一個描述獨異變換之下代數曲面上秩二的層的模空間動機變換公式

Abstract

本篇論文主要宗旨係在推廣[LQ98a,LQ98b]中所做出來的一個從Vafa與Witten從S-對偶猜想中預測出來的一個描述代數曲面上穩定秩二層的模空間的不變量在獨異變換之下的公式。 該兩篇論文所考慮的不變量為virtual霍奇多項式；我們想要將這些結果作到動機的版本。 在這篇論文中，我們驗證[LQ99,LQ98]當中的一些證明可以推廣到動機的設定之下，並在這樣的框架之下我們藉由[Moz19]中的一個Quot概型的動機生成函數公式回答了一個於[LQ98]中提出的帶有組合味道的猜想。我們也簡化了[LQ98]當中的一些計算。

The purpose of this thesis is to generalize a collection of results in [LQ99,LQ98] concerning change of invariants of moduli space of rank-2 stable sheaves over an algebraic surface under blowup, which is a set of formulas predicted by Vafa and Witten in the context of the S-duality conjecture. In these two papers, the invariants the authors considered are the virtual Hodge polynomials, and our goal is to refine these invariants to the settings of Grothendieck's motivic ring of varieties. In this paper, we verified that some of the proofs given in [LQ99,LQ98] can be generalized to the motivic setting, and by working in the motivic ring of varieties, we are able to answer a conjectural combinatorial formula posed in [LQ98], by using a formulae concerning motivic generating series of Quot schemes given in [Moz19]. We also simplified some of the calculations in [LQ98].