法諾三維多樣體的導出範疇之研究

Abstract

本文探討兩個皮卡數為一的平滑法諾三維空間之導出範疇。第一個是指數為一、次數為十四的三維空間（下稱「第一空間」）；第二個是指數為二、次數為三的三維空間（下稱「第二空間」）。對這兩個空間，我們各自選取其有界導出範疇中的一個特別子範疇。就第一空間而言，該子範疇─常被稱作庫茲涅佐夫成分─是由一對標準例外對象（包含一個秩二向量束與結構層）的右正交補所構成；對第二空間，則取由結構層與超平面類線束組成之例外對的右正交補。 二零零四年，庫茲涅佐夫建立了一個對應，聯繫這兩類法諾三維空間的模空間，並證明：對每一個平滑的第一空間，都存在一個平滑的第二空間，使得它們各自的庫茲涅佐夫成分彼此等價。本論文有兩個主要目標：第一，證明上述兩個子範疇皆不含例外對象；第二，說明庫茲涅佐夫的等價可以實現為傅立葉─穆凱變換。

In this article, we study the derived categories of two smooth Fano threefolds with Picard number one. The first is the threefold of index one and degree fourteen, which we call“X14＂; the second is the threefold of index two and degree three, referred to as“Y3＂. For each variety we consider a distinguished subcategory of its bounded derived category of coherent sheaves. In the case of X14, this subcategory—often called the Kuznetsov component—is defined as the right orthogonal to the standard exceptional pair consisting of a rank-two vector bundle and the structure sheaf. For Y3, the analogous subcategory is the right orthogonal to the pair formed by the structure sheaf and the line bundle associated with the hyperplane class. In a 2004 paper, Alexander Kuznetsov constructed a correspondence between the moduli stacks that classify these two families of Fano threefolds. More precisely, for every smooth Fano threefold X14 there exists a smooth Fano threefold Y3 such that their Kuznetsov components are equivalent. This thesis has two main goals: first, to prove that the two chosen subcategories contain no exceptional objects; and second, to show that Kuznetsov’s equivalence can be realized as a Fourier–Mukai transform.