Hermitian Line Bundle Research Articles

Effective bounds of linear series on algebraic varieties and arithmetic varieties

Abstract In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert–Samuel formula and the arithmetic Hilbert–Samuel formula. The treatments are high-dimensional generalizations of [25] and [26]. Similar results are obtained independently by Huayi Chen [7] with less explicit error terms.

On the invariant spectrum on

Motivated by the work of Abreu and Freitas , we study the invariant spectrum of the Laplace operator associated to hermitian line bundles endowed with invariant metrics over .

Realization of an equivariant holomorphic Hermitian line bundle as a Quillen determinant bundle

Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,H)$ on $X\times M$ (the action of $G$ on $X$ is trivial), such that the corresponding Quillen determinant line bundle $({\mathcal Q}, h_Q)$, which is a $G$--equivariant holomorphic Hermitian line bundle on $M$, is isomorphic to the given $G$--equivariant holomorphic Hermitian line bundle $({\mathcal L}\,,h)$. This proves a conjecture of Dey and Mathai in \cite{DM}.

Integrable systems, toric degenerations and Okounkov bodies

Let $$X$$ be a smooth projective variety of dimension $$n$$ over $$\mathbb {C}$$ equipped with a very ample Hermitian line bundle $$\mathcal {L}$$ . In the first part of the paper, we show that if there exists a toric degeneration of $$X$$ satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from $$X$$ to the special fiber $$X_0$$ which is a symplectomorphism on an open dense subset $$U$$ . From this we are then able to construct a completely integrable system on $$X$$ in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions $$\{H_1, \ldots , H_n\}$$ on $$X$$ which are continuous on all of $$X$$ , smooth on an open dense subset $$U$$ of $$X$$ , and pairwise Poisson-commute on $$U$$ . Moreover, our integrable system in fact generates a Hamiltonian torus action on $$U$$ . In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the ‘moment map’ $$\mu = (H_1, \ldots , H_n): X \rightarrow \mathbb {R}^n$$ is precisely the Newton-Okounkov body $$\Delta = \Delta (R, v)$$ associated to the homogeneous coordinate ring $$R$$ of $$X$$ , and an appropriate choice of a valuation $$v$$ on $$R$$ . Our main technical tools come from algebraic geometry, differential (Kähler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Łojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties $$X$$ , this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.

An arithmetic Hilbert–Samuel theorem for singular hermitian line bundles and cusp forms

AbstractWe prove arithmetic Hilbert–Samuel type theorems for semi-positive singular hermitian line bundles of finite height. This includes the log-singular metrics of Burgos–Kramer–Kühn. The results apply in particular to line bundles of modular forms on some non-compact Shimura varieties. As an example, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions.

Chern–Simons line bundle on Teichmüller space

Let $X$ be a non-compact geometrically finite hyperbolic 3-manifold without cusps of rank 1. The deformation space $\mc{H}$ of $X$ can be identified with the Teichmuller space $\mc{T}$ of the conformal boundary of $X$ as the graph of a section in $T^*\mc{T}$. We construct a Hermitian holomorphic line bundle $\mc{L}$ on $\mc{T}$, with curvature equal to a multiple of the Weil-Petersson symplectic form. This bundle has a canonical holomorphic section defined by $e^{\frac{1}{\pi}{\rm Vol}_R(X)+2\pi i\CS(X)}$ where ${\rm Vol}_R(X)$ is the renormalized volume of $X$ and $\CS(X)$ is the Chern-Simons invariant of $X$. This section is parallel on $\mc{H}$ for the Hermitian connection modified by the $(1,0)$ component of the Liouville form on $T^*\mc{T}$. As applications, we deduce that $\mc{H}$ is Lagrangian in $T^*\mc{T}$, and that ${\rm Vol}_R(X)$ is a Kahler potential for the Weil-Petersson metric on $\mc{T}$ and on its quotient by a certain subgroup of the mapping class group. For the Schottky uniformisation, we use a formula of Zograf to construct an explicit isomorphism of holomorphic Hermitian line bundles between $\mc{L}^{-1}$ and the sixth power of the determinant line bundle.

The Hamiltonian geometry of the space of unitary connections with symplectic curvature

Let $\mathcal{L} \to \mathcal{M}$ be a Hermitian line bundle over a compact manifold. Write $\mathcal{S}$ for the space of all unitary connections in $\mathcal{L}$ whose curvatures define symplectic forms on $\mathcal{M}$ and $\mathcal{G}$ for the identity component of the group of unitary bundle isometries of $\mathcal{L}$, which acts on $\mathcal{S}$ by pullback. The main observation of this note is that $\mathcal{S}$ carries a $\mathcal{G}$-invariant symplectic structure, there is a moment map for the $\mathcal{G}$-action and that this embeds the components of $\mathcal{S}$ as $\mathcal{G}$-coadjoint orbits. Restricting to the subgroup of $\mathcal{G}$ which covers the identity on $\mathcal{M}$, we see that prescribing the volume form of a symplectic structure can be seen as finding a zero of a moment map. When $\mathcal{M}$ is a Kahler manifold, this gives a moment-map interpretation of the Calabi conjecture. We also describe some directions for future research based upon the picture outlined here.

Asymptotics of orbifold Bergman kernels: mixed curvature case

We derive some asymptotic expansion formulas of the Bergman kernel of high tensor powers of an Hermitian orbifold line bundle with mixed curvature tensored with an orbifold vector bundle on a compact symplectic orbifold. In particular, when the orbifold has isolated singularities, we get an explicit formula for the asymptotic expansion of the Bergman kernel in the distribution sense. Finally, by applying our results to the complex case, we get a Riemann–Roch–Kawasaki type formula.

Effective bound of linear series on arithmetic surfaces

We prove effective upper bounds on the number of effective sections of a Hermitian line bundle over an arithmetic surface. It is an effective version of the arithmetic Hilbert–Samuel formula in the nef case. As a consequence, we obtain effective lower bounds on the Faltings height and on the self-intersection of the canonical bundle in terms of the number of singular points on fibers of the arithmetic surface.

The Second Coefficient of the Asymptotic Expansion of the Bergman Kernel of the Hodge–Dolbeault Operator

We calculate the second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge–Dolbeault operator associated with high powers of a Hermitian line bundle with non-degenerate curvature, using the method of formal power series developed by Ma and Marinescu.

Semi-Abelian Schemes and Heights of Cycles in Moduli Spaces of Abelian Varieties

We study extension properties of Barsotti-Tate groups and we establish diophantine inequalities involving heights of cycles with respect to logarithmically singular hermitian line bundles on arithmetic varieties. We apply these results to bound heights of cycles on moduli spaces of abelian varieties, induced by quotients of abelian varieties by levels of Barsotti-Tate subgroups, over function fields over number fields. To achieve this aim, we combine our results with an effective version of Rumely's theorem on integral points on possibly open arithmetic surfaces and with Faltings' theorems on heights of abelian varieties in isogeny classes.

Equidistribution of Zeros of Holomorphic Sections in the Non-compact Setting

We consider N-tensor powers of a positive Hermitian line bundle L over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed with respect to the natural measure coming from the curvature of L, as N tends to infinity. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL2(Z) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors.

A vanishing result for the Spin c Dirac operator defined along the leaves of a foliation

In the setting of a closed Riemannian manifold endowed with a smooth, non-necessarily integrable distribution, we extend a Lichnerowicz type formula which is known to work in the particular case of a transverse bundle associated to a Riemannian foliation. Interesting settings in which non-integrable distributions appear naturally are emphasized. As an application, we consider the distribution as being even dimensional and integrable; we consider also a hermitian line bundle, with a hermitian connection, such that the induced curvature tensor is non-degenerate, and an arbitrary hermitian bundle endowed also with a hermitian connection. Taking the k power of the line bundle and canonically constructing a Spinc Dirac operator defined along the leaves of the foliation generated by the distribution, we prove a vanishing result for the half kernel of this operator.

Positivity of relative canonical bundles and applications

Given a family $f:\mathcal{X} \to S$ of canonically polarized manifolds, the unique Kähler–Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $\mathcal{K}_{\mathcal{X}/S}$ . We use a global elliptic equation to show that this metric is strictly positive on $\mathcal{X}$ , unless the family is infinitesimally trivial. For degenerating families we show that the curvature form on the total space can be extended as a (semi-)positive closed current. By fiber integration it follows that the generalized Weil–Petersson form on the base possesses an extension as a positive current. We prove an extension theorem for hermitian line bundles, whose curvature forms have this property. This theorem can be applied to a determinant line bundle associated to the relative canonical bundle on the total space. As an application the quasi-projectivity of the moduli space $\mathcal{M}_{\text{can}}$ of canonically polarized varieties follows. The direct images $R^{n-p}f_{*}\Omega^{p}_{\mathcal{X}/S}(\mathcal {K}_{\mathcal{X}/S}^{\otimes m})$ , m>0, carry natural hermitian metrics. We prove an explicit formula for the curvature tensor of these direct images. We apply it to the morphisms $S^{p}\mathcal{T}_{S} \to R^{p}f_{*}\Lambda^{p}\mathcal{T}_{\mathcal{X}/S}$ that are induced by the Kodaira–Spencer map and obtain a differential geometric proof for hyperbolicity properties of $\mathcal{M}_{\text{can}}$ .

Resolution of singularities for a class of Hilbert modules

Let M be the completion of the polynomial ring C(z) under bar] with respect to some inner product, and for any ideal I subset of C (z) under bar], let I] be the closure of I in M. For a homogeneous ideal I, the joint kernel of the submodule I] subset of M is shown, after imposing some mild conditions on M, to be the linear span of the set of vectors {p(i)(partial derivative/partial derivative(w) over bar (1),...,partial derivative/partial derivative(w) over bar (m)) K-I] (., w)vertical bar(w=0), 1 <= i <= t}, where K-I] is the reproducing kernel for the submodule 2] and p(1),..., p(t) is some minimal ``canonical set of generators'' for the ideal I. The proof includes an algorithm for constructing this canonical set of generators, which is determined uniquely modulo linear relations, for homogeneous ideals. A short proof of the ``Rigidity Theorem'' using the sheaf model for Hilbert modules over polynomial rings is given. We describe, via the monoidal transformation, the construction of a Hermitian holomorphic line bundle for a large class of Hilbert modules of the form I]. We show that the curvature, or even its restriction to the exceptional set, of this line bundle is an invariant for the unitary equivalence class of I]. Several examples are given to illustrate the explicit computation of these invariants.

Differentiability of the arithmetic volume function

We introduce the positive intersection product in Arakelov geometry and prove that the arithmetic volume function is continuously differentiable. As applications, we compute the distribution function of the asymptotic measure of a Hermitian line bundle and several other arithmetic invariants.

Chern–Simons pre-quantizations over four-manifolds

We endow the space of connections on an SU ( n ) -principal bundle over a four-manifold with a pre-symplectic structure and define a Hamiltonian action on it of the group of gauge transformations that are trivial on the boundary. Then we consider the trivial SU ( n ) -principal bundle for n ⩾ 3 over the four-manifold that is a submanifold of a null-cobordant four-manifold, and we construct on the moduli space of flat connections a hermitian line bundle with connection whose curvature is given by the pre-symplectic form. This is the Chern–Simons pre-quantization of moduli spaces. The group of gauge transformations on the boundary of the four-manifold acts on the moduli space of flat connections by an infinitesimally symplectic way. When the four-manifold is a 4-dimensional disc we show that this action is lifted to the pre-quantization by its Lie group extension. The geometric description of the latter is related to the 4-dimensional Wess–Zumino–Witten model.

On the L 2-extension of twisted holomorphic sections of singular hermitian line bundles

We prove a sharp Ohsawa–Takegoshi–Manivel type L 2-extension result for twisted holomorphic sections of singular hermitian line bundles over almost Stein manifolds. We establish as corollaries some extension results for pluri-twisted holomorphic sections of singular hermitian line bundles over projective manifolds.

A remark on the embedding theorem associated to complex connections of mixed type

Let $M$ be a compact complex manifold and let $(L,H)$ be a holomorphic Hermitian line bundle over $M$ such that the curvature form of $h$ is nondegenerate and splits into the difference $\Theta_{+} - \Theta_{-}$ of two semipositive forms $\Theta_{+}$ and $\Theta_{-}$ whose null spaces define mutually transverse holomorphic foliations $\mathcal{F}_{-}$ and $\mathcal{F}_{+}$, respectively. Then $L^{m}$ admits, for sufficiently large $m \in \mathbb{N}$, $C^{\infty}$ sections whose ratio embeds $M$ into $\mathbb{CP}^{N}$ holomorphically (resp. antiholomorphically) along $\mathcal{F}_{+}$ (resp. along $\mathcal{F}_{-}$).

Heights and metrics with logarithmic singularities

We prove lower bound and finiteness properties for arakelovian heights with respect to pre-log-log hermitian ample line bundles. These heights were introduced by Burgos, Kramer and Kühn in [J. I. Burgos, J. Kramer and U. Kühn, Cohomological arithmetic Chow rings, J. Inst. Math. Jussieu 6 (2007), 1–172.], in their extension of the arithmetic intersection theory of Gillet and Soulé [H. Gillet and C. Soulé, Arithmetic intersection theory, Publ. Math. IHES 72 (1990), 94–174.], aimed to deal with hermitian vector bundles equipped with metrics admitting suitable logarithmic singularities. Our results generalize the corresponding properties for the heights of cycles in Bost-Gillet-Soulé [J.-B. Bost, H. Gillet and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), 903–1027.], as well as the properties established by Faltings [G. Faltings, Finiteness theorems for abelian varieties over number fields, Arithmetic Geometry, G. Cornell and J. H. Silverman, eds., Springer-Verlag (1986), 9–27.] for heights of points attached to hermitian ample line bundles whose metrics have logarithmic singularities. We also discuss various geometric constructions where such pre-log-log hermitian ample line bundles naturally arise.