Conormal Bundle Research Articles

On varieties with Ulrich twisted conormal bundles

We study varieties X ⊂ P r X \subset \mathbb {P}^r such that N X ∗ ( k ) N_X^*(k) is an Ulrich vector bundle for some integer k k . We first prove that such an X X must be a curve. Then we give several examples of curves with N X ∗ ( k ) N_X^*(k) an Ulrich vector bundle.

Tangent, cotangent, normal and conormal bundles are almost never instanton bundles

In this very short note we give an elementary characteristic free proof of the result claimed in the title (see Theorem 1.2 for a more precise formulation), generalizing a recent result proved by Benedetti et al. 2023 (see reference list) for Ulrich bundles over the complex field. Moreover, we also give a similar result about the twists of the cotangent bundle and make some comments about the possibility to obtain an analogous result for twists of the tangent bundle.

Arthur packets for 𝑝-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples

In this article we propose a geometric description of Arthur packets for p p -adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of p p -adic groups and propose a geometric description of the transfer coefficients that appear in Arthur’s main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan’s work, including the prediction that Arthur packets are ABV-packets for p p -adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

On Fredholm boundary conditions on manifolds with corners, I: Global corner cycles obstructions

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with corners $X$ of any codimension, there is a natural and explicit morphism $$K_*(\mathcal{K}_b(X)) \stackrel{T}{\longrightarrow} H^{pcn}_*(X,\mathbb{Q})$$ between the $K-$theory group of the algebra $\mathcal{K}_b(X)$ of $b$-compact operators for $X$ and the periodic conormal homology group with rational coeficients, and that $T$ is a rational isomorphism. As shown by the first two authors in a previous paper this computation implies that the rational groups $H^{pcn}_{ev}(X,\mathbb{Q})$ provide an obstruction to the Fredholm perturbation property for compact connected manifold with corners. The difference with respect to the previous article of the first two authors in which they solve this problem for low codimensions is that we overcome in the present article the problem of computing the higher spectral sequence K-theory differentials associated to the canonical filtration by codimension by introducing an explicit topological space whose singular cohomology is canonically isomorphic to the conormal homology and whose K-theory is naturally isomorphic to the $K-$theory groups of the algebra $\mathcal{K}_b(X)$.

Smoothing of multiple structures on embedded Enriques manifolds

We show that given an embedding of an Enriques manifold of index d in a large enough projective space, there will exist embedded multiple structures with conormal bundle isomorphic to the trace zero module of the universal covering map, the universal cover being either a hyperkähler or a Calabi–Yau manifold. We then show that these multiple structures (also known as d-ropes) can be smoothed to smooth hyperkähler or Calabi–Yau manifolds respectively. Hence we obtain a flat family of hyperkähler (or Calabi–Yau) manifolds embedded in the same projective space which degenerates to an embedded d-rope structure on the given Enriques manifold of index d. The above shows that these d-rope structures on the embedded Enriques manifold are points of the Hilbert scheme containing the fibres of the above family. We show that they are smooth points of the Hilbert scheme when \(d=2\).

A Classification of Twisted Austere 3-Folds

A twisted-austere $k$-fold $(M, \mu)$ in $\mathbb R^n$ consists of a $k$-dimensional submanifold $M$ of $\mathbb R^n$ together with a closed $1$-form $\mu$ on $M$ such that the `twisted conormal bundle' $N^* M + \mu$ is a special Lagrangian submanifold of $\mathbb C^n$. The 1-form $\mu$ and the second fundamental form of $M$ must satisfy a particular system of coupled nonlinear second order PDE. We first review these twisted-austere conditions and give an explicit example. Then we focus on twisted-austere 3-folds, giving a geometric description of all solutions when the base $M$ is a cylinder and when $M$ is austere. Finally, we prove that, other than the case of a generalized helicoid in $\mathbb R^5$ discovered by Bryant, there are no other possibilities for the base $M$. This gives a complete classification of twisted-austere $3$-folds in $\mathbb R^n$.

Fano manifolds containing a negative divisor isomorphic to a rational homogeneous space of Picard number one

Let [Formula: see text] be an [Formula: see text]-dimensional complex Fano manifold [Formula: see text]. Assume that [Formula: see text] contains a divisor [Formula: see text], which is isomorphic to a rational homogeneous space with Picard number one, such that the conormal bundle [Formula: see text] is ample over [Formula: see text]. Building on the works of Tsukioka, Watanabe and Casagrande–Druel, we give a complete classification of such pairs [Formula: see text].

A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Pantev, Toen, Vaqui\'e and Vezzosi arXiv:1111.3209 defined $k$-shifted symplectic derived schemes and stacks ${\bf X}$ for $k\in\mathbb Z$, and Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in them. They have important applications to Calabi-Yau geometry and quantization. Bussi, Brav and Joyce arXiv:1305.6302 proved a 'Darboux Theorem' giving explicit Zariski or \'etale local models for $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$ presenting them as twisted shifted cotangent bundles. We prove a 'Lagrangian Neighbourhood Theorem' giving explicit Zariski or etale local models for Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$, relative to the Bussi-Brav-Joyce 'Darboux form' local models for ${\bf X}$. That is, locally such Lagrangians can be presented as twisted shifted conormal bundles. We also give a partial result when $k=0$. We expect our results will have future applications to $k$-shifted Poisson geometry (see arXiv:1506.03699), to defining 'Fukaya categories' of complex or algebraic symplectic manifolds, and to categorifying Donaldson-Thomas theory of Calabi-Yau 3-folds and 'Cohomological Hall algebras'.

Simple sheaves for knot conormals

We classify the simple sheaves microsupported along the conormal bundle of a knot. We also establish a correspondence between simple sheaves up to local systems and augmentations, explaining the underlying reason why knot contact homology representations detect augmentations.

Grassmann–Grassmann conormal varieties, integrability, and plane partitions

We give a conjectural formula for sheaves supported on (irreducible) conormal varieties inside the cotangent bundle of the Grassmannian, such that their equivariant K-class is given by the partition function of an integrable loop model, and furthermore their K-theoretic pushforward to a point is a solution of the level 1 quantum Knizhnik–Zamolodchikov equation. We prove these results in the case that the Lagrangian is smooth (hence is the conormal bundle to a subGrassmannian). To compute the pushforward to a point, or equivalently to the affinization, we simultaneously degenerate the Lagrangian and sheaf (over the affinization); the sheaf degenerates to a direct sum of cyclic modules over the geometric components, which are in bijection with plane partitions, giving a geometric interpretation to the Razumov–Stroganov correspondence satisfied by the loop model.

Blow-ups in generalized complex geometry

Generalized complex geometry is a theory that unifies complex geometry and symplectic geometry into one single framework. It was introduced by Hitchin and Gualtieri around 2002. In this thesis we address the following question: given a generalized complex manifold together with a submanifold, does the blow-up of that submanifold admit again a generalized complex structure? We give the following answers to this question: If the directions that are normal to the submanifold are purely complex, then the blow-up admits a generalized complex structure for which the blow-down map is generalized holomorphic if and only if the induced Lie algebra on the conormal bundle of the submanifold is degenerate. This is a rather restrictive condition, but it is always satisfied if the submanifold is of complex codimension two. On the other extreme, if all the normal directions are symplectic, then we show that the blow-up always admits a generalized complex structure, provided the submanifold is compact. We then also give an example of a submanifold that is not of the above two types, and show that it can not be blown up. Finally, we also address the analogous question in generalized Kahler geometry. Given a generalized Kahler manifold together with a submanifold, we show that under certain geometric constraints on the submanifold the blow-up is again generalized Kahler.

Specialness and Isotriviality for Regular Algebraic Foliations

We show that an everywhere regular foliation $\mathcal F$ with compact canonically polarized leaves on a quasi-projective manifold $X$ has isotrivial family of leaves when the orbifold base of this family is special. By a recent work of Berndtsson, Paun and Wang, the same proof works in the case where the leaves have trivial canonical bundle. The specialness condition means that the $p$-th exterior power of the logarithmic extension of its conormal bundle does not contain any rank-one subsheaf of maximal Kodaira dimension $p$, for any $p>0$. This condition is satisfied, for example, in the very particular case when the Kodaira dimension of the determinant of the Logarithmic extension of the conormal bundle vanishes. Motivating examples are given by the `algebraically coisotropic' submanifolds of irreducible hyperk\ahler projective manifolds.

A microlocal approach to eigenfunction concentration

We describe a new approach to understanding averages of high energy Laplace eigenfunctions, $u_h$, over submanifolds, $$ \Big|\int _H u_hd\sigma_H\Big| $$ where $H\subset M$ is a submanifold and $\sigma_H$ the induced by the Riemannian metric on $M$. This approach can be applied uniformly to submanifolds of codimension $1\leq k\leq n$ and in particular, gives a new approach to understanding $\|u_h\|_{L^\infty(M)}$. The method, developed in the author's recent work together with Y. Canzani and J. Toth, relies on estimating averages by the behavior of $u_h$ microlocally near the conormal bundle to $H$. By doing this, we are able to obtain quantitative improvements on eigenfunction averages under certain uniform non-recurrent conditions on the conormal directions to $H$. In particular, we do not require any global assumptions on the manifold $(M,g)$.

Waves on the Free Surface Described by Linearized Equations of Hydrodynamics with Localized Right-Hand Sides

A linearized system of equations of hydrodynamics with time-dependent spatially localized right-hand side placed both on the free surface (and on the bottom of the basin) and also in the layer of the liquid is considered in a layer of variable depth with a given basic plane-parallel flow. A method of constructing asymptotic solutions of this problem is suggested; it consists of two stages: (1) a reduction of the three-dimensional problem to a two-dimensional inhomogeneous pseudodifferential equation on the nonperturbed free surface of the liquid, (2) a representation of the localized right-hand side in the form of a Maslov canonical operator on a special Lagrangian manifold and the subsequent application of a generalization to evolution problems of an approach, which was recently suggested in the paper [A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and M. Rouleux, Dokl. Ross. Akad. Nauk 475 (6), 624–628 (2017); Engl. transl.: Dokl. Math. 96 (1), 406–410 (2017)], to solving stationary problems with localized right-hand sides and its combination with “nonstandard” characteristics. A method of calculation (generalizing long-standing results of Dobrokhotov and Zhevandrov) of an analog of the Kelvin wedge and the wave fields inside the wedge and in its neighborhood is suggested, which uses the consideration that this method is the projection to the extended configuration space of a Lagrangian manifold formed by the trajectories of the Hamiltonian vector field issuing from the intersection of the set of zeros of the extended Hamiltonian of the problem with conormal bundle to the graph of the vector function defining the trajectory of motion of an equivalent source on the surface of the liquid.

Open Gromov-Witten Invariants from the Augmentation Polynomial

A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X = O P 1 ( − 1 , − 1 ) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds L K ⊂ X obtained from the conormal bundles of knots K ⊂ S 3 . This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For ( r , s ) torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bott localization. Using this result for the unknot and the ( 3 , 2 ) torus knot, we show that the augmentation polynomial can be derived from these open Gromov-Witten invariants.

Stability for the Lens Rigidity Problem

Let g be a Riemannian metric for \({\mathbb{R}^d}\) (\({d \geqq 3}\)) which differs from the Euclidean metric only in a smooth and strictly convex bounded domain M. The lens rigidity problem is concerned with recovering the metric g inside M from the corresponding lens relation on the boundary \({\partial M}\). In this paper, the stability of the lens rigidity problem is investigated for metrics which are a priori close to a given non-trapping metric satisfying the “strong fold-regular” condition. A metric g is called strong fold-regular if for each point \({x\in M}\), there exists a set of geodesics passing through x whose conormal bundle covers \({T^*_{x}M}\). Moreover, these geodesics contain either no conjugate points or only fold conjugate points with a non-degeneracy condition. Examples of strong fold-regular metrics are constructed. Our main result gives the first stability result for the lens rigidity problem in the case of anisotropic metrics which allow conjugate points. The approach is based on the study of the linearized inverse problem of recovering a metric from its induced geodesic flow, which is a weighted geodesic X-ray transform problem for symmetric 2-tensor fields. A key ingredient is to show that the kernel of the X-ray transform on symmetric solenoidal 2-tensor fields is of finite dimension. It remains open whether the kernel space is trivial or not.

Piunikhin-Salamon-Schwarz isomorphisms and spectral invariants for conormal bundle

We give a construction of the Piunikhin-Salamon-Schwarz isomorphism between the Morse homology and the Floer homology generated by Hamiltonian orbits starting at the zero section and ending at the conormal bundle. We also prove that this isomorphism is natural in the sense that it commutes with the isomorphisms between the Morse homology for different choices of the Morse function and the Floer homology for different choices of the Hamiltonian. We define a product on the Floer homology and prove triangle inequality for conormal spectral invariants with respect to this product.

On some transverse geometrical structures of lifted foliation to its conormal bundle.

We consider the lift of a foliation to its conormal bundle and some transverse geometrical structures associated with this foliation are studied. We introduce a good vertical connection on the conormal bundle and, moreover, if the conormal bundle is endowed with a transversal Cartan metric, we obtain that the lifted foliation to its conormal bundle is a Riemannian one. Also, some transversally framed f(3, ε)-structures of corank 2 on the normal bundle of lifted foliation to its conormal bundle are introduced and an almost (para)contact structure on a transverse Liouville distribution is obtained.

Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions

We consider the conormal bundle of a Schubert variety \(S_I\) in the cotangent bundle \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) of the Grassmannian \({{\mathrm{\mathrm {Gr}}}}\) of \(k\)-planes in \({{\mathrm{\mathbb {C}}}}^n\). This conormal bundle has a fundamental class \({\kappa _I}\) in the equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\). Here \({{\mathrm{\mathbb T}}}=({{\mathrm{\mathbb {C}}}}^*)^n\times {{\mathrm{\mathbb {C}}}}^*\). The torus \(({{\mathrm{\mathbb {C}}}}^*)^n\) acts on \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) in the standard way and the last factor \({{\mathrm{\mathbb {C}}}}^*\) acts by multiplication on fibers of the bundle. We express this fundamental class as a sum \(Y_I\) of the Yangian \(Y(\mathfrak {gl}_2)\) weight functions \((W_J)_J\). We describe a relation of \(Y_I\) with the double Schur polynomial \([S_I]\). A modified version of the \(\kappa _I\) classes, named \(\kappa '_I\), satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold \(T^*\!{{\mathrm{\mathrm {Gr}}}}\). This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on \({{\mathrm{\mathrm {Gr}}}}\). The classes \((\kappa '_I)_I\) form a basis in the suitably localized equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\). This basis depends on the choice of the coordinate flag in \({{\mathrm{\mathbb {C}}}}^n\). We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.

Sensitivity analysis of an inverse problem for the wave equation with caustics

The paper investigates the sensitivity of the inverse problem of recovering the velocity field in a bounded domain from the boundary dynamic Dirichlet-to-Neumann map (DDtN) for the wave equation. Three main results are obtained: (1) assuming that two velocity fields are non-trapping and are equal to a constant near the boundary, it is shown that the two induced scattering relations must be identical if their corresponding DDtN maps are sufficiently close; (2) a geodesic X-ray transform operator with matrix-valued weight is introduced by linearizing the operator which associates each velocity field with its induced Hamiltonian flow. A selected set of geodesics whose conormal bundle can cover the cotangent space at an interior point is used to recover the singularity of the X-ray transformed function at the point; a local stability estimate is established for this case. Although fold caustics are allowed along these geodesics, it is required that these caustics contribute to a smoother term in the transform than the point itself. The existence of such a set of geodesics is guaranteed under some natural assumptions in dimensions greater than or equal to three by the classification result on caustics and regularity theory of Fourier Integral Operators. The interior point with the above required set of geodesics is called “fold-regular”. (3) Assuming that a background velocity field with every interior point fold-regular is fixed and another velocity field is sufficiently close to it and satisfies a certain orthogonality condition, it is shown that if the two corresponding DDtN maps are sufficiently close then they must be equal.