Hermitian Line Bundle Research Articles

The Yang–Mills–Higgs functional on complex line bundles: Asymptotics for critical points

Abstract We consider a gauge-invariant Ginzburg–Landau functional (also known as Abelian Yang–Mills–Higgs model), on Hermitian line bundles over closed Riemannian manifolds of dimension n ≥ 3 {n\geq 3} . Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, thanks to a suitable monotonicity formula, we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, converge to the weight measure of a stationary, rectifiable varifold of codimension 2.

Geometry of analytic continuation on complex manifolds – history, survey, and report

Abstract Beginning with the state of art around 1953, solutions of the Levi problem on complex manifolds will be recalled at first up to Takayama’s result in 1998. Then, the activity of extending the results by the L 2 {L}^{2} method in these decades will be reported. The method is by exploiting the finite dimensionality of certain L 2 {L}^{2} ∂ ¯ \bar{\partial } -cohomology groups to prove that a Hermitian holomorphic line bundle L L over a complex manifold M M is bimeromorphically equivalent to an ample bundle when it is restricted to a bounded locally pseudoconvex domain Ω ⋐ M \Omega \hspace{0.15em}\Subset \hspace{0.15em}M under the positivity of L ∣ ∂ Ω {L| }_{\partial \Omega } and the regularity of ∂ Ω \partial \Omega .

Asymptotic normality of divisors of random holomorphic sections on non-compact complex manifolds

We prove a central limit theorem for divisors of Gaussian i.i.d. centered holomorphic sections of tensor powers of a Hermitian holomorphic line bundle over a non-compact Hermitian manifold.

The asymptotic behavior of Bergman kernels

Let (X,d,p) be the pointed Gromov-Hausdorff limit of a sequence of pointed complete polarized Kähler manifolds (Ml,ωl,Ll,hl,pl) with Ric(hl)=2πωl, Ric(ωl)≥−Λωl and Vol(B1(pl))≥v, ∀l∈N, where Λ,v>0 are constants. Then X is a normal complex space [33].In this paper, we discuss the convergence of the Hermitian line bundles (Ll,hl) and the Bergman kernels. In particular, we show that the Kähler forms ωl converge to a unique closed positive current ωX on Xreg. By establishing a version of L2 estimate on the limit line bundle on X, we give a convergence result of Fubini-Study currents on X. Then we prove that the convergence of Bergman kernels implies a uniform Lp asymptotic expansion of Bergman kernel on the collection of n-dimensional polarized Kähler manifolds (M,ω,L,h) with Ricci lower bound −Λ and non-collapsing condition Vol(B1(x))≥v>0. Under the additional orthogonal bisectional curvature lower bound, we will also give a uniform C0 asymptotic estimate of Bergman kernel for all sufficiently large m, which improves a theorem of Jiang [28]. By calculating the Bergman kernels on orbifolds, we disprove a conjecture of Donaldson-Sun in [20].

Convergence of the self‐dual U(1)‐Yang–Mills–Higgs energies to the (n−2)$(n-2)$‐area functional

AbstractGiven a hermitian line bundle on a closed Riemannian manifold , the self‐dual Yang–Mills–Higgs energies are a natural family of functionals defined for couples consisting of a section and a hermitian connection ∇ with curvature . While the critical points of these functionals have been well‐studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen–Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Γ‐convergence of to (2π times) the codimension two area: more precisely, given a family of couples with , we prove that a suitable gauge invariant Jacobian converges to an integral ‐cycle Γ, in the homology class dual to the Euler class , with mass . We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min‐max values for the ‐area from the Almgren–Pitts theory with those obtained from the Yang–Mills–Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken‐type monotonicity result along the gradient flow of .

Kähler-Einstein metrics and obstruction flatness of circle bundles

Obstruction flatness of a strongly pseudoconvex hypersurface Σ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of Σ, complete up to Σ, has a potential −log⁡u such that u is C∞-smooth up to Σ. In general, u has only a finite degree of smoothness up to Σ. In this paper, we study obstruction flatness of hypersurfaces Σ that arise as unit circle bundles S(L) of negative Hermitian line bundles (L,h) over Kähler manifolds (M,g). We prove that if (M,g) has constant Ricci eigenvalues, then S(L) is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and (M,g) is complete, then we show that the corresponding disk bundle admits a complete Kähler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of S(L) when (M,g) is a Kähler surface (dim⁡M=2) with constant scalar curvature.

The theta invariants and the volume function on arithmetic varieties

We introduce a new arithmetic invariant for hermitian line bundles on arithmetic varieties. We use this invariant to measure the variation of the volume function with respect to the metric. We apply the theory developed here to the study of the arithmetic geometry of toric varieties. As an application, we obtain a generalized Hodge index theorem for hermitian line bundles which are not necessarily toric. When the metrics are toric, we recover some results due to Burgos, Phillippon, Sombra and Moriwaki.

Mirror of volume functionals on manifolds with special holonomy

We can define the “volume” V for Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold X, which can be considered to be the “mirror” of the standard volume for submanifolds. This is called the Dirac-Born-Infeld (DBI) action in physics.In this paper, (1) we introduce the negative gradient flow of V, which we call the line bundle mean curvature flow. Then, we show the short-time existence and uniqueness of this flow. When X is Kähler, we relate the negative gradient of V to the angle function and deduce the mean curvature for Hermitian metrics on a holomorphic line bundle defined by Jacob and Yau.(2) We relate the functional V to a deformed Hermitian Yang–Mills (dHYM) connection, a deformed Donaldson–Thomas connection for a G2-manifold (a G2-dDT connection), a deformed Donaldson–Thomas connection for a Spin(7)-manifold (a Spin(7)-dDT connection), which are considered to be the “mirror” of special Lagrangian, (co)associative and Cayley submanifolds, respectively. When X is a compact Spin(7)-manifold, we prove the “mirror” of the Cayley equality, which implies the following. (a) Any Spin(7)-dDT connection is a global minimizer of V and its value is topological. (b) Any Spin(7)-dDT connection is flat on a flat line bundle. (c) If X is a product of S1 and a compact G2-manifold Y, any Spin(7)-dDT connection on the pullback of the Hermitian complex line bundle over Y is the pullback of a G2-dDT connection modulo closed 1-forms.We also prove analogous statements for G2-manifolds and Kähler manifolds of dimension 3 or 4.

Deformation Theory of Deformed Hermitian Yang–Mills Connections and Deformed Donaldson–Thomas Connections

A deformed Donaldson-Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a $G_2$-manifold $X$ satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. The dDT connection is an analogue of a deformed Hermitian Yang-Mills (dHYM) connection which is extensively studied recently. In this paper, we study the moduli spaces of dDT and dHYM connections. In the former half, we prove that the deformation of dDT connections is controlled by a subcomplex of the canonical complex, an elliptic complex defined by Reyes Carri\'on, by introducing a new coclosed $G_2$-structure. If the deformation is unobstructed, we also show that the connected component is a $b^{1}$-dimensional torus, where $b^{1}$ is the first Betti number of $X$. A canonical orientation on the moduli space is also given. We also prove that the obstruction of the deformation vanishes if we perturb the $G_2$-structure generically under some mild assumptions. In the latter half, we prove that the moduli space of dHYM connections, if it is nonempty, is a $b^{1}$-dimensional torus, especially, it is connected and orientable. We also prove the existence of a family of moduli spaces along a deformation of underlying structures if some necessary conditions are satisfied.

Chern–Weil and Hilbert–Samuel formulae for singular Hermitian line bundles

We show a Chern–Weil type statement and a Hilbert–Samuel formula for a large class of singular plurisubharmonic metrics on a line bundle over a smooth projective complex variety. For this we use the theory of b-divisors and the so-called multiplier ideal volume function. We apply our results to the line bundle of Siegel-Jacobi forms over the universal abelian variety endowed with its canonical invariant metric. This generalizes the results of [ J. I. Burgos Gil et al., Lond. Math. Soc. Lect. Note Ser. 427, 45–77 (2016; Zbl 1378.14027)] to higher degrees.

Heat Kernels Estimates for Hermitian Line Bundles on Manifolds of Bounded Geometry

We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian upper bound for the associated heat kernel. The proof is based on some tools from the theory of operator semigroups in a Hilbert space, results on Sobolev spaces adapted to the current setting, and weighted estimates with appropriate exponential weights.

Dynamical obstructions to classification by (co)homology and other TSI-group invariants

In the spirit of Hjorth’s turbulence theory, we introduce “unbalancedness”: a new dynamical obstruction to classifying orbit equivalence relations by actions of Polish groups which admit a two-sided invariant metric (TSI). Since abelian groups are TSI, unbalancedness can be used for identifying which classification problems cannot be solved by classical homology and cohomology theories. In terms of applications, we show that Morita equivalence of continuous-trace C ∗ C^* -algebras, as well as isomorphism of Hermitian line bundles, are not classifiable by actions of TSI groups. In the process, we show that the Wreath product of any two non-compact subgroups of S ∞ S_{\infty } admits an action whose orbit equivalence relation is generically ergodic against any action of a TSI group and we deduce that there is an orbit equivalence relation of a CLI group which is not classifiable by actions of TSI groups.

Rational versus transcendental points on analytic Riemann surfaces

Let (X, L) be a polarized variety over a number field K. We suppose that L is an hermitian line bundle. Let M be a non compact Riemann Surface and $$U\subset M$$ be a relatively compact open set. Let $$\varphi :M\rightarrow X(\mathbf{C})$$ be a holomorphic map. For every positive real number T, let $$A_U(T)$$ be the cardinality of the set of $$z\in U$$ such that $$\varphi (z)\in X(K)$$ and $$h_L(\varphi (z))\le T$$ . After a revisitation of the proof of the sub exponential bound for $$A_U(T)$$ , obtained by Bombieri and Pila, we show that there are intervals of the reals such that for T in these intervals, $$A_U(T)$$ is upper bounded by a polynomial in T. We then introduce subsets of type S with respect of $$\varphi $$ . These are compact subsets of M for which an inequality similar to Liouville inequality on algebraic points holds. We show that, if M contains a subset of type S, then, for every value of T the number $$A_U(T)$$ is bounded by a polynomial in T. As a consequence, we show that if M is a smooth leaf of an algebraic foliation in curves defined over K then $$A_U(T)$$ is bounded by a polynomial in T. Let S(X) be the subset (full for the Lebesgue measure) of points which verify some kind of Liouville inequalities. In the second part we prove that $$\varphi ^{-1}(S(X))\ne \emptyset $$ if and only if $$\varphi ^{-1}(S(X))$$ is full for the Lebesgue measure on M.

Deformation Theory of Deformed Donaldson–Thomas Connections for $${\text {Spin}(7)}$$-manifolds

A deformed Donaldson-Thomas connection for a manifold with a ${\rm Spin}(7)$-structure, which we call a ${\rm Spin}(7)$-dDT connection, is a Hermitian connection on a Hermitian line bundle $L$ over a manifold with a ${\rm Spin}(7)$-structure defined by fully nonlinear PDEs. It was first introduced by Lee and Leung as a mirror object of a Cayley cycle obtained by the real Fourier-Mukai transform and its alternative definition was suggested in our other paper. As the name indicates, a ${\rm Spin}(7)$-dDT connection can also be considered as an analogue of a Donaldson-Thomas connection (${\rm Spin}(7)$-instanton). In this paper, using our definition, we show that the moduli space $\mathcal{M}_{{\rm Spin}(7)}$ of ${\rm Spin}(7)$-dDT connections has similar properties to these objects. That is, we show the following for an open subset $\mathcal{M}'_{{\rm Spin}(7)} \subset \mathcal{M}_{{\rm Spin}(7)}$. (1) Deformations of elements of $\mathcal{M}'_{{\rm Spin}(7)}$ are controlled by a subcomplex of the canonical complex introduced by Reyes Carri\'on by introducing a new ${\rm Spin}(7)$-structure from the initial ${\rm Spin}(7)$-structure and a ${\rm Spin}(7)$-dDT connection. (2) The expected dimension of $\mathcal{M}'_{{\rm Spin}(7)}$ is finite. It is $b^1$, the first Betti number of the base manifold, if the initial ${\rm Spin}(7)$-structure is torsion-free. (3) Under some mild assumptions, $\mathcal{M}'_{{\rm Spin}(7)}$ is smooth if we perturb the initial ${\rm Spin}(7)$-structure generically. (4) The space $\mathcal{M}'_{{\rm Spin}(7)}$ admits a canonical orientation if all deformations are unobstructed.

On the slopes of the lattice of sections of hermitian line bundles

In this paper we apply Arakelov theory to study the distribution of the Petersson norms of classical cusp forms as well as the distribution of the sup norms of rational functions on adelic subsets of curves. The method in both cases is to study the limiting distribution of the successive minima of norms of global sections of powers of a metrized ample line bundle as one takes increasing powers of the bundle. We develop a general method for computing the measure associated to this distribution. We also study measures associated to the zeros of sections which have small norm.

Stable Solutions to the Abelian Yang–Mills–Higgs Equations on $$S^2$$ and $$T^2$$

We show under natural assumptions that stable solutions to the abelian Yang–Mills–Higgs equations on Hermitian line bundles over the round 2-sphere actually satisfy the vortex equations, which are a first-order reduction of the (second-order) abelian Yang–Mills–Higgs equations. We also obtain a similar result for stable solutions on a flat 2-torus. Our method of proof comes from the work of Bourguignon–Lawson (Commun Math Phys 79(2):189–230, 1981) concerning stable SU(2) Yang–Mills connections on compact homogeneous 4-manifolds.

Curvature and $$L^p$$ Bergman Spaces on Complex Submanifolds in $$\pmb {{\mathbb {C}}^N}$$

Let M be a closed complex submanifold in $${\mathbb {C}}^N$$ with the complete Kahler metric induced by the Euclidean metric. Several finiteness theorems on the $$L^p$$ Bergman space of holomorphic sections of a given Hermitian line bundle L over M and the associated $$L^2$$ cohomology groups are obtained. Some infiniteness theorems are also given in order to test the accuracy of finiteness theorems. As applications we obtain some rigidity results concerning growth of curvatures.

A Natural Hermitian Line Bundle on the Moduli Space of Semistable Representations of a Quiver

This paper describes the construction of a natural Hermitian holomorphic line bundle on the stratified moduli space of complex representations of a finite quiver, which are semistable with respect to a fixed rational weight and have a fixed type. It is shown that the curvature of this Hermitian line bundle on each stratum of the moduli space is essentially the Kahler form of that stratum.

Finite TYCZ expansions and cscK metrics

Let (M,g) be a Kähler manifold whose associated Kähler form ω is integral and let (L,h)→(M,ω) be a quantization hermitian line bundle. In this paper we study those Kähler manifolds (M,g) admitting a finite TYCZ expansion, namely those for which the associated Kempf distortion function Tmg is of the form:Tmg(p)=fs(p)ms+fs−1(p)ms−1+⋯+fr(p)mr,fj∈C∞(M),s,r∈Z. We show that if the TYCZ expansion is finite then Tmg is indeed a polynomial in m of degree n, n=dimC⁡M, and the log-term of the Szegö kernel of the disc bundle D⊂L⁎ vanishes (where L⁎ is the dual bundle of L). Moreover, we provide a complete classification of the Kähler manifolds admitting finite TYCZ expansion either when M is a complex curve or when M is a complex surface with a cscK metric which admits a radial Kähler potential.

Transcendental Liouville Inequalities on Projective Varieties

Abstract Let $p$ be an algebraic point of a projective variety $X$ defined over a number field. Liouville inequality tells us that the norm at $p$ of a non-vanishing integral global section of a hermitian line bundle over $X$ is zero or it cannot be too small with respect to the $\sup $ norm of the section itself. We study inequalities similar to Liouville’s for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture by Chudnovsky and give two applications to the growth of the number of rational points of bounded height on the image of an analytic map from a disk to a projective variety.