Let π : M → B be a Lagrangian torus fibration with singularities such that the fibers are of Maslov index zero, and unobstructed. The paper constructs a rigid analytic space M ∨ 0 over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus B0 ⊂ B of π. Transition functions of M ∨ 0 are obtained via A∞ homomorphisms which captures the wall-crossing phenomenon of moduli spaces of holomorphic disks. To see this we use the fact that the space of ω-tamed almost complex structures is contractible. Hence there exist a two-parameter family of almost complex structures J x0,x1,x2 ((x 0 , x 1 , x 2 ) ∈ ∆ 2 ) such that J 0,x1,x2 = J jk ; J x0,0,x2 = J ik ; J x0,x1,0 = J ij .
Let h ⊂ g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module U(g)/U(g)h whose associated graded h-module is isomorphic to S(n). We give a necessary and sufficient condition for the existence of a splitting of this filtration. In turn such a splitting yields an isomorphism between the h-modules U(g)/U(g)h and S(n). For the diagonal embedding h ⊂ h ⊕ h the condition is automatically satisfied and we recover the classical Poincaré-Birkhoff-Witt theorem.The main theorem and its proof are direct translations of results in algebraic geometry, obtained using an ad hoc dictionary. This suggests the existence of a unified framework allowing the simultaneous study of Lie algebras and of algebraic varieties, and a closely related work in this direction is on the way.
Abstract. We give a construction of NC-smooth thickenings (a notion defined by Kapranov [17]) of a smooth variety equipped with a torsion free connection. We show that a twisted version of this construction realizes all NC-smooth thickenings as 0th cohomology of a differential graded sheaf of algebras, similarly to Fedosov's construction in [12]. We use this dg resolution to construct and study sheaves on NC-smooth thickenings. In particular, we construct an NC version of the Fourier-Mukai transform from coherent sheaves on a (commutative) curve to perfect complexes on the canonical NC-smooth thickening of its Jacobian. We also define and study analytic NC-manifolds. We prove NC-versions of some of GAGA theorems, and give a C ∞ -construction of analytic NC-thickenings that can be used in particular for Kähler manifolds with constant holomorphic sectional curvature. Finally, we describe an analytic NC-thickening of the Poincaré line bundle for the Jacobian of a curve, and the corresponding Fourier-Mukai functor, in terms of A ∞ -structures.
Abstract. Let i : X ֒→ Y be a closed embedding of smooth algebraic varieties. Denote by N the normal bundle of X in Y. The present paper contains two constructions of certain Lie structure on the shifted normal bundle N[−1] encoding the information of the formal neighborhood of X in Y. We also present a few applications of these Lie theoretic constructions in understanding the algebraic geometry of embeddings.
In this paper, we introduce a new class of structured spaces which is locally modeled by Costello's L-infinity spaces. This provides an alternative approach to study the derived geometric structures in the algebraic, analytic, or smooth category. Our main result asserts that on a compact complex manifold, the moduli space of simple holomorphic structures on a complex vector bundle with a fixed determinant bundle, admits a natural analytic homotopy L-infinity enhancement.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.