Minimal Fano threefolds: exceptional sets and vanishing cycles
Let \(F\) be a Fano manifold of dimension \(m\) and \(m\) is odd. Suppose \(F\) is \textit{minimal} (i.e., \(H^i(F,\mathbb{Z}) = \mathbb{Z}\) for even \(i\) and \(= 0\) for odd \(i\)). Then the bounded derived category \(\mathcal{D}^b(F)\) of coherent sheaves on \(F\) possesses a \textit{complete exceptional collection} \(E_1,\dots,E_n\in\text{Ob}(\mathcal{D}^b(F))\), denoted by \(\mathbf{E}\) in what follows (i.e., \(\text{Ext}^i(E_j,E_k) = 0 = \text{Ext}^{i+1}(E_j,E_j)\) and \(\text{Ext}^0(E_j,E_j)=\mathbb{C}\) for all \(i\geq 0,j>k\), and \(E_i\)'s generate \(\mathcal{D}^b(F)\)). One can introduce a bilinear (Riemann--Roch) form \(\chi\) on \(\left<E_1,\dots,E_n\right>\subseteq K^0\otimes\mathbb{Q}\) via \(\chi(E_j,E_k) := \sum_i(-1)^i\dim \text{Ext}^i(E_j,E_k)\) for all \(j,k\). Put \(V := (K^0\otimes\mathbb{Q})/\text{Ker}(\chi^s)\), where \(\chi^s\) is the symmetrization of \(\chi\). Further, let \(M\) be the matrix of quantum multiplication by \(-K_F\), and \(\eta\) be the \textit{anticanonical quantum \(D\)-module} on the torus \(G_m = \text{Spec}~\mathbb{C}[t,t^{-1}]\), given by \(t\frac{\partial}{\partial t}\eta = \eta M\). One then takes the Fourier transform of the mean extension of \(\eta\) to \(\mathbb{A}^1\), called the \textit{regularized quantum \(D\)-module}. The author of the paper under review conjectures the following (\textit{ESVC Conjecture}, see 2.1 in the text): (i) regularized quantum \(D\)-module has regular singularities and finite singularities \(x_1,\dots,x_n\) are simple; (ii) there exists \(\mathbf{E}\) as above such that the representation \(\pi_1(\mathbb{A}^1\setminus{\{x_1,\dots,x_n\}}) \to O(V,\chi^s)\), mapping the loop around each \(x_i\) to the reflection w.\,r.\,t. the corresponding \(E_i\), is isomorphic to the monodromy of the regularized quantum \(D\)-module. The ESVC Conjecture is verified for the following (minimal) Fano \(3\)-folds: \(\mathbb{P}^3\), quadric \(Q\), \(V_{22}\) and \(V_5\). One starts with the modular curve \(X_0(N)^{\circ}\) minus the cusps and elliptic points. Let \(\varphi: \pi_1(X_0(N)^{\circ})\to PSL_2(\mathbb{Z})\) be the tautological representation (on \(H^1\) of the elliptic curve over \(X_0(N)\)). One then considers the action of \(W:=\bigl(\begin{smallmatrix} 0 & 1 \\ -N & 0 \end{smallmatrix}\bigr)\) (the \textit{Atkin--Lehner} involution) on \(X_0(N)\), with a unique fixed point \(O\), for which the quotient \(X_0(N)^{W^{\circ}}:=\big(X_0(N)^{\circ}\setminus{\{O\}}\big)/W\) will have \(\pi_1(X_0(N)^{W^{\circ}})\) generated by \(\pi_1(X_0(N)^{\circ})\) and a loop \(\iota\) around \(O\). It turns out (cf. \textit{V.\,V. Golyshev} [London Mathematical Society Lecture Note Series 338, 88--121 (2007; Zbl 1114.14024)]) that for \(N = -K_F^3/2d^2\) and the (Fano) index \(d=1\) (with \(-K_F = dH\) for some ample \(H\in H^2(F,\mathbb{Z})\)), there exists a representation of \(\pi_1(X_0(N)^{W^{\circ}})\) in \(\text{Sym}^2\varphi\), extending \(\varphi\) above, which coincides with the monodromy of the regularized quantum \(D\)-module. Then for \(V_{22}\), the image of the \(\text{Sym}^2\)-representation of \(\pi_1(X_0(N)^{W^{\circ}})\) is generated by reflections w.\,r.\,t. some vectors \(v_1,\dots,v_4\), for which the Gram matrix \(\{(v_i,v_j)\}_{1\leq i,j\leq 4}\) coincides with \(\chi^s\) w.\,r.\,t. some \(\mathbf{E}\) as above, constructed in \textit{A.\,G. Kuznetsov} [Mosc. Univ. Math. Bull. 51, No. 3, 35--37 (1996); translation from Vestn. Mosk. Univ., Ser. I 1996, No. 3, 41--44 (1996; Zbl 0913.14010)]. Further, the arguments for \(V_5\) are similar to the preceding ones, with the only modification that one passes to an index \(2\) subgroup of \(\pi_1(X_0(N)^{W^{\circ}})\) (as \(d=2\)), to construct the monodromy, and uses the result of \textit{D.\,O. Orlov} [Mosc. Univ. Math. Bull. 46, No. 5, 48--50 (1991); translation from Vestn. Mosk. Univ., Ser. I 1991, No. 5, 69--71 (1991; Zbl 0784.14010)] to find the required \(\chi^s\) and \(\mathbf{E}\). Finally, the cases of \(\mathbb{P}^3\) and \(Q\) are treated similarly, for \(\mathbf{E}\) and \(\chi\) being known due to \textit{D.\,Yu. Nogin} [Ann. Sci. Éc. Norm. Supér. (4) 27, No. 2, 129--172 (1994; Zbl 0845.14012)] and the Kodaira vanishing.
scientific article; zbMATH DE number 6106374
scientific article