论文标题
连贯的束带的模量空间的定向数据在卡拉比远3倍上
Orientation data for moduli spaces of coherent sheaves over Calabi-Yau 3-folds
论文作者
论文摘要
令$ x $为紧凑的calabi-yau 3倍，并写入$ \ mathcal m，\ bar {\ mathcal m} $，用于coh $（x），d^b $ coh $（x）$中的对象的Moduli堆栈。有天然线包$ k _ {\ mathcal m} \ to \ mathcal m $，$ k _ {\ bar {\ mathcal m}}} \ to \ bar {\ mathcal m} $，类似物捆绑包的类似物。 $ \ Mathcal M，\ bar {\ Mathcal M} $上的方向数据是平方根线捆绑包的同态类别类别$ k _ {\ Mathcal M}^{1/2}，k _ {\ bar {\ bar {\ Mathcal M}}}^{1/2} $ compatience and compatience and compatience and compatience and compatience。它是由Kontsevich和Soibelman Arxiv引入的：1006.270在他们的动机唐纳森 - 托马斯不变的理论中，对使用不良滑轮对Donaldson-Thomas理论进行分类很重要。 我们表明，可以为所有紧凑型卡拉比（Calabi-Yau）构建自然取向数据3倍，也可以使用紧凑型的连贯的滑轮和完美的calabi-yau 3倍$ x $，并带有旋转光滑的投影型射击射击压实$ x \ jkookrightarrow y $。这证明了唐纳森 - 托马斯理论的长期猜想。 这些是更普遍的结果的特殊情况。令$ x $为3倍的旋转式投影。使用旋转结构，我们构造线捆绑$ k _ {\ Mathcal M} \ to \ Mathcal M $，$ K _ {\ bar {\ Mathcal M}} \ to \ bar {\ Mathcal M} $。我们将旋转结构定义为$ \ Mathcal M，\ bar {\ Mathcal m} $作为正方根的同构类别$ k _ {\ Mathcal M}^{1/2}，k _ {\ bar {\ bar {\ Mathcal M}}}}}^{1/2} $。我们证明自然自旋结构存在于$ \ Mathcal M，\ bar {\ Mathcal M} $上。当$ x $是微不足道的旋转结构的3倍时，它们等效于方向数据。 我们使用以前的论文ARXIV：1908.03524证明了这一点，该论文在微分几何模量上构建了“旋转结构”（某个复杂线束的平方根$ k_p \ to \ mathcal b_p $）堆叠$ \ \ \ \ \ \ \ Mathcal b_p b_p b_p $ $ $ $ $ $ $ p \ $ p \ yif x x x x x x x x $ \ $ p_p $
Let $X$ be a compact Calabi-Yau 3-fold, and write $\mathcal M,\bar{\mathcal M}$ for the moduli stacks of objects in coh$(X),D^b$coh$(X)$. There are natural line bundles $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal M}$, analogues of canonical bundles. Orientation data on $\mathcal M,\bar{\mathcal M}$ is an isomorphism class of square root line bundles $K_{\mathcal M}^{1/2},K_{\bar{\mathcal M}}^{1/2}$, satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman arXiv:1006.270 in their theory of motivic Donaldson-Thomas invariants, and is important in categorifying Donaldson-Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi-Yau 3-folds, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds $X$ with a spin smooth projective compactification $X\hookrightarrow Y$. This proves a long-standing conjecture in Donaldson-Thomas theory. These are special cases of a more general result. Let $X$ be a spin smooth projective 3-fold. Using the spin structure we construct line bundles $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal M}$. We define spin structures on $\mathcal M,\bar{\mathcal M}$ to be isomorphism classes of square roots $K_{\mathcal M}^{1/2},K_{\bar{\mathcal M}}^{1/2}$. We prove that natural spin structures exist on $\mathcal M,\bar{\mathcal M}$. They are equivalent to orientation data when $X$ is a Calabi-Yau 3-fold with the trivial spin structure. We prove this using our previous paper arXiv:1908.03524, which constructs 'spin structures' (square roots of a certain complex line bundle $K_P\to\mathcal B_P$) on differential-geometric moduli stacks $\mathcal B_P$ of connections on a principal U$(m)$-bundle $P\to X$ over a compact spin 6-manifold $X$.
