The authors of the paper under review relate the base spaces of the semiuniversal unfoldings of certain weighted homogeneous singularities to the moduli space of \(A_{\infty}\)-structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. Using these relations, they prove homological mirror symmetry for an \(n\)-dimensional affine hypersurface of degree \(n+2\) and the double cover of the \(n\)-dimensional affine space branched along a degree \(2n+2\) hypersurface. These homological mirror symmetry results answer for Seidel’s conjecture that homological mirror symmetry could be obtained as deformations of homological mirror symmetry at the large volume limit. As a byproduct, they compute symplectic cohomology of certain Milnor fibers of isolated hypersurface singularities via taking Hochschild cohomology of the corresponding category under the homological mirror symmetry.
Let us review the main theorems after explaining the geometric settings as follows. Take a sequence \((d_{1},\cdots , d_{n};h)\) of positive integers satisfying \(h>max\{d_{1},\cdots, d_{n}\}\) and \(gcd(d_{1},\cdots,d_{n},h)=1\). Consider a weighted homogeneous polynomial \(w(x_{1},\cdots, x_{n})\in k[x_{1},\cdots, x_{n}]\) in \(n\) variables of weight \((d_{1},\cdots, d_{n};h)\); \[ w(t^{d_{1}}x_{1},\cdots, t^{d_{n}}x_{n})=t^{h}w(x_{1},\cdots, x_{n}), t\in \mathbb{G}_{m}. \] Consider a set \(I_{w}:=\{\mathbf{i}=(i_{1},\cdots, i_{n})\in \mathbb{Z}_{\geq 0}^{\oplus n}: d_{1}i_{1}+\cdots+d_{n}i_{n}=h\}\). Then the weighted homogenous polynomial can be written as the sum of monomials with index set \(I_{w}\); \(w(x_{1},\cdots, x_{n})=\sum_{I_{w}}c_{\mathbf{i}}x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}, c_{\mathbf{i}}\in \mathbb{G}_{m}\).
Consider a commutative algebraic subgroup \(\Gamma_{w}\) of \(\mathbb{G}_{m}^{n+1}\) defined by \(\{(t_{1},\cdots, t_{n+1})\in \mathbb{G}_{m}^{n+1}|t_{1}^{i_{1}}\cdots t_{n}^{i_{n}}=t_{n+1} \text{ for all } (i_{1},\cdots, i_{n})\in I_{w}\}\). The group \(\widehat{\Gamma}_{w}:=\mathrm{Hom}(\Gamma_{w},\mathbb{G}_{m})\) of characters of \(\Gamma_{w}\) is generated by \(\chi_{i}:\Gamma_{w}\to \mathbb{G}_{m}\), by \(\xi(t_{1},\cdots, t_{n+1})=t_{i}\) for \(1\leq i\leq n+1\), with the relations generated by \(i_{1}\chi_{1}+\cdots+i_{n}\chi_{n}-\chi_{n+1}=0, \mathbf{i}\in I_{w}\). Since the last coordinate of the elements of \(\Gamma_{w}\) is determined by the relations, we can consider \(\Gamma_{w}\) as a subgroup of \(\mathbb{G}_{m}^{n}\) by taking first \(n\) coordinates and denote \(\chi_{w}:=\chi_{n+1}, \chi_{w}(t_{1},\cdots,t_{n}):=t_{1}^{i_{1}}\cdots t_{n}^{i_{n}}\), the last coordinate separately. Then the group \(\Gamma_{w}\subset \mathbb{G}_{m}^{n}\) coinsides with the group of diagonal transformations of \(\mathbb{A}^{n}\) which keep \(w\) semi-invariant as follows; \(w(t\cdot (x_{1},\cdots,x_{n}))=\chi_{w}(t)w(x_{1},\cdots,x_{n})\), for \(t\in \Gamma_{w}, \mathbf{x}\in \mathbb{A}^{n}\). Then the injective homomorphism \(\phi:\mathbb{G}_{m}\to \Gamma_{w}, \phi(t):=(t^{d_{1}}\cdots, t^{d_{n}})\) has the cokernel \(\{(t_{1},\cdots,t_{n})\in\mathbb{G}_{m}^{n}|t_{1}^{i_{1}}\cdots t_{n}^{i_{n}}=1\}/<j_{w}:=(e^{2\pi\sqrt{-1}d_{1}/h},\cdots, e^{2\pi\sqrt{-1}d_{n}/h})>\). Here, \(<j_{w}>\) is the cyclic group \(\ker \chi_{w}\cap \phi(\mathbb{G}_{m})\) of order \(h\). Define a new group \(\Gamma\) which is a subgroup of \(\Gamma_{w}\) containing \(\phi(\mathbb{G}_{m})\) as a subgroup of finite index. Consider the restricted map \(\chi:=\chi_{w}|_{\Gamma}:\Gamma\subset \Gamma_{w}\to\mathbb{G}_{m}\), then the kernel of \(\chi\) is a finite group. The set of such subgroup \(\Gamma\) with \(\phi(\mathbb{G}_{m})\subset \Gamma \subset \Gamma_{w}\) with \([\Gamma:\phi(\mathbb{G}_{m})]<\infty\) is in bijection with the set of finite subgroups of \(ker\chi_{w}\) containing \(j_{w}\). Consider a ring \(\overline{R}:=k[x_{1},\cdots,x_{n}]/(w)\) and the group \(\Gamma\) acts on \(\mathrm{Spec}(\overline{R})\). Define a stack \(\mathcal{X}:=[(\mathrm{Spec}\overline{R} \setminus 0)/\Gamma]\). Consider \(R:=k[x_{0},x_{1},\cdots,x_{n}]/(w)\). Define \(\chi_{0}:=\chi-\chi_{1}-\cdots-\chi_{n}\) and extend the diagonal action \(\Gamma\) on \(\mathrm{Spec}(k[x_{1},\cdots,x_{n}])\) to \(\mathrm{Spec}(k[x_{0},x_{1},\cdots,x_{n}])\) via \(\chi_{0}\oplus \chi_{1}\oplus \cdots\oplus \chi_{n}\). Assume \(d_{0}:=h-d_{1}-\cdots-d_{n}>0\). The stack \(\mathcal{Y}_{0}:=[(\mathrm{Spec}\,R \setminus0)/\Gamma]\) is a projective cone over \(\mathcal{X}\) obtained from \([\mathrm{Spec}\, \overline{R}/\ker\chi_{0}]\) by adding \(\mathcal{X}\).
Assume that \(w:\mathbb{A}^{n}\to \mathbb{A}\) has an isolated singularity at the origin. Consider the Jacobi algebra \(\mathrm{Jac}_{w}:=k[x_{1},\cdots,x_{n}]/(\partial_{1}w,\cdots,\partial_{n}w)\) and the Milnor number \(\mu:=\dim_{k}\mathrm{Jac}_{w}\). Define the set \(J_{w}\) of exponents of monomials representing a basis of \(\mathrm{Jac}_{w}\). Consider a seminuniversal unfolding \(\widetilde{w}:\mathbb{A}^{n}\times \mathrm{Spec}(k[u_{1},\cdots,u_{\mu}])\to \mathbb{A}^{1}\) by \(\widetilde{w}(x_{1},\cdots,x_{n},u_{j})=w(x_{1},\cdots,x_{n})+\sum_{(j_{1},\cdots,j_{n})\in J_{w}}u_{j}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}\). Denote the base by \(\widetilde{U}:=\mathrm{Spec}(k[u_{1},\cdots,u_{\mu}])\). Define \[ U:=\{u_{j}\in \widetilde{U}|u_{j}\neq0 \implies [\exists w_{j} \text{ such that }\chi=w_{j}\chi_{0}+j_{1}\chi_{1}+\cdots +j_{n}\chi_{n}]\}. \] Consider the restricted function \(W:=\widetilde{w}:\mathbb{A}^{n+1}\times U\to \mathbb{A}^{1}\). Then there is a family \(\pi_{\mathcal{Y}}:\mathcal{Y}:=[(W^{-1}(0)\setminus (0\times U))/\Gamma]\to U\) of stacks over \(U\), whose fiber over \(u\in U\) is denoted by \(\mathcal{Y}_{u}\). The \(\mathbb{G}_{m}\)-action on \(\mathbb{A}^{n+1}\times U\) given by \(t\cdot(x_{0},x_{1},\cdots,x_{n},u_{j})=(t^{-1}x_{0},x_{1},\cdots,x_{n},t^{w_{j}}u_{j})\), which induces the actions on both \(\mathcal{Y}\) and \(U\) which makes the map \(\pi_{\mathcal{Y}}\) equivariant.
The following quasi-equivalence of two dg categories has been known since Orlov proved in [Y. Lekili and T. Perutz, “Arithmetic mirror symmetry for the 2-torus”, Preprint, arXiv:1211.4632]; \(\mathrm{Coh}([(\text{Spec}R\setminus 0)/\Gamma])\cong mf([\mathbb{A}^{n+1}/\Gamma],W),\) where \(\mathrm{Coh}(\cdot)\) means the bounded derived category of cohorent sheaves on the stack and \(mf([\mathbb{A}^{n+1}/\Gamma],W)\) means the idempotent-complete dg category of \(\Gamma\)-equivariant matrix factorizations. Taking Hochschild cohomology on both sides, \(HH^{*}([(\text{Spec}R\setminus 0)/\Gamma])\cong HH^{*}(\mathbb{A}^{n+1},\Gamma,W)\) and the right-hand side \(HH^{*}(\mathbb{A}^{n+1},\Gamma,W)\) can be computed as explained in the section 3.
Assume that the category \(mf([\mathbb{A}^{n}/\Gamma,w])\) has a tilting object \(E\), which means that the cohomologies of the endomorphism dg algebra \(\mathrm{End}(E)\) is concentrated in degree \(0\) and \(mf([\mathbb{A}^{n}/\Gamma], w)=<E>\), by shifts, cones and direct summands. Let \(\mathcal{E}\) be the pullback of \(E\) to \(mf([\mathbb{A}_{U}^{n}/\Gamma],w)\), then \(\mathrm{End}(\mathcal{E})\cong\mathrm{End}(E)\otimes k[U]\). Let \(\mathcal{S}\) be the pushforward of \(\mathcal{E}\) further to \(mf([\mathbb{A}_{U}^{n+1}/\Gamma],w)\). Then \(mf([\mathbb{A}_{U}^{n+1}/\Gamma],W)\cong coh\mathcal{Y}\) and \(\mathcal{S}\) split-generates \(\mathrm{perf} \mathcal{Y}\), where \(\mathcal{Y}:=[(W^{-1}(0)\setminus(0\times U))/\Gamma]\) was defined above, [D. Orlov, Prog. Math. 270, 503–531 (2009; Zbl 1200.18007), Theorem 16] and [Theorem 4.1 of the paper under review]. By taking a section of \(\omega_{\mathbb{A}_{U}^{n+1}/U}(\chi)\), we get an isomorphism \(\mathrm{End}(\mathcal{S})\cong A\otimes \mathcal{O}_{U}\),where the algebra \(A:=A^{0}\oplus \mathrm{Hom}_{k}(A^{0},k)[-(n-1)]\), by the degree \(n-1\) trivial extension algebra of \(A^{0}\) with the multiplication is defined by \((a,f)\cdot (b,g):=(ab,ag+fb)\).
Let \(A\) be a graded associative \(k\)-algebra. A minimal \(A_{\infty}\)-structure on \(A\) is an \(A_{\infty}\)-structure \((\mu^{k})_{k=1}^{\infty}\) on the graded vector space underlying the algebra \(A\) satisfying that \(\mu^{1}=0\) and \(\mu^{2}=\bullet_{A}\), the product structrue on the algebra \(A\). A minimal \(A_{\infty}\)-structure is called to be formal if the higher strcutures are vanishing, i.e., \(\mu^{k}=0, \text{ for } k>2\). If \(HH^{1}(A)_{<0}=0\), the functor sending a \(k\)-algebra \(R\) to the set of gauge equivalence classes of minimal \(A_{\infty}\)-structures on \(A\otimes R\) is represented by the affine affine scheme, \(\mathcal{U}_{\infty}(A)\), [A. Polishchuk, Duke Math. J. 166, No. 15, 2871–2924 (2017; Zbl 1401.14099), Corollary 3.2.5]. Then there is a natural \(\mathbb{G}_{m}\)-action on \(\mathcal{U}_{\infty}(A)\) given by \(t\cdot (\mu^{d})_{d=2}^{\infty}=(t^{d-2}\mu^{d})_{d=2}^{\infty}\) and the fixed points of this \(\mathbb{G}_{m}\)-action are the formal \(A_{\infty}\)-structures.
Let \(\mathcal{A}\) be the minimal model of the Yoneda dg algebra \(\mathrm{End}(\mathcal{S})\). Then \(Q\mathrm{Coh}(\mathcal{Y})\cong\mathrm{Mod}(\mathcal{A})\). Let \(\mathcal{A}_{0}:=\mathcal{A}\otimes_{k} k\) be the \(A_{\infty}\)-algebra over \(k\) obtained by restricting \(\mathcal{A}\) to the origin \(0\in U\). Considering \(\mathbb{G}_{m}\)-action, \(\mathcal{A}_{0}\) is a formal \(A_{\infty}\)-algebra. By [Y. Lekili and T. Perutz, “Arithmetic mirror symmetry for the 2-torus”, Preprint, arXiv:1211.4632], the natural dg enhancement of \(\mathrm{End}(S)\) gives a family of minimal \(A_{\infty}\)-algebra structures on \(A\) over \(U\). Therefore, we get a morphism \(U\to \mathcal{U}_{\infty}(A)\). The statement of Theorem 1.6 of the paper under review is that for a weighted homogeneous polynomial with the properties we explaned above, the morphism \(U\to \mathcal{U}_{\infty}(A)\) is \(\mathbb{G}_{m}\)-equivariant isomorphism sending the origin \(0\in U\) to the formal \(A_{\infty}\)-structure on \(A\). One of ideas of the proof is to reconstruct the \(\mathbb{Z}\)-algebra \((\mathrm{Hom}^{0}(\mathcal{O}_{\mathcal{Y}}(i),\mathcal{O}_{\mathcal{Y}}(i)))_{i,j\in\mathbb{Z}}\) up to isomorphism from the family of \(A_{\infty}\)-algebras using the Theorem 4.1, which says that \(\mathcal{S}\) split-generates perf \(\mathcal{Y}\). Moreover, by proposition A. 6 of [E. Looijenga, Math. Ann. 269, 357–387 (1984; Zbl 0568.14003)], \(U\) is the fine moduli space of \((\oplus_{i=0}^{\infty}H^{0}(\mathcal{O}(iX))/t\cdot \oplus_{i=0}^{\infty}H^{0}(\mathcal{O}(iX)))\)-polarized schemes and the universal family if given by the coarse moduli scheme of \(\mathcal{Y}\).