Chern Form Research Articles

Regularity results for pluriclosed flow

In [8] the authors introduced a parabolic flow of pluriclosed metrics. New advancements in this flow are given, including improved regularity results, a gradient property and expanding entropy functional, and a conjectural picture of optimal existence results and their consequences. Let (M2n, J) be a complex manifold, and let ω denote a Hermitian metric on M . The metric ω is pluriclosed if ∂∂ω = 0. Consider the initial value problem ∂ ∂t ω = ∂∂∗ω + ∂∂∗ω + √−1 2 ∂∂ log det g ω(0) = ω0. (1) This equation was introduced in [8] as a tool for understanding complex, non-Kahler manifolds. Equation (1) falls into a general class of flows of Hermitian metrics, and as shown in [7], solutions to (1) exist as long as the Chern curvature, torsion, and covariant derivative of torsion are bounded. In this note we will describe many results which refine the existence and singularity formation of solutions to (1), which appear in [9]. The starting point is to interpret equation (1) using the Bismut connection. This is a small but crucial observation which clarifies many aspects of (1). Let us recall the Bismut connection. Let (M2n, ω, J) be a complex manifold with pluriclosed metric, and let D denote the Levi Civita connection. Then the Bismut connection ∇ is defined by 〈∇XY, Z〉 = 〈DXY, Z〉+ 12d ω(X, Y, Z) where dcω(X,Y, Z) := dω(JX, JY, JZ). Let Ω denote the curvature of this connection, and let P denote the Chern form of this connection, i.e. in complex coordinates Pij = Ω k ijk Finally, let PC denote the Ricci form associated to the Chern connection. One can calculate that P = P − dd∗ω. In particular, this implies that a solution to (1) may be expressed as ∂ ∂t ω = − P 1,1 (2) where P 1,1 denotes the projection of P onto (1, 1)-forms. There is another geometric evolution equation naturally associated to a connection with torsion given by a closed Date: August 16, 2010. 1 2 JEFFREY STREETS AND GANG TIAN three-form, namely the B-field renormalization group flow of string theory. A remarkable fact is that (2) is equivalent to this B-field flow, after a change by diffeomorphism. Thus Perelman-type monotone quantities which had previously been discovered for the B-field flow, specifically an energy functional [4] and expanding entropy functional [6], carry over to solutions to (1). To describe the energy functional, first generalize the notation slightly and let (Mn, g) be a Riemannian manifold, and let T denote a three-form on M . Let

Localization of Atiyah classes

We construct the Atiyah classes of holomorphic vector bundles using (1,0)-connections and developing a Chern–Weil type theory, allowing us to effectively compare Chern and Atiyah forms. Combining this point of view with the Čech–Dolbeault cohomology, we prove several results about vanishing and localization of Atiyah classes, and give some applications. In particular, we prove a Bott type vanishing theorem for (not necessarily involutive) holomorphic distributions. As an example we also present an explicit computation of the residue of a singular distribution on the normal bundle of an invariant submanifold that arises from the Camacho–Sad type localization.

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.

Axion Response in Gapless Systems

The strong topological insulator in 3D is expected to realize a quantized magnetoelectric response, the so-called axion response. However, many of the materials predicted to be topological insulators have turned out to be metallic, with bulk Fermi surfaces. Following the result of Bergman and Refael [Phys. Rev. B 82, 195417 (2010)] that the surface states of the topological insulator persist even when the band structure gap is closed, we explore the fate of the magnetoelectric response in such systems. We find that a nonquantized magnetoelectric coupling remains once a bulk Fermi surface opens. More generally, we find higher-dimensional analogs of the intrinsic anomalous Hall effect for all Chern forms-quantized transport coefficients in the gapped case become nonquantized when the gap is closed. In particular, the nonquantized magnetoelectric response in 3D descends from the intrinsic anomalous Hall effect analog in 4D.

Chern-flat and Ricci-flat invariant almost Hermitian structures

We study left-invariant almost Hermitian structures on homogeneous spaces having either flat Chern connection or flat Ricci–Chern form. Many examples are carefully described, and a classification is given in low dimensions.

Horizontal Forms of Chern Type on Complex Finsler Bundles

The aim of this paper is to construct horizontal Chern forms of a holomorphic vector bundle using complex Finsler structures. Also, some properties of these forms are studied.

The Spectral Density Function of a Toric Variety

For a Kahler manifold (X, \omega) with a holomorphic line bundle L and metric h such that the Chern form of L is \omega, the spectral measures are the measures \mu_N = \sum |s_{N,i}|^2 \nu, where \{s_{N,i}\}_i is an L^2-orthonormal basis for H^0(X, L^{\otimes N}), and \nu is Liouville measure. We study the asymptotics in N of \mu_N for (X, L) a Hamiltonian toric manifold, and give a precise expansion in terms of powers 1/N^j and data on the moment polytope \Delta of the Hamiltonian torus K acting on X. In addition, for an infinitesimal character k of K and the unique unit eigensection s_{Nk} for the character Nk of the torus action on H^0(X, L^N), we give a similar expansion for the measures \mu_{Nk} = |s_{Nk}|^2 \nu. A final remark shows that the eigenbasis \{s_{k}, k \in \Delta \cap \mathbb{Z}^{\dim K} \} is a Bohr-Sommerfeld basis in the sense of Tyurin. Some of the present results are related to work of Shiffman, Tate and Zelditch. The present paper uses no microlocal analysis, but rather an Euler-Maclaurin formula for Delzant polytopes.

Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant $f1$ for the first Chern class case becomes an obstruction to the existence of transverse Kähler metric of constant scalar curvature. We prove the existence of transverse Kähler-Ricci solitons (or Sasaki-Ricci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if $S$ is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain Sasaki-Einstein metrics on the $U(1)$-bundles associated with the canonical line bundles of toric Fano manifolds, including as a special case an irregular toric Sasaki-Einstein metrics on the unit circle bundle associated with the canonical bundle of the two-point blow-up of the complex projective plane.

Affine Hermitian-Einstein Metrics

Here c1(E, h) is the first Chern form of E with respect to a Hermitian metric h. The famous theorem of Donaldson [7, 8] (for algebraic manifolds only) and Uhlenbeck-Yau [24, 25] says that an irreducible vector bundle E → N is ω-stable if and only if it admits a HermitianEinstein metric (i.e. a metric whose curvature, when the 2-form part is contracted with the metric on N , is a constant times the identity endomorphism on E). This correspondence between stable bundles and Hermitian-Einstein metrics is often called the Kobayashi-Hitchin correspondence. An important generalization of this theorem is provided by Li-Yau [15] for complex manifolds (and subsequently due to Buchdahl by a different method for surfaces [3]). The major insight for this extension is the fact that the degree is well-defined as long as the Hermitian form ω on N satisfies only ∂∂ωn−1 = 0. This is because

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.

A generalized Poincaré-Lelong formula

We prove a generalization of the classical Poincaré-Lelong formula. Given a holomorphic section $f$, with zero set $Z$, of a Hermitian vector bundle $E\to X$, let $S$ be the line bundle over $X\setminus Z$ spanned by $f$ and let $Q=E/S$. Then the Chern form $c(D_Q)$ is locally integrable and closed in $X$ and there is a current $W$ such that ${dd}^cW=c(D_E)-c(D_Q)-M,$ where $M$ is a current with support on $Z$. In particular, the top Bott-Chern class is represented by a current with support on $Z$. We discuss positivity of these currents, and we also reveal a close relation with principal value and residue currents of Cauchy-Fantappiè-Leray type.

The first Chern form on moduli of parabolic bundles

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen’s metric and interprete it as a local index theorem for the family of \(\overline\partial\)-operators in the associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kahler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein–Maass series. The cuspidal defect is explicitly expressed through the curvature forms of certain natural line bundles on the moduli space related to the parabolic structure. We also compare our result with Witten’s volume computation.

Infinite-Dimensional Lie Theory

Nowadays infinite-dimensional Lie theory is a core area of modern mathematics, covering a broad range of branches, such as the structure and classification theory of infinite-dimensional Lie algebras, geometry of infinite-dimensional Lie groups and their homogeneous spaces, and representation theory of infinite-dimensional Lie groups, Lie algebras and Lie-superalgebras. The focus of this workshop was on recent developments in all of these areas with particular emphasis on connections with other branches of mathematics, such as algebraic groups and Galois cohomology. The meeting was attended by 52 participants from many European countries, Canada, the USA, Brazil, Japan and Australia. The meeting was organized around a series of 23 lectures each of 50 minutes duration representing the major recent advances in the area. We feel that the meeting was exciting and highly successful. The quality of the lectures, several of which surveyed recent developments, was outstanding. The exceptional atmosphere of the Oberwolfach Institute provided an optimal environment for bringing people working in algebraically, geometrically or analytically oriented areas of infinite-dimensional Lie theory together, and to create an atmosphere of scientific interaction and cross-fertilization. Without going too much into detail, let us mention some important new developments that were showcased during the workshop. In the structure theory of infinite-dimensional Lie algebras, the classification of extended affine Lie algebras, based on the notion of a Lie torus has now reached a mature state (Neher). In the classification theory of infinite-dimensional Lie algebras, several deep results were obtained recently with Galois cohomology methods exhibiting exciting connections between forms of multiloop algebras and the Galois theory of forms of algebras over rings (Allison, Gille, Chernousov). This branch of structure theory is complemented by the connection between the classification of generalized Kac–Moody algebras and automorphic forms (Scheithauer). In the representation theory of infinite-dimensional Lie algebras, the most exciting new developments concern various kinds of categories of representations of current algebras and Kac–Moody–Lie (super-)algebras (Benkart, Chari, Futorny, Gorelik, Kumar, Littelmann, Serganova). Another interesting, recently very active direction of Kac–Moody theory are Kac–Moody groups over finite fields, which leads to a new class of infinite simple groups (Caprace). On geometric and analytic Lie theory, we had exiting talks on new geometric directions in the representation theory of Banach–Lie groups, related to Banach–Lie–Poisson spaces (Ratiu), and applications of heat kernel measures in the representation theory of loop groups (Pickrell). On the opposite side of the spectrum of Lie group theory, namely direct limit theory, crucial progress has been made on direct limits of unitary representations, as well as in the context of direct limits of infinite-dimensional groups (Wolf, Glöckner). We further had several contributions dealing with geometric aspects such as Chern forms, gerbes and generalized projective geometries (Paycha, Schweigert, Bertram). Finally, we had several exciting talks about several more particular results, dealing with vertex operator algebras, polyzeta values and quantization (Billig, Mathieu, Omori). More specific information is contained in the abstracts which follow in this volume.

Cusps and the family hyperbolic metric

The hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces, there is a unique germ for the isometry class of a complete hyperbolic metric at a cusp. The renormalization of the metric for the punctured unit disc provides a renormalization for a hyperbolic metric at a cusp. For a holomorphic family of punctured Riemann surfaces, the family of (co)tangent spaces along a puncture defines a tautological holomorphic line bundle over the base of the family. The Hermitian connection and Chern form for the renormalized metric are determined. Connections to the works of M. Mirzakhani [Mi1], [Mi2] and L. Takhtajan and P. Zograf [TZ2] and to intersection numbers for the moduli space of punctured Riemann surfaces studied by E. Witten [Wi1], [Wi2] are presented

Renormalised Chern–Weil forms associated with families of Dirac operators

We provide local expressions for Chern–Weil type forms built from superconnections associated with families of Dirac operators previously investigated in [S. Scott, Zeta–Chern forms and the local family index theorem, Trans. Amer. Math. Soc. (in press). arXiv: math.DG/0406294] and later in [S. Paycha, S. Scott, Chern–Weil forms associated with superconnections, in: B. Booss-Bavnbeck, S. Klimek, M. Lesch, W. Zhang (Eds.), Analysis, Geometry and Topology of Elliptic Operators, World Scientific, 2006]. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as renormalised Chern–Weil forms in as far as they coincide with regularised Chern–Weil forms up to residue correction terms. Similarly, a new formula for the curvature of the local fermionic vacuum line bundles is derived using a residue correction term added to the naive curvature formula. We interpret the odd degree Chern–Weil type forms built from superconnections as Wodzicki residues and establish a transgression formula along the lines of known transgression formulae for η -forms.

Harmonic total Chern forms and stability

In this paper we will perturb the scalar curvature of compact Kähler manifolds by incorporating it with higher Chern forms, and then show that the perturbed scalar curvature has many common properties with the unperturbed scalar curvature. In particular the perturbed scalar curvature becomes a moment map, with respect to a perturbed symplectic structure, on the space of all complex structures on a fixed symplectic manifold, which extends the results of Donaldson and Fujiki on the unperturbed case.

Chern Forms on Mapping Spaces

We state a Chern–Weil type theorem which is a generalization of a Chern–Weil type theorem for Fredholm structures stated by Freed in [4]. Using this result, we investigate Chern forms on based manifold of maps \(Map_b(M,N)\) following two approaches, the first one using the Wodzicki residue, and the second one using renormalized traces of pseudo-differential operators along the lines of [1, 19, 20]. We specialize to the case \(N=G\) to study current groups. Finally, we apply these results to a class of holomorphic connections on the loop group \(H^{1/2}_b(S^{1},G)\). In this last example, we precise Freed's construction [5] on the loop group: The cohomology of the first Chern form of any holomorphic connection in the class considered is given by the Kahler form.

Some invariant and equivariant cohomology classes of the space of Kähler metrics

Invariant and equivariant cohomology classes on the space of Kähler forms are defined. Relations to the obstructions to the existence of Kähler-Einstein metrics and Kähler metrics of harmonic Chern forms are discussed.

INNER STRUCTURE OF GAUSS–BONNET–CHERN THEOREM AND THE MORSE THEORY

We define a new one-form HA based on the second fundamental tensor [Formula: see text], the Gauss–Bonnet–Chern form can be novelly expressed with this one-form. Using the ϕ-mapping theory we find that the Gauss–Bonnet–Chern density can be expressed in terms of the δ-function δ(ϕ) and the relationship between the Gauss–Bonnet–Chern theorem and Hopf–Poincaré theorem is given straightforwardly. The topological current of the Gauss–Bonnet–Chern theorem and its topological structure are discussed in details. At last, the Morse theory formula of the Euler characteristic is generalized.

Curvature on determinant bundles and first Chern forms

The Quillen–Bismut–Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut–Freed connection on the determinant bundle. In finite dimensions, these forms agree (up to sign), but in infinite dimensions there is a correction term, which we express in terms of Wodzicki residues. We illustrate these results with a string theory computation. There is a natural super vector bundle over the manifold of smooth almost complex structures on a Riemannian surface. The Bismut–Freed superconnection is identified with classical Teichmüller theory connections, and its curvature and regularized first Chern form are computed.