}\cong\bigoplus_{g\in\mu_{d}}\mathcal{O}_{\Gamma_{g}}$ as $\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}$ sheaves. Note that the decomposition of the object $\operatorname{Ind}_{\mathbb{C}^{\ast}}^{\mathbb{C}^{\ast}\times_{\mathbb{C}^{\ast}}\mathbb{C}^{\ast}}\Delta_{\ast}\mathcal{O}_{\mathbb{A}^{n+1}}$ into a direct sum corresponds exactly to the decomposition in Proposition 2.4, which identifies different Hom-spaces with the pieces of Jacobian rings.

Let $a=(f_{1},f_{\gamma},\cdots,f_{\gamma^{j}},\cdots f_{\gamma^{d-1}})\in\mathrm{Hom}(\Delta,\Delta(m_{1})[t_{1}])\cong\bigoplus^{d-1}_{j=0}\mathrm{Jac}(\omega_{\gamma^{j}})_{m_{1}-k_{\gamma^{j}}+d\frac{t_{1}-rkW_{\gamma^{j}}}{2}}$, and $b=~{}(g_{1},g_{\gamma},\cdots,g_{\gamma^{j}},\cdots,g_{\gamma^{d-1}})\in\mathrm{Hom}(\Delta,\Delta(m_{2})[t_{2}])\cong\bigoplus^{d-1}_{j=0}\mathrm{Jac}(\omega_{\gamma^{j}})_{m_{2}-k_{\gamma^{j}}+d\frac{t_{2}-rkW_{\gamma^{j}}}{2}}$.