Chern Form Research Articles

Existence of higher extremal Kähler metrics on a minimal ruled surface

In this paper we prove that on a special type of minimal ruled surface, which is an example of a ‘pseudo-Hirzebruch surface’, every Kähler class admits a certain kind of ‘higher extremal Kähler metric’, which is a Kähler metric whose corresponding top Chern form and volume form satisfy a nice equation motivated by analogy with the equation characterizing an extremal Kähler metric. From an already proven result, it will follow that this specific higher extremal Kähler metric cannot be a ‘higher constant scalar curvature Kähler (hcscK) metric’, which is defined, again by analogy with the definition of a constant scalar curvature Kähler (cscK) metric, to be a Kähler metric whose top Chern form is harmonic. By doing a certain set of computations involving the top Bando-Futaki invariant we will conclude that hcscK metrics do not exist in any Kähler class on this surface.

Higher Chern-Simons based on (2-)crossed modules

We present higher Chern-Simons theories based on (2-)crossed modules. We start from the generalized differential forms in Generalized Differential Calculus and define the corresponding generalized connections which consist of higher connections. Then we establish the generalized Chern-Simons forms to get the higher Chern-Simons actions. Finally, we develop the higher second Chern forms and Chern-Weil theorems.

Bott–Chern harmonic forms and primitive decompositions on compact almost Kähler manifolds

Let (X,J,omega ) be a compact 2n-dimensional almost Kähler manifold. We prove primitive decompositions for Bott–Chern and Aeppli harmonic forms in special bidegrees and show that such bidegrees are optimal. We also show how the spaces of primitive Bott–Chern, Aeppli, Dolbeault and partial -harmonic forms on (X,J,omega ) are related.

Equidistribution of Hodge loci II

Let $\mathbb {V}$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, the pull–push form. In particular, it is always analytically dense when the pull–push form does not vanish. When the weight is two, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q$th Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal {A}_g$, equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal {A}_g$. These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull–push form appears in this greater generality, we provide several tools to determine it, and we compute it in many examples.

Pointwise universal Gysin formulae and applications towards Griffiths’ conjecture

Let $X$ be a complex manifold, $(E,h)\to X$ be a rank $r$ holomorphic hermitian vector bundle, and $\rho$ be a sequence of dimensions $0 = \rho_0 < \rho_1 < \cdots < \rho_m = r$. Let $Q_{\rho,j}$, $j=1,\dots,m$, be the tautological line bundles over the (possibly incomplete) flag bundle $\mathbb{F}_{\rho}(E) \to X$ associated to $\rho$, endowed with the natural metrics induced by that of $E$, with Chern curvatures $\Xi_{\rho,j}$. We show that the universal Gysin formula \textsl{\`{a} la} Darondeau--Pragacz for the push-forward of a homogeneous polynomial in the Chern classes of the $Q_{\rho,j}$'s also hold pointwise at the level of the Chern forms $\Xi_{\rho,j}$ in this hermitianized situation. As an application, we show the positivity of several polynomials in the Chern forms of a Griffiths (semi)positive vector bundle not previously known, thus giving some new evidences towards a conjecture by Griffiths, which in turn can be seen as a pointwise hermitianized version of the Fulton--Lazarsfeld Theorem on numerically positive polynomials for ample vector bundles.

Chern currents of coherent sheaves

Given a finite locally free resolution of a coherent analytic sheaf $\mathcal F$, equipped with Hermitian metrics and connections, we construct an explicit current, obtained as the limit of certain smooth Chern forms of $\mathcal F$, that represents the Chern class of $\mathcal F$ and has support on the support of $\mathcal F$. If the connections are $(1,0)$-connections and $\mathcal F$ has pure dimension, then the first nontrivial component of this Chern current coincides with (a constant times) the fundamental cycle of $\mathcal F$. The proof of this goes through a generalized Poincar\'e-Lelong formula, previously obtained by the authors, and a result that relates the Chern current to the residue current associated with the locally free resolution.

On characteristic forms of positive vector bundles, mixed discriminants, and pushforward identities

We prove that Schur polynomials in Chern forms of Nakano and dual Nakano positive vector bundles are positive as differential forms. Moreover, modulo a statement about the positivity of a “double mixed discriminant” of linear operators on matrices, which preserve the cone of positive definite matrices, we establish that Schur polynomials in Chern forms of Griffiths positive vector bundles are weakly positive as differential forms. This provides differential-geometric versions of Fulton–Lazarsfeld inequalities for ample vector bundles. An interpretation of positivity conditions for vector bundles through operator theory lies in the core of our approach. Another important step in our proof is to establish a certain pushforward identity for characteristic forms, refining the determinantal formula of Kempf–Laksov for holomorphic vector bundles on the level of differential forms. In the same vein, we establish a local version of Jacobi–Trudi identity. Then we study the inverse problem and show that already for vector bundles over complex surfaces, one cannot characterize Griffiths positivity (and even ampleness) through the positivity of Schur polynomials, even if one takes into consideration all quotients of a vector bundle.

A note on Griffiths' conjecture about the positivity of Chern–Weil forms

Let (E,h) be a Griffiths semipositive Hermitian holomorphic vector bundle of rank 3 over a complex manifold. In this paper, we prove the positivity of the characteristic differential form c1(E,h)∧c2(E,h)−c3(E,h), thus providing a new evidence towards a conjecture by Griffiths about the positivity of the Schur polynomials in the Chern forms of Griffiths semipositive vector bundles. As a consequence, we establish a new chain of inequalities between Chern forms. Moreover, we point out how to obtain the positivity of the second Chern form c2(E,h) in any rank, starting from the well-known positivity of such form if (E,h) is just Griffiths positive of rank 2. The final part of the paper gives an overview on the state of the art of Griffiths' conjecture, collecting several remarks and open questions.

Chern forms of Hermitian metrics with analytic singularities on vector bundles

We define Chern and Segre forms, or rather currents, associated with a Griffiths positive singular hermitian metric $h$ with analytic singularities on a holomorphic vector bundle $E$. The currents are constructed as pushforwards of generalized Monge-Amp\`ere products on the projectivization of $E$. The Chern and Segre currents represent the Chern and Segre classes of $E$, respectively, and coincide with the Chern and Segre forms of $E$ and $h$, where $h$ is smooth. Moreover, our currents coincide with the Chern and Segre forms constructed by the first three authors and Ruppenthal in the cases when these are defined.

Phonon Magnetic Moment from Electronic Topological Magnetization.

The traditional theory of magnetic moments for chiral phonons is based on the picture of the circular motion of the Born effective charge, typically yielding a small fractional value of the nuclear magneton. Here we investigate the adiabatic evolution of electronic states induced by the lattice vibration of a chiral phonon and obtain an electronic orbital magnetization in the form of a topological second Chern form. We find that the traditional theory needs to be refined by introducing a k resolved Born effective charge, and identify another contribution from the phonon-modified electronic energy together with the momentum-space Berry curvature. The second Chern form can diverge when there is a Yang's monopole near the parameter space of interest as illustrated by considering a phonon at the Brillouin zone corner in a gapped graphene model. We also find large magnetic moments for the optical phonon in bulk topological materials where nontopological contribution is also important. Our results agree with recent observations in experiments.

Nonproper intersection products and generalized cycles

We develop intersection theory in terms of the {{mathscr {B}}}-group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the {{mathscr {B}}}-classes have well-defined multiplicities at each point. We focus on a {{mathscr {B}}}-analogue of the intersection theory based on the Stückrad–Vogel procedure and the join construction in projective space. Our approach provides global {{mathscr {B}}}-classes which satisfy a Bézout theorem and have the expected local intersection numbers. We also introduce {{mathscr {B}}}-analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a {{mathscr {B}}}-variant of van Gastel’s formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.

An analytic application of Geometric Invariant Theory

Given a compact Kähler manifold, Geometric Invariant Theory is applied to construct analytic GIT-quotients that are local models for a classifying space of (poly)stable holomorphic vector bundles containing the coarse moduli space of stable bundles as an open subspace. For local models invariant generalized Weil-Petersson forms exist on the parameter spaces, which are restrictions of symplectic forms on smooth ambient spaces. If the underlying Kähler manifold is of Hodge type, then the Weil-Petersson form on the moduli space of stable vector bundles is known to be the Chern form of a certain determinant line bundle equipped with a Quillen metric. It gives rise to a holomorphic line bundle on the classifying GIT space together with a continuous hermitian metric.

The Weil–Petersson current on Douady spaces

AbstractThe Douady space of compact subvarieties of a Kähler manifold is equipped with the Weil–Petersson current, which is everywhere positive with local continuous potentials, and of class when restricted to the locus of smooth fibers. There a Quillen metric is known to exist, whose Chern form is equal to the Weil–Petersson form. In the algebraic case, we show that the Quillen metric can be extended to the determinant line bundle as a singular hermitian metric. On the other hand the determinant line bundle can be extended in such a way that the Quillen metric yields a singular hermitian metric whose Chern form is equal to the Weil–Petersson current. We show a general theorem comparing holomorphic line bundles equipped with singular hermitian metrics which are isomorphic over the complement of a snc divisor B. They differ by a line bundle arising from the divisor and a flat line bundle. The Chern forms differ by a current of integration with support in B and a further current related to its normal bundle. The latter current is equal to zero in the case of Douady spaces due to a theorem of Yoshikawa on Quillen metrics for singular families over curves.

On \u03b1\u2032-effects from D-branes in 4d mathcal{N} = 1

In this work we study type IIB Calabi-Yau orientifold compactifications in the presence of space-time filling D7-branes and O7-planes. In particular, we conclude that α′2gs-corrections to their DBI actions lead to a modification of the four-dimensional mathcal{N} = 1 Kähler potential and coordinates. We focus on the one-modulus case of the geometric background i.e. h1,1 = 1 where we find that the α′2gs-correction is of topological nature. It depends on the first Chern form of the four-cycle of the Calabi-Yau orientifold which is wrapped by the D7-branes and O7-plane. This is in agreement with our previous F-theory analysis and provides further evidence for a potential breaking of the no-scale structure at order α′2gs. Corrected background solutions for the dilaton, the warp-factor as well as the internal space metric are derived. Additionally, we briefly discuss α′-corrections from other Dp-branes.

Spectral 𝜁-invariants lifted to coverings

The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral ζ \zeta -invariants using lifted defect formulae which express discrepancies of ζ \zeta -regularised traces in terms of Wodzicki residues. We derive Atiyah’s L 2 L^2 -index theorem as an instance of the Z 2 \mathbb {Z}_2 -graded generalisation of the canonical lift of spectral ζ \zeta -invariants and we show that certain lifted spectral ζ \zeta -invariants for geometric operators are integrals of Pontryagin and Chern forms.

Nonnegative Hermitian vector bundles and Chern numbers

We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern numbers of such a holomorphic vector bundle are nonnegative and can be bounded below and above respectively by two special Chern numbers. As applications, we obtain a family of new results on compact connected complex manifolds which are homogeneous or can be holomorphically immersed into complex tori, some of which improve several classical results.

A vector bundle version of the Monge-Ampère equation

We introduce a vector bundle version of the complex Monge-Ampère equation motivated by a desire to study stability conditions involving higher Chern forms. We then restrict ourselves to complex surfaces, provide a moment map interpretation of it, and define a positivity condition (MA-positivity) which is necessary for the infinite-dimensional symplectic form to be Kähler. On rank-2 bundles on compact complex surfaces, we prove two consequences of the existence of a “positively curved” solution to this equation - Stability (involving the second Chern character) and a Kobayashi-Lübke-Bogomolov-Miyaoka-Yau type inequality. Finally, we prove a Kobayashi-Hitchin correspondence for a dimensional reduction of the aforementioned equation.

Curvature Currents and Chern Forms of Singular Hermitian Metrics on Holomorphic Vector Bundles

We study curvature currents of singular Hermitian metrics on holomorphic vector bundles. It is known that curvature currents of singular Hermitian metrics on vector bundles are generally not defined with measure coefficients. In this paper, we give some sufficient conditions that curvature currents can be defined with measure coefficients. Moreover, we investigate Chern forms associated with singular Hermitian metrics.

Topological terms, AdS2n gravity, and renormalized entanglement entropy of holographic CFTs

We extend our topological renormalization scheme for Entanglement Entropy to holographic CFTs of arbitrary odd dimensions in the context of the AdS/CFT correspondence. The procedure consists in adding the Chern form as a boundary term to the area functional of the Ryu-Takayanagi minimal surface. The renormalized Entanglement Entropy thus obtained can be rewritten in terms of the Euler characteristic and the AdS curvature of the minimal surface. This prescription considers the use of the Replica Trick to express the renormalized Entanglement Entropy in terms of the renormalized gravitational action evaluated on the conically-singular replica manifold extended to the bulk. This renormalized action is obtained in turn by adding the Chern form as the counterterm at the boundary of the 2n-dimensional asymptotically AdS bulk manifold. We explicitly show that, up to next-to-leading order in the holographic radial coordinate, the addition of this boundary term cancels the divergent part of the Entanglement Entropy. We discuss possible applications of the method for studying CFT parameters like central charges.

Renormalization of entanglement entropy from topological terms

We propose a renormalization scheme for entanglement entropy of three-dimensional CFTs with a four-dimensional asymptotically AdS gravity dual in the context of the gauge/gravity correspondence. The procedure consists in adding the Chern form as a boundary term to the area functional of the Ryu-Takayanagi minimal surface. We provide an explicit prescription for the renormalized entanglement entropy, which is derived via the replica trick. This is achieved by considering a Euclidean gravitational action renormalized by the addition of the Chern form at the spacetime boundary, evaluated in the conically-singular replica manifold. We show that the addition of this boundary term cancels the divergent part of the entanglement entropy, recovering the results obtained by Taylor and Woodhead. We comment on how this prescription for renormalizing the entanglement entropy is in line with the general program of topological renormalization in asymptotically AdS gravity.