اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLagrangian cobordism functor in microlocal sheaf theory
72
0
0.0
(
0
)
تأليف
Wenyuan Li
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Given a Lagrangian cobordism $L$ of Legendrian submanifolds from $Lambda_-$ to $Lambda_+$, we construct a functor $Phi_L^*: Sh^c_{Lambda_+}(M) rightarrow Sh^c_{Lambda_-}(M) otimes_{C_{-*}(Omega_*Lambda_-)} C_{-*}(Omega_*L)$ between sheaf categories of compact objects with singular support on $Lambda_pm$ and its adjoint on sheaf categories of proper objects, using Nadler-Shendes work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high dimensional Legendrian submanifolds.
قيم البحث
اقرأ أيضاً
We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstruc
tions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.
In this paper we introduce the following new ingredients: (1) rework on part of the Lagrangian surgery theory; (2) constructions of Lagrangian cobordisms on product symplectic manifolds; (3) extending Biran-Cornea Lagrangian cobordism theory to the i
mmersed category. As a result, we manifest Seidels exact sequences (both the Lagrangian version and the symplectomorphism version), as well as Wehrheim-Woodwards family Dehn twist sequence (including the codimension-1 case missing in the literature) as consequences of our surgery/cobordism constructions. Moreover, we obtain an expression of the autoequivalence of Fukaya category induced by Dehn twists along Lagrangian $mathbb{RP}^n$, $mathbb{CP}^n$ and $mathbb{HP}^n$, which matches Huybrechts-Thomass mirror prediction of the $mathbb{CP}^n$ case modulo connecting maps. We also prove the split generation of any symplectomorphism by Dehn twists in $ADE$-type Milnor fibers.
Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known elementary building blocks for Lagrangian cobordisms that are smoothl
y the attachment of $0$- and $1$-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is decomposable into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms.
The Nadler-Zaslow correspondence famously identifies the finite-dimensional Floer homology groups between Lagrangians in cotangent bundles with the finite-dimensional Hom spaces between corresponding constructible sheaves. We generalize this correspo
ndence to incorporate the infinite-dimensional spaces of morphisms at infinity, given on the Floer side by Reeb trajectories (also known as wrapping) and on the sheaf side by allowing unbounded infinite rank sheaves which are categorically compact. When combined with existing sheaf theoretic computations, our results confirm many new instances of homological mirror symmetry. More precisely, given a real analytic manifold $M$ and a subanalytic isotropic subset $Lambda$ of its co-sphere bundle $S^*M$, we show that the partially wrapped Fukaya category of $T^*M$ stopped at $Lambda$ is equivalent to the category of compact objects in the unbounded derived category of sheaves on $M$ with microsupport inside $Lambda$. By an embedding trick, we also deduce a sheaf theoretic description of the wrapped Fukaya category of any Weinstein sector admitting a stable polarization.
In this paper, we discuss Floer homology of Lagrangian submanifolds in an open symplectic manifold given as the complement of a smooth divisor. Firstly, a compactification of moduli spaces of holomorphic strips in a smooth divisor complement is intro
duced. Next, this compactification is used to define Lagrangian Floer homology of two Lagrangians in the divisor complement. The main new feature of this paper is that we do not make any assumption on positivity or negativity of the divisor.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
