Minimal Fano threefolds: exceptional sets and vanishing cycles
type: Item
instance of: scholarly article
title: Minimal Fano threefolds: exceptional sets and vanishing cycles
zbMATH Open document ID: 1255.14033
DOI: 10.1134/S1064562409010062
published in: Doklady Mathematics
publication date: 2012-11-15T00:00:00Z
review text: 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…
reviewed by: Ilya Karzhemanov
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6106374
zbMATH Keywords: minimal Fano threefold
zbMATH Keywords: exceptional set
zbMATH Keywords: vanishing cycle
author: Vasily Golyshev
MaRDI profile type: Publication
cites work: Stokes matrices and monodromy of the quantum cohomology of projective spaces
cites work: Helices on some Fano threefolds : constructivity of semiorthogonal bases of $K_0$
cites work: Solitons and helices: The search for a math-physics bridge
full work available at URL: s1064562409010062
OpenAlex ID: W2083331681
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description: scientific article; zbMATH DE number 6106374