论文标题
dt不变的方向在准标记calabi-yau上4倍
Orientations for DT invariants on quasi-projective Calabi-Yau 4-folds
论文作者
论文摘要
对于calabi-yau 4倍$（x，ω）$，其中$ x $是准主机，$ω$是其规范捆绑$ k_x $的无处一节，（派生的）模量是紧凑的完美支持的完美综合体$ \ nathcal {m} _x _x _x _x _x _x $ -2 $ -2 $ -2 $ -2 $ -2 $ -2 $ -2 $ - $ o^ω\ to \ Mathcal {M} _x $在Borisov-Joyce Arxiv的意义上：1504.00690定义$ x $的Donaldson-Thomas类型不变性所必需的。我们首先将CAO-Gross-Joyce Arxiv的可矫正性结果扩展到4倍的投影旋转。然后，对于任何光滑的投影紧凑型$ \ bar {x} $，以便$ d = \ bar {x} \ backslash x $是严格的正常交叉，我们在堆栈上定义了方向捆绑包$ \ MATHCAL {M} _ {\ bar {X}} \ times _ {\ Mathcal {\ Mathcal {M} _D} \ Mathcal {M} _ {\ bar {x}}} $，并表达这些作为$ \ \ Mathbb {Z} _2 $ -Bund的poteriontor in of $ \ mathbb { $ x $。结果，我们将方向束$ o^ω\与\ Mathcal {M} _x $与紧凑型K理论的分类空间上的量学理论方向相关联。使用后者的可定位性，我们获得了$ \ Mathcal {m} _x $的可尊重性。我们还证明了稳定对的模量空间和适当亚物种的希尔伯特方案的定义性。最后，我们考虑直接总和下方向的兼容性。
For a Calabi-Yau 4-fold $(X,ω)$, where $X$ is quasi-projective and $ω$ is a nowhere vanishing section of its canonical bundle $K_X$, the (derived) moduli stack of compactly supported perfect complexes $\mathcal{M}_X$ is $-2$-shifted symplectic and thus has an orientation bundle $O^ω\to \mathcal{M}_X$ in the sense of Borisov-Joyce arXiv:1504.00690 necessary for defining Donaldson-Thomas type invariants of $X$. We extend first the orientability result of Cao-Gross-Joyce arXiv:1811.09658 to projective spin 4-folds. Then for any smooth projective compactification $\bar{X}$, such that $D=\bar{X}\backslash X$ is strictly normal crossing, we define orientation bundles on the stack $\mathcal{M}_{\bar{X}}\times_{\mathcal{M}_D}\mathcal{M}_{\bar{X}}$ and express these as pullbacks of $\mathbb{Z}_2$-bundles in gauge theory constructed using positive Dirac operators on the double of $X$. As a result, we relate the orientation bundle $O^ω\to \mathcal{M}_X$ to a gauge-theoretic orientation on the classifying space of compactly supported K-theory. Using orientability of the latter, we obtain orientability of $\mathcal{M}_X$. We also prove orientability of moduli spaces of stable pairs and Hilbert schemes of proper subschemes. Finally, we consider the compatibility of orientations under direct sums.
