AI 中文总结
本文通过BBDJS极小模型构造平移接触叠上的单值周遍层，证明Legendrian局部基本类的消失条件，提出接触Joyce猜想类比，并建立接触不变量与欧拉示性数的联系。
AI 中文摘要
将BBDJS极小模型应用于$-1$-平移接触导出Artin叠的导出辛化，并沿结构自由$\u001b[G_m$-作用代数地下推，我们在任意有向的此类叠上构造了一个$\u001b[ell$-进周遍层，并利用Verdier关于单值层的特殊化等价性为其配备了驯顺的扭曲单值算子$\u001b[theta$。我们证明Legendrian子簇$L$的局部基本类满足$\u001b[theta \u001b[circ \u001b[mu_L = (-1)^{\u001b[mathrm{vdim} L}\u001b[mu_L$，因此在奇数虚维数的Legendrian上，该类因$\u001b[theta$-不变性而消失。此外，两种奇偶性在$A_1$坐标卡上均已出现，而一个奇数维Legendrian可以承载非零局部类。我们进一步提出了关于分级定向的Joyce猜想的接触类比，其中定向数据被虚维数的奇偶性所扭曲。在假设单值细化辛猜想成立的前提下，我们通过$\u001b[ell$-进拉推函子构造了范畴化的Legendrian 2-范畴$\u001b[mathfrak{L}\u001b[mathcal{F}_c(X)$和$LLeg_0$。最后，我们证明接触Behrend函数恒等于$1$，从而相关的Donaldson-Thomas不变量是经典截断的紧支撑étale欧拉示性数，并且$\u001b[theta$的高阶迹恢复了第一迹所丢弃的奇点类型。作为应用，我们证明余切丛中锥形Lagrangian的导出交的辛不变量恒为零，而接触不变量计算了射影化交的欧拉示性数，并给出了余法丛的显式公式。
英文摘要
Applying the BBDJS minimal model to the derived symplectification of a $-1$-shifted contact derived Artin stack and descending algebraically along the structural free $\mathbb{G}_m$-action, we construct an $\ell$-adic perverse sheaf on any oriented such stack, and use Verdier's specialization equivalence for monodromic sheaves to equip it with a tame twisted monodromy operator $θ$. We show that the local fundamental class of a Legendrian $L$ satisfies $θ\circ μ_L = (-1)^{\mathrm{vdim} L}μ_L$, so that on Legendrians of odd virtual dimension, the class vanishes due to $θ$-invariance. Moreover, both parities occur already on the $A_1$ chart while an odd Legendrian can carry a nonzero local class. We further formulate a contact analogue of Joyce's conjecture for a graded orientation, in which the orientation datum is twisted by the parity of the virtual dimension. Under the assumption of a monodromic refinement of the symplectic conjecture, we construct the categorified Legendrian 2-categories $\mathfrak{L}\mathcal{F}_c(X)$ and $LLeg_0$ via $\ell$-adic pull-push functors. Finally, we show that the contact Behrend function is identically $1$, so that the associated Donaldson-Thomas invariant is the compactly supported étale Euler characteristic of the classical truncation, and that the higher traces of $θ$ recover the singularity type that the first trace discards. As an application, we show that the symplectic invariant of a derived intersection of conic Lagrangians in a cotangent bundle vanishes identically, while the contact invariant computes the Euler characteristic of the projectivized intersection, with an explicit formula for conormal bundles.
Comments53 pages, comments welcome! :)
