Trivial Line Bundle Research Articles

Affine Subspace Concentration Conditions

We define a new notion of affine subspace concentration conditions for lattice polytopes, and prove that they hold for smooth and reflexive polytopes with barycenter at the origin. Our proof involves considering the slope stability of the canonical extension of the tangent bundle by the trivial line bundle and with the extension class $c_1(\mathcal{T}_X)$ on Fano toric varieties.

Non-degeneracy of Cohomological Traces for General Landau–Ginzburg Models

We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is a non-compact complex manifold with trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$, assuming only that the critical locus of $W$ is compact (but may not consist of isolated points). These results can be viewed as certain "deformed" versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of $W$, while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the $\mathbb{Z}_2$-graded category of topological D-branes of such models.

Sheaves on non-reduced curves in a projective surface

Sheaves on non-reduced curves can appear in moduli spaces of 1-dimensional semistable sheaves over a surface and moduli spaces of Higgs bundles as well. We estimate the dimension of the stack Mx (nC, χ) of pure sheaves supported at the non-reduced curve nC (n ≽ 2) with C an integral curve on X. We prove that the Hilbert-Chow morphism $${h_{L,\chi}}:{\cal M}_X^H\left({L,\chi} \right) \to \left| L \right|$$ sending each semistable 1-dimensional sheaf to its support has all its fibers of the same dimension for X Fano or with the trivial canonical line bundle and |L| contains integral curves.

Picard groups of certain compact complex parallelizable manifolds and related spaces

Let G be a complex simply connected semisimple Lie group and let Γ be a torsionless uniform irreducible lattice in G. Then Γ﹨G is a compact complex non-Kähler manifold whose tangent bundle is holomorphically trivial. In this note we compute the Picard group of Γ﹨G when rank(G)≥3. When rank(G)<3, we determine the group Pic0(Γ﹨G)⊂Pic(Γ﹨G) of topologically trivial holomorphic line bundles. When rank(G)≥2, we also show that Pic0(PΓ) is isomorphic to Pic0(Y) where PΓ is a Γ﹨G-bundle associated to a principal G-bundle over a compact connected complex manifold Y, and, when rank(G)≥3, we show that Pic(Y)→Pic(PΓ) is injective with finite cokernel.

Visible actions of compact Lie groups on complex spherical varieties

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible actions on complex manifolds. He proved that for a Lie group $U$ the space of global sections of a $U$-equivariant holomorphic vector bundle $\mathcal{W}$ is multiplicity-free if the $U$-action on the base space $X$ is strongly visible and if the isotropy representations on the fibers are multiplicity-free under some compatibility condition on an anti-holomorphic diffeomorphism of $\mathcal{W}$. Especially in the case of the trivial line bundle, his theorem says that if a Lie group acts on the base space strongly visibly then the space of holomorphic functions is multiplicity-free. In short, the visibility is a geometric condition that assures the multiplicity-freeness property. In this article we consider the converse direction when $U$ is a compact real form of a connected complex reductive algebraic group $G$ and $X$ is an irreducible complex algebraic $G$-variety. In this setting the multiplicity-freeness property of the space of algebraic sections of arbitrary $G$-line bundle is equivalent to the sphericity of the $G$-action on $X$. Then our main result says that the multiplicity-freeness property implies the visibility, namely, if $X$ is a $G$-spherical variety then the $U$-action on $X$ is strongly visible.

Generic Dynamical Properties of Connections on Vector Bundles

Abstract Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $\nabla ^{\mathcal{E}}$ on the vector bundle $\mathcal{E}$. First of all, we show that twisted conformal Killing tensors (CKTs) are generically trivial when $\dim (M) \geq 3$, answering an open question of Guillarmou–Paternain–Salo–Uhlmann [ 14]. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations, which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $\nabla ^{\textrm{End}({\operatorname{{\mathcal{E}}}})}$ on the endomorphism bundle $\textrm{End}({\operatorname{{\mathcal{E}}}})$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e., the geodesic flow is Anosov on the unit tangent bundle), the connections are generically opaque, namely that generically there are no non-trivial subbundles of $\mathcal{E}$ that are preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called operators of uniform divergence type, and on perturbative arguments from spectral theory (especially on the theory of Pollicott–Ruelle resonances in the Anosov case).

The arithmetic Hodge index theorem and rigidity of dynamical systems over function fields

In one of the fundamental results of Arakelov's arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge Index Theorem for arithmetic surfaces by relating the intersection pairing to the negative of the N\'eron-Tate height pairing. More recently, Moriwaki and Yuan-Zhang generalized this to higher dimension. In this work, we extend these results to projective varieties over transcendence degree one function fields. The new challenge is dealing with non-constant but numerically trivial line bundles coming from the constant field via Chow's $K/k$-trace functor. As an application of the Hodge Index Theorem, we also prove a rigidity theorem for the set of canonical height zero points of polarized algebraic dynamical systems over function fields. For function fields over finite fields, this gives a rigidity theorem for preperiodic points, generalizing previous work of Mimar, Baker-DeMarco, and Yuan-Zhang.

Calderón problem for Yang–Mills connections

We consider the problem of identifying a unitary Yang–Mills connection \nabla on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian \nabla^*\nabla over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in the analytic category, by using geometric analysis methods. Moreover, by using a Runge-type approximation argument along curves to recover holonomy, we are able to uniquely determine both the bundle structure and the connection. Also, we prove that the DNmap is an elliptic pseudodifferential operator of order one on the restriction of the vector bundle to the boundary, whose full symbol determines the complete Taylor series of an arbitrary connection, metric and an associated potential at the boundary.

The Local Structure of Generalized Contact Bundles

AbstractGeneralized contact bundles are odd-dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular, in a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.

Manifold properties of planar polygon spaces

We prove that the tangent bundle of a generic space of planar n-gons with specified side lengths, identified under isometry, plus a trivial line bundle is isomorphic to (n−2) times a canonical line bundle. We then discuss consequences for orientability, cobordism class, immersions, and parallelizability.

Dirac’s magnetic monopole and the Kontsevich star product

We examine relationships between various quantization schemes for an electrically charged particle in the field of a magnetic monopole. Quantization maps are defined in invariant geometrical terms, appropriate to the case of nontrivial topology, and are constructed for two operator representations. In the first setting, the quantum operators act on the Hilbert space of sections of a nontrivial complex line bundle associated with the Hopf bundle, whereas the second approach uses instead a quaternionic Hilbert module of sections of a trivial quaternionic line bundle. We show that these two quantizations are naturally related by a bundle morphism and, as a consequence, induce the same phase-space star product. We obtain explicit expressions for the integral kernels of star-products corresponding to various operator orderings and calculate their asymptotic expansions up to the third order in the Planck constant . We also show that the differential form of the magnetic Weyl product corresponding to the symmetric ordering agrees completely with the Kontsevich formula for deformation quantization of Poisson structures and can be represented by Kontsevich’s graphs.

Non locally trivializable CR line bundles over compact Lorentzian CR manifolds

We consider compact CR manifolds of arbitrary CR codimension that satisfy certain geometric conditions in terms of their Levi form. Over these compact CR manifolds, we construct a deformation of the trivial CR line bundle over M which is topologically trivial over M but fails to be even locally CR trivializable over any open subset of M. In particular, our results apply to compact Lorentzian CR manifolds of hypersurface type.

Varieties with $\mathbb {P}$-units

We study the class of compact Kahler manifolds with trivial canonical bundle and the property that the cohomology of the trivial line bundle is generated by one element. If the square of the generator is zero, we get the class of strict Calabi--Yau manifolds. If the generator is of degree 2, we get the class of compact hyperkahler manifolds. We provide some examples and structure results for the cases where the generator is of higher nilpotency index and degree. In particular, we show that varieties of this type are closely related to higher-dimensional Enriques varieties.

Partial Fourier–Mukai transform for integrable systems with applications to Hitchin fibration

Let X be an abelian scheme over a scheme B. The Fourier--Mukai transform gives an equivalence between the derived category of X and the derived category of the dual abelian scheme. We partially extend this to certain schemes X over B (which we call degenerate abelian schemes) whose generic fiber is an abelian variety, while special fibers are singular. Our main result provides a fully faithful functor from a twist of the derived category of Pic$^\tau$(X/B) to the derived category of X. Here Pic$^\tau$(X/B) is the algebraic space classifying fiberwise numerically trivial line bundles. Next, we show that every algebraically integrable system gives rise to a degenerate abelian scheme and discuss applications to Hitchin systems.

Twisted cohomology of the complement of theta divisors in an abelian surface

Let [Formula: see text] be an abelian surface, and [Formula: see text] be the sum of [Formula: see text] distinct theta divisors having normal crossings. We set [Formula: see text]. We study the structure of the nonvanishing twisted cohomology group [Formula: see text], where [Formula: see text] denotes a locally constant sheaf over [Formula: see text] defined by a multiplicative meromorphic function on [Formula: see text] infinitely ramified just along the divisor [Formula: see text] (as will be seen below, we will take as such a function a product of complex powers of theta functions). The de Rham complex on [Formula: see text] with logarithmic poles along [Formula: see text], associated to the twisted cohomology groups [Formula: see text], is [Formula: see text]-valued, where [Formula: see text] denotes a topologically trivial (i.e. Chern class zero) line bundle over [Formula: see text] determined by the locally constant sheaf [Formula: see text]. Therefore the main results of this paper, which give us information on the order of poles of meromorphic 2-forms on [Formula: see text] generating the cohomology group [Formula: see text], are divided into Theorems 4.5 and 4.6, according as the de Rham complex on [Formula: see text] with logarithmic poles along [Formula: see text] takes the values in a holomorphically nontrivial line bundle [Formula: see text] or a holomorphically trivial one [Formula: see text] ([Formula: see text] denoting the holomorphically trivial line bundle [Formula: see text]). Such a phenomenon does not occur in the case of the twisted cohomology of complex projective space with hyperplane arrangement.

Some results on deformations of sections of vector bundles

Let E be a vector bundle on a smooth complex projective variety X. We study the family of sections $$s_t\in H^0(E\otimes L_t)$$ where $$L_t\in Pic^0(X)$$ is a family of topologically trivial line bundle and $$L_0={\mathcal {O}}_X,$$ that is, we study deformations of $$s=s_0$$ . By applying the approximation theorem of Artin (Invent Math 5:277–291, 1968) we give a transversality condition that generalizes the semi-regularity of an effective Cartier divisor. Moreover, we obtain another proof of the Severi–Kodaira–Spencer theorem (Bloch In Invent Math 17:51–66, 1972). We apply our results to give a lower bound to the continuous rank of a vector bundle as defined by Miguel Barja (Duke Math J 164(3):541–568, 2015) and a proof of a piece of the generic vanishing theorems (Green and Lazarsfeld, Invent Math 90:389–407, 1987) and (Green and Lazarsfeld, J Am Math Soc 4:87–103, 1991) for the canonical bundle. We extend also to higher dimension a result given in (Mendes-Lopes et al. In Geo Topol 17:1205:1223, 2013) on the base locus of the paracanonical base locus for surfaces.

An Analytic Description of Local Intersection Numbers at Non-Archimedian Places for Products of Semi-Stable Curves

Nous généralisons une formule de Shou-Wu Zhang [8, Thm 3.4.2], qui donne une description en termes d’analyse élémentaire pour le nombre d’intersections arithmétiques locales de trois diviseurs de Cartier à support dans la fibre spéciale sur l’auto-produit d’une surface arithmétique semi-stable. Par un argument d’approximation, Zhang étend sa formule à une formule pour les nombres d’intersections arithmétiques locaux de trois fibrés en droites avec des métriques adéliques sur l’auto-produit d’une courbe sous condition que le fibré en droites sous-jacent soit trivial. En utilisant les résultats en théorie de l’intersection de [5] nous généralisons les résultats de Zhang aux d-iemes auto-produits pour un nombre naturel arbitraire d. Pour que les approximations convergent, nous devons supposer que le nombre naturel d satisfait une certaine condition d’annulation [5, 4.7]. Cette condition est satisfaite au moins pour d∈{2,3,4,5} 2

Toward a higher codimensional Ueda theory

Ueda’s theory is a theory on a flatness criterion around a smooth hypersurface of a certain type of topologically trivial holomorphic line bundles. We propose a codimension two analogue of Ueda’s theory. As an application, we give a sufficient condition for the anti-canonical bundle of the blow-up of the three dimensional projective space at 8 points to be non semi-ample however admit a smooth Hermitian metric with semi-positive curvature.

A Stein Criterion Via Divisors for Domains Over Stein Manifolds

It is shown that a domain $X$ over a Stein manifold is Stein if the following two conditions are fulfilled: a) the cohomology group $H^i(X,\mathscr{O})$ vanishes for $i \geq 2$ and b) every topologically trivial holomorphic line bundle over $X$ admits a non-trivial meromorphic section. As a consequence we recover, with a different proof, a known result due to Siu stating that a domain $X$ over a Stein manifold $Y$ is Stein provided that $H^i(X,\mathscr{O})=0$ for $i \geq 1$.

A Riemann–Roch theorem for the noncommutative two torus

We prove the analogue of the Riemann–Roch formula for the noncommutative two torus A θ = C ( T θ 2 ) equipped with an arbitrary translation invariant complex structure and a Weyl factor represented by a positive element k ∈ C ∞ ( T θ 2 ) . We consider a topologically trivial line bundle equipped with a general holomorphic structure and the corresponding twisted Dolbeault Laplacians. We define a spectral triple ( A θ , H , D ) that encodes the twisted Dolbeault complex of A θ and whose index gives the left hand side of the Riemann–Roch formula. Using Connes’ pseudodifferential calculus and heat equation techniques, we explicitly compute the b 2 terms of the asymptotic expansion of Tr ( e − t D 2 ) . We find that the curvature term on the right hand side of the Riemann–Roch formula coincides with the scalar curvature of the noncommutative torus recently defined and computed in Connes and Moscovici (2014) and independently computed in Fathizadeh and Khalkhali (2014).