Configurations in abelian categories. V. Changing stability conditions
Configurations in abelian categories. V. Changing stability conditions
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阿贝尔范畴中的配置。
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发表时间：
2004
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通讯作者：
D. joyce
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作者：
D. joyce
这是数学系列论文的第四篇。AG / 0312190,数学。AG / 0503029,数学。给定一个有限偏序集合（I,<），一个(I,<)-组态是a中满足某些公理的对象和态射的有限集合。配置描述了A中的对象X如何分解成子对象。第一篇论文是数学。AG/0312190利用Artin堆栈理论，定义了构型，研究了对象的模空间Obj_A， M(I,<)_A和A中的(I,<)-构型。第二道数学题。AG/0503029考虑了Obj_A上可构造函数和“堆栈函数”的代数，使用数学中发展的理论。AG / 0403305, math.AG / 0509722。第三个数学。AG/0410267在A上引入了稳定性条件（t, t, <），并证明了A中A类中t-半稳定对象的模空间Obj_{ss}^ A (t)是Obj_A中的可构造集合，因此其特征函数d_{ss}^ A (t)是可构造的。证明了d_{ss}^a(t)等可构造函数和堆栈函数上的许多恒等式。本文首先通过将d_{ss}^a(t)写成d_{ss}^b(t)的乘积的和，研究了当稳定性条件（t, t, <）变为（t', t', <）时Obj_{ss}^a(t)的变化。然后讨论了不变量I_{ss}^a(t)或I_{ss}(I,<,k,t) ‘计数’ t-半稳定对象和构型，满足恒等式和从（t, t, <）到（t', t', <）的变换规律。当A是光滑投影曲线P上相干束的颤振Q或coh(P)表示的范畴模- kq时，我们计算了不变量。对于具有K_P^{-1} nef或P A Calabi-Yau 3-fold的P A曲面，当A=coh(P)时，我们发现了不变量的特殊性质。
This is the fourth in a series of papers math.AG/0312190, math.AG/0503029, math.AG/0410267 on configurations in an abelian category A. Given a finite partially ordered set (I,<), an (I,<)-configuration is a finite collection of objects and morphisms in A satisfying some axioms. Configurations describe how an object X in A decomposes into subobjects. The first paper math.AG/0312190 defined configurations and studied moduli spaces Obj_A, M(I,<)_A of objects and (I,<)-configurations in A, using the theory of Artin stacks. The second math.AG/0503029 considered algebras of constructible functions and "stack functions" on Obj_A, using the theories developed in math.AG/0403305, math.AG/0509722. The third math.AG/0410267 introduced stability conditions (t,T,<) on A, and showed the moduli space Obj_{ss}^a(t) of t-semistable objects in class a in A is a constructible set in Obj_A, so its characteristic function d_{ss}^a(t) is constructible. It proved many identities on constructible and stack functions such as d_{ss}^a(t). This paper first studies how Obj_{ss}^a(t) changes as we vary the stability condition (t,T,<) to (t',T',<), by writing d_{ss}^a(t') as a sum of products of d_{ss}^b(t). Then we discuss invariants I_{ss}^a(t) or I_{ss}(I,<,k,t) 'counting' t-semistable objects and configurations in A, satisfying identities and transformation laws from (t,T,<) to (t',T',<). We compute the invariants when A is a category mod-KQ of representations of a quiver Q or coh(P) of coherent sheaves on a smooth projective curve P. We find special properties of the invariants when A=coh(P) for P a surface with K_P^{-1} nef, or P a Calabi-Yau 3-fold.