[数学] 广义Donaldson-Thomas不变量的一个理论 [推广有奖]

摘要翻译：
设X是C上的Calabi-Yau的3倍。X的Donaldson-Thomas不变量是关于Gieseker稳定条件t的整数Dt^a(t),它对X上具有Chern特征a的稳定束进行计数。它们仅针对X上不存在严格半可定束的Chern特征a而定义，具有在X变形下不变的良好性质，但它们在稳定条件t变化时的行为直到现在才被理解。本书定义并研究了唐纳森-托马斯不变量的一个推广。我们的新不变量\bar{DT}^a(t)是有理数，定义于所有Chern字符a,如果在a类中没有严格半可定束，则等于DT^a(t)。它们是变形不变的，在稳定条件的变化下具有已知的变换规律。为了证明这一切，我们研究了X上相干束的模叠加M的局部结构。我们证明了M的图谱可以局部地写成复流形上的全纯函数f的Crit(f)，并利用这一点推导了M的Behrend函数的恒等式。我们在例子中计算了我们的不变量，并对它们的完整性性质进行了猜想。我们将该理论推广到具有来自超势的关系的颤振表示的阿贝尔范畴，并将我们的思想与Szendroi的“非交换Donaldson-Thomas不变量”和Reineke等人的工作联系起来。这本书在论文ARXIV:0910.0105中进行了调查。
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英文标题：
《A theory of generalized Donaldson-Thomas invariants》
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作者：
Dominic Joyce and Yinan Song
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最新提交年份：
2010
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Physics        物理学
二级分类：High Energy Physics - Theory        高能物理-理论
分类描述：Formal aspects of quantum field theory. String theory, supersymmetry and supergravity.
量子场论的形式方面。弦理论，超对称性和超引力。
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英文摘要：
  Let X be a Calabi-Yau 3-fold over C. The Donaldson-Thomas invariants of X are integers DT^a(t) which count stable sheaves with Chern character a on X, with respect to a Gieseker stability condition t. They are defined only for Chern characters a for which there are no strictly semistable sheaves on X. They have the good property that they are unchanged under deformations of X. Their behaviour under change of stability condition t was not understood until now.   This book defines and studies a generalization of Donaldson-Thomas invariants. Our new invariants \bar{DT}^a(t) are rational numbers, defined for all Chern characters a, and are equal to DT^a(t) if there are no strictly semistable sheaves in class a. They are deformation-invariant, and have a known transformation law under change of stability condition.   To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for M may be written locally as Crit(f) for f a holomorphic function on a complex manifold, and use this to deduce identities on the Behrend function of M.   We compute our invariants in examples, and make a conjecture about their integrality properties. We extend the theory to abelian categories of representations of a quiver with relations coming from a superpotential, and connect our ideas with Szendroi's "noncommutative Donaldson-Thomas invariants" and work by Reineke and others.   This book is surveyed in the paper arXiv:0910.0105.
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PDF链接：
https://arxiv.org/pdf/0810.5645