##
**A category of kernels for equivariant factorizations and its implications for Hodge theory.**
*(English)*
Zbl 1401.14086

In the paper under review the authors prove that, roughly speaking, the continuous internal Hom between dg-categories of equivariant factorizations in the homotopy category of \(k\)-linear dg-categories is again a category of factorizations.

Let \(X\) be a smooth variety over an algebraically closed field \(k\) of characteristic zero equipped with an action of an affine algebraic group \(G\). Write \(\pi. G\times X\to X\) for the projection and \(\sigma: G\times X\to X\) for the action. Recall that a \(G\)-equivariant quasi-coherent sheaf on \(X\) is a quasi-coherent sheaf \(\mathcal{F}\) together with an isomorphism \(\theta: \sigma^*\mathcal{F}\to \pi^*\mathcal{F}\) satisfying a cocycle condition. This gives an abelian category \(\mathrm{Qcoh}_GX\). If \(\mathcal{F}\) is coherent or locally free, we accordingly call \((\mathcal{F},\theta)\) coherent or locally free. Let \(w\) be a \(G\)-invariant global section of an invertible equivariant sheaf \(\mathcal{L}\). A quasi-coherent matrix factorization of \(w\) is given by equivariant quasi-coherent sheaves \(\mathcal{E}_{-1}\), \(\mathcal{E}_0\) together with maps \(\phi_0^{\mathcal E}: \mathcal E_{-1}\to \mathcal{E}_{0}\) and \(\phi_{-1}^{\mathcal E}: \mathcal E_{0}\to \mathcal{E}_{-1}\otimes \mathcal{L}\) satisfying \[ \phi_{-1}^{\mathcal E}\circ\phi_{0}^{\mathcal E}=w=(\phi_{0}^{\mathcal E}\otimes\mathcal{L})\circ\phi_{-1}^{\mathcal E}. \] Denote such an object by \(\mathcal{E}\). The morphisms between two matrix factorizations form a complex, where, for instance, \[ \operatorname{Hom}^{2l}(\mathcal{E},\mathcal{F}):=\mathrm{Hom}(\mathcal{E}_{-1},\mathcal{F}_{-1}\otimes\mathcal{L}^l)\oplus {\operatorname{Hom}}(\mathcal{E}_{0},\mathcal{F}_{0}\otimes\mathcal{L}^l) \] for \(l\in \mathbb{Z}\) and the differential is given by an explicit formula. Hence, matrix factorizations form a dg-category, denoted by \(\mathrm{Fact}(X,G,w)\). Considering matrix factorizations with injective components gives a dg-category \(\mathrm{Inj}(X,G,w)\).

The main result of the paper under review can be described as follows. Let \((X,G,w)\) be as above. Let \(H\) be an affine algebraic group acting on a smooth variety \(Y\) and let \(v\) be a \(G\)-invariant section of an \(H\)-equivariant line bundle \(\mathcal{L}'\) on \(Y\). Let \(U(\mathcal{L})\) be the complement of the zero section in the geometric vector bundle corresponding to \(\mathcal{L}\), and denote the same construction for \(\mathcal{L}'\) by \(U(\mathcal{L}')\). Denote the functions induced by \(w\) and \(v\) on \(U(\mathcal{L})\) and \(U(\mathcal{L}')\) by \(f_w\) and \(f_v\), respectively. Set \(-f_w\boxplus f_v=-f_w\otimes_k 1+1\otimes_k f_v\). Note that \(G\times H\) acts on \(U(\mathcal{L})\times U(\mathcal{L}')\) and allow \(\mathbb{G}_m\) to scale the fibres of \(U(\mathcal{L})\times U(\mathcal{L}')\) diagonally. The authors prove that there is an equivalence \[ \mathrm{RHom}_c(\mathrm{Inj}(X,G,w),\mathrm{Inj}(Y,H,v))\cong\mathrm{Inj}(U(\mathcal{L})\times U(\mathcal{L}'), G\times H\times \mathbb{G}_m,-f_w\boxplus f_v) \] in the homotopy category of \(k\)-linear dg-categories. So, roughly speaking, the functors between categories of matrix factorizations are again given by matrix factorizations. The above result is used to compute the (extended) Hochschild cohomology of \((X,G,w)\) when \(X\) is affine and \(G\), \(w\) satisfy some technical assumptions.

The story begins with a thorough introduction to equivariant sheaves. The authors define the abelian category of quasi-coherent equivariant sheaves, restriction and inflation functors, pullback etc. They also study the global dimension of \(\mathrm{Qcoh}_GX\).

Section 3 is devoted to equivariant factorizations. The dg-categories mentioned above and their variants (for instance, one involving coherent sheaves) are defined and studied. For example, the authors define a dg-functor \(\mathrm{Fact}(X,G,w)\otimes_k \mathrm{Fact}(X,G,v)\to \mathrm{Fact}(X,G,w+v)\), a version of the \(\mathcal{H}om\)-functor in this setting, an appropriate notion of box product, restriction and inflation functors for the case of a subgroup \(H\) of \(G\), and establish results relating various of these functors.

In the same section the authors also introduce and study the absolute derived category of matrix factorizations, following work of Positselski. The idea of the construction is roughly as follows. To the dg-category \(\mathrm{Fact}(X,G,w)\) one can associate an abelian category having the same objects, but where the morphisms between two factorizations are given by closed degree-zero morphisms between them in \(\mathrm{Fact}(X,G,w)\). The resulting category is denoted by \(Z^0\mathrm{Fact}(X,G,w)=\mathcal{A}\). Given a complex with objects from \(\mathcal{A}\), one can define a matrix factorization, called its totalization. Considering the subcategory of \(\mathrm{Fact}(X,G,w)\) consisting of totalizations of bounded exact complexes from \(\mathcal{A}\) gives the subcategory of acyclic factorizations. The absolute derived category \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) is then defined as the Verdier quotient of \([\mathrm{Fact}(X,G,w)]\), the homotopy category of \(\mathrm{Fact}(X,G,w)\) (which is a triangulated category), by the homotopy category of acyclic factorizations. Of course, there are variants of this definition if one works with coherent or locally free factorizations.

One of the good properties \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) has is that it is a compactly-generated triangulated category and equivalent to \([\mathrm{Inj}(X,G,w)]\). Furthermore, the authors show that the idempotent completion of the coherent version \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) is equivalent to the homotopy category of those factorizations of \(\mathrm{Inj}(X,G,w)\) which are compact in \([\mathrm{Inj}(X,G,w)]\). Another advantage of the absolute derived category is the existence of an essentially surjective functor from a certain singularity category to it, allowing one to use geometry when proving statements about the absolute derived category. More precisely, let \(Y\) be the vanishing locus of \(w\). Under some technical conditions, there is an essentially surjective functor \[ \mathrm{D}^{\mathrm{sg}}_G(Y):=\mathrm{D}^{\mathrm{b}}(\mathrm{Coh}_GY)/\mathrm{Perf}_GY\to \mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)], \] where \(\mathrm{Perf}_GY\) is the subcategory of perfect complexes, that is, bounded complexes of locally free \(G\)-equivariant sheaves of finite rank. The functor actually exists in greater generality, since one can consider categories supported on a closed \(G\)-invariant subset of \(Y\).

In Section 4 generation of equivariant derived categories is studied. More precisely, one wants to find a set of generators for the bounded derived category of coherent \(G\)-equivariant sheaves \(\mathrm{D}^{\mathrm{b}}(\mathrm{Coh}_G X)\), where \(X\) is a singular variety equipped with a \(G\)-action. The rough idea is to focus on \(\mathrm{D}^{\mathrm{b}}(\mathrm{Qcoh}_G X)\). Using the essentially surjective functor mentioned above, the authors can then produce a set of generators for an appropriate absolute derived category.

In the following section the main result is proved. For this the authors need to recall some facts from the theory of dg-categories and, in particular, results by Toën concerning the construction of the internal (continuous) Hom in the homotopy category of dg-categories. One of the main steps towards the proof of the main result is an equivalence between some absolute derived category and the derived category of dg-modules over the tensor product of matrix factorizations categories, and this is where the generators from the previous section come in, since the proof involves showing that a compact generating set is sent to a compact generating set and is fully faithful on these. In the same section the authors define and study the (extended) Hochschild (co)homology and compute it in the case mentioned above.

The last section is devoted to two applications. Firstly, combining the computation of the extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov relating the bounded derived category of coherent sheaves on a smooth complex hypersurface \(Z\) in projective space with equivariant matrix factorizations allows the authors to recover Griffiths’ description of the primitive cohomology of \(Z\) in terms of homogeneous pieces of the Jacobian algebra of the polynomial defining \(Z\). As a second application the authors give a new proof of the Hodge conjecture for self-products of a particular \(K3\) surface closely related to the Fermat cubic fourfold.

Let \(X\) be a smooth variety over an algebraically closed field \(k\) of characteristic zero equipped with an action of an affine algebraic group \(G\). Write \(\pi. G\times X\to X\) for the projection and \(\sigma: G\times X\to X\) for the action. Recall that a \(G\)-equivariant quasi-coherent sheaf on \(X\) is a quasi-coherent sheaf \(\mathcal{F}\) together with an isomorphism \(\theta: \sigma^*\mathcal{F}\to \pi^*\mathcal{F}\) satisfying a cocycle condition. This gives an abelian category \(\mathrm{Qcoh}_GX\). If \(\mathcal{F}\) is coherent or locally free, we accordingly call \((\mathcal{F},\theta)\) coherent or locally free. Let \(w\) be a \(G\)-invariant global section of an invertible equivariant sheaf \(\mathcal{L}\). A quasi-coherent matrix factorization of \(w\) is given by equivariant quasi-coherent sheaves \(\mathcal{E}_{-1}\), \(\mathcal{E}_0\) together with maps \(\phi_0^{\mathcal E}: \mathcal E_{-1}\to \mathcal{E}_{0}\) and \(\phi_{-1}^{\mathcal E}: \mathcal E_{0}\to \mathcal{E}_{-1}\otimes \mathcal{L}\) satisfying \[ \phi_{-1}^{\mathcal E}\circ\phi_{0}^{\mathcal E}=w=(\phi_{0}^{\mathcal E}\otimes\mathcal{L})\circ\phi_{-1}^{\mathcal E}. \] Denote such an object by \(\mathcal{E}\). The morphisms between two matrix factorizations form a complex, where, for instance, \[ \operatorname{Hom}^{2l}(\mathcal{E},\mathcal{F}):=\mathrm{Hom}(\mathcal{E}_{-1},\mathcal{F}_{-1}\otimes\mathcal{L}^l)\oplus {\operatorname{Hom}}(\mathcal{E}_{0},\mathcal{F}_{0}\otimes\mathcal{L}^l) \] for \(l\in \mathbb{Z}\) and the differential is given by an explicit formula. Hence, matrix factorizations form a dg-category, denoted by \(\mathrm{Fact}(X,G,w)\). Considering matrix factorizations with injective components gives a dg-category \(\mathrm{Inj}(X,G,w)\).

The main result of the paper under review can be described as follows. Let \((X,G,w)\) be as above. Let \(H\) be an affine algebraic group acting on a smooth variety \(Y\) and let \(v\) be a \(G\)-invariant section of an \(H\)-equivariant line bundle \(\mathcal{L}'\) on \(Y\). Let \(U(\mathcal{L})\) be the complement of the zero section in the geometric vector bundle corresponding to \(\mathcal{L}\), and denote the same construction for \(\mathcal{L}'\) by \(U(\mathcal{L}')\). Denote the functions induced by \(w\) and \(v\) on \(U(\mathcal{L})\) and \(U(\mathcal{L}')\) by \(f_w\) and \(f_v\), respectively. Set \(-f_w\boxplus f_v=-f_w\otimes_k 1+1\otimes_k f_v\). Note that \(G\times H\) acts on \(U(\mathcal{L})\times U(\mathcal{L}')\) and allow \(\mathbb{G}_m\) to scale the fibres of \(U(\mathcal{L})\times U(\mathcal{L}')\) diagonally. The authors prove that there is an equivalence \[ \mathrm{RHom}_c(\mathrm{Inj}(X,G,w),\mathrm{Inj}(Y,H,v))\cong\mathrm{Inj}(U(\mathcal{L})\times U(\mathcal{L}'), G\times H\times \mathbb{G}_m,-f_w\boxplus f_v) \] in the homotopy category of \(k\)-linear dg-categories. So, roughly speaking, the functors between categories of matrix factorizations are again given by matrix factorizations. The above result is used to compute the (extended) Hochschild cohomology of \((X,G,w)\) when \(X\) is affine and \(G\), \(w\) satisfy some technical assumptions.

The story begins with a thorough introduction to equivariant sheaves. The authors define the abelian category of quasi-coherent equivariant sheaves, restriction and inflation functors, pullback etc. They also study the global dimension of \(\mathrm{Qcoh}_GX\).

Section 3 is devoted to equivariant factorizations. The dg-categories mentioned above and their variants (for instance, one involving coherent sheaves) are defined and studied. For example, the authors define a dg-functor \(\mathrm{Fact}(X,G,w)\otimes_k \mathrm{Fact}(X,G,v)\to \mathrm{Fact}(X,G,w+v)\), a version of the \(\mathcal{H}om\)-functor in this setting, an appropriate notion of box product, restriction and inflation functors for the case of a subgroup \(H\) of \(G\), and establish results relating various of these functors.

In the same section the authors also introduce and study the absolute derived category of matrix factorizations, following work of Positselski. The idea of the construction is roughly as follows. To the dg-category \(\mathrm{Fact}(X,G,w)\) one can associate an abelian category having the same objects, but where the morphisms between two factorizations are given by closed degree-zero morphisms between them in \(\mathrm{Fact}(X,G,w)\). The resulting category is denoted by \(Z^0\mathrm{Fact}(X,G,w)=\mathcal{A}\). Given a complex with objects from \(\mathcal{A}\), one can define a matrix factorization, called its totalization. Considering the subcategory of \(\mathrm{Fact}(X,G,w)\) consisting of totalizations of bounded exact complexes from \(\mathcal{A}\) gives the subcategory of acyclic factorizations. The absolute derived category \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) is then defined as the Verdier quotient of \([\mathrm{Fact}(X,G,w)]\), the homotopy category of \(\mathrm{Fact}(X,G,w)\) (which is a triangulated category), by the homotopy category of acyclic factorizations. Of course, there are variants of this definition if one works with coherent or locally free factorizations.

One of the good properties \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) has is that it is a compactly-generated triangulated category and equivalent to \([\mathrm{Inj}(X,G,w)]\). Furthermore, the authors show that the idempotent completion of the coherent version \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) is equivalent to the homotopy category of those factorizations of \(\mathrm{Inj}(X,G,w)\) which are compact in \([\mathrm{Inj}(X,G,w)]\). Another advantage of the absolute derived category is the existence of an essentially surjective functor from a certain singularity category to it, allowing one to use geometry when proving statements about the absolute derived category. More precisely, let \(Y\) be the vanishing locus of \(w\). Under some technical conditions, there is an essentially surjective functor \[ \mathrm{D}^{\mathrm{sg}}_G(Y):=\mathrm{D}^{\mathrm{b}}(\mathrm{Coh}_GY)/\mathrm{Perf}_GY\to \mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)], \] where \(\mathrm{Perf}_GY\) is the subcategory of perfect complexes, that is, bounded complexes of locally free \(G\)-equivariant sheaves of finite rank. The functor actually exists in greater generality, since one can consider categories supported on a closed \(G\)-invariant subset of \(Y\).

In Section 4 generation of equivariant derived categories is studied. More precisely, one wants to find a set of generators for the bounded derived category of coherent \(G\)-equivariant sheaves \(\mathrm{D}^{\mathrm{b}}(\mathrm{Coh}_G X)\), where \(X\) is a singular variety equipped with a \(G\)-action. The rough idea is to focus on \(\mathrm{D}^{\mathrm{b}}(\mathrm{Qcoh}_G X)\). Using the essentially surjective functor mentioned above, the authors can then produce a set of generators for an appropriate absolute derived category.

In the following section the main result is proved. For this the authors need to recall some facts from the theory of dg-categories and, in particular, results by Toën concerning the construction of the internal (continuous) Hom in the homotopy category of dg-categories. One of the main steps towards the proof of the main result is an equivalence between some absolute derived category and the derived category of dg-modules over the tensor product of matrix factorizations categories, and this is where the generators from the previous section come in, since the proof involves showing that a compact generating set is sent to a compact generating set and is fully faithful on these. In the same section the authors define and study the (extended) Hochschild (co)homology and compute it in the case mentioned above.

The last section is devoted to two applications. Firstly, combining the computation of the extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov relating the bounded derived category of coherent sheaves on a smooth complex hypersurface \(Z\) in projective space with equivariant matrix factorizations allows the authors to recover Griffiths’ description of the primitive cohomology of \(Z\) in terms of homogeneous pieces of the Jacobian algebra of the polynomial defining \(Z\). As a second application the authors give a new proof of the Hodge conjecture for self-products of a particular \(K3\) surface closely related to the Fermat cubic fourfold.

Reviewer: Pawel Sosna (Hamburg) (MR3270588)

### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

18E30 | Derived categories, triangulated categories (MSC2010) |

18G55 | Nonabelian homotopical algebra (MSC2010) |

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\textit{M. Ballard} et al., Publ. Math., Inst. Hautes Étud. Sci. 120, 1--111 (2014; Zbl 1401.14086)

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