Semistability of invariant bundles over $G/Γ$

K-semistability is equivariant volume minimization

This is a continuation to the paper [arXiv:1511.08164] in which a problem of minimizing normalized volumes over $\mathbb{Q}$-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-Einstein metrics on Fano varieties. In particular, we prove that for a $\mathbb{Q}$-Fano variety $V$, the K-semistability of $(V, -K_V)$ is equivalent to the condition that the normalized volume is minimized at the canonical valuation ${\rm ord}_V$ among all $\mathbb{C}^*$-invariant valuations on the cone associated to any po…

A necessary condition for Chow semistability of polarized toric manifolds

Let Δ\subset \mathbb{R}^n be an n-dimensional Delzant polytope. It is well-known that there exist the n-dimensional compact toric manifold X_Δand the very ample (\mathbb{C}^\times)^n-equivariant line bundle L_Δon X_Δassociated with Δ. In the present paper, we show that if (X_Δ,L_Δ^i) is Chow semistable then the sum of integer points in iΔis the constant multiple of the barycenter of Δ. Using this result we get a necessary condition for the polarized toric manifold (X_Δ,L_Δ) being asymptotically Chow semistable. Moreover we can generalize the result of Futaki, Sano and the author to the case wh…

On semistable principal bundles over a complex projective manifold

Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character χof P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle vanishes in rational cohomology if and only if the line bundle over E/P defined by χis numerically effective. Similar results remain valid for principal bundles with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka where he…

On vector bundles over hyperkähler twistor spaces

We study the holomorphic vector bundles E over the twistor space Tw(M) of a compact simply connected hyperkähler manifold $M$. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection π:Tw(M)\rightarrow CP^1. It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on Tw(M) of prime rank, we prove that its restriction to the generic fibre of πis stable. On the other hand,…

Scalar curvature and asymptotic Chow stability of projective bundles and blowups

The holomorphic invariants introduced by Futaki as obstruction to the asymptotic Chow semistability are studied by an algebraic-geometric point of view and are shown to be the Mumford weights of suitable line bundles on the Hilbert scheme. These invariants are calculated in two special cases. The first is a projective bundle over a curve of genus at least two, and it is shown that it is asymptotically Chow polystable (with every polarization) if and only the underlying vector bundle is slope polystable. This proves a conjecture of Morrison with the extra assumption that the involved polarizati…

Projectivity of moment map quotients

Let G be a complex reductive group and K a maximal compact subgroup. If X is a smooth projective G-variety, with a fixed (not necessarily integral) K-invariant Kaehler form, then the K-action is Hamiltonian. Let M be the zero fiber of the corresponding moment map. It is well known that the quotient M/K is a complex space in a natural way. We prove that M/K is a projective variety. In particular, it follows that semistability with respect to a moment map is equivalent to semistability in the sense of Mumford.

The Donaldson-Futaki invariant for sequences of test configurations

In this note, given a polarized algebraic manifold $(X,L)$, we define the Donaldson-Futaki invariant for a sequence of test configurations for $(X,L)$ with exponents tending to infinity. This then allows us to define a strong version of K-stability or K-semistability for $(X,L)$. In particular, $(X,L)$ will be shown to be K-semistable in this strong sense if the polarization class $c_1(L)$ admits a constant scalar curvature Kaehler metric.

Fano型射影丛与射影锥的Delta不变量

Delta invariants of projective bundles and projective cones of Fano type

本文针对Fano型射影丛与射影锥，建立了计算其Delta不变量的精确公式，为相关几何量的确定提供了明确工具。

Gromov-Witten invariants of the moduli of bundles on a surface

We produce an equality between the Gromov-Witten invariants of the moduli space M of rank two odd degree stable vector bundles over a Riemann surface $Σ$ and the Donaldson invariants of the algebraic surface $Σ\times P^1$. We discuss on to how extent the Quantum cohomology of M determines its Gromow-Witten invariants. Finally we find the isomorphism between the usual cohomology and the Quantum cohomology for the moduli space M over a Riemann surface of genus g=3.

Einstein-Hilbert泛函与Donaldson-Futaki不变量

The Einstein-Hilbert functional and the Donaldson-Futaki invariant

本文研究极化Kähler流形上Einstein-Hilbert泛函，证明其临界点为常数量曲率Sasaki结构，并揭示其与Donaldson-Futaki不变量的联系，由此给出K-半稳定性的新证明。

Singularities of metrics on Hodge bundles and their topological invariants

We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds. We also describe corresponding invariants for more general degenerating families in the case of the Quillen metric.

Invariant connections and invariant holomorphic bundles on homogeneous manifolds

Let $X$ be a differentiable manifold endowed with a transitive action $α:A\times X\longrightarrow X$ of a Lie group $A$. Let $K$ be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: {enumerate} equivalence classes of $α$-invariant $K$-connections on $X$, $α$-invariant gauge classes of $K$-connections on $X$, and $α$-invariant isomorphism classes of pairs $(Q,P)$ consisting of a holomorphic $K^\C$-bundle $Q\longrightarrow X$ and a $K$-reduction $P$ of $Q$ (when $X$ has an $…

$η$-invariant and flat vector bundles

We present an alternate definition of the mod {\bf Z} component of the Atiyah-Patodi-Singer $η$ invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi and Singer.

稳定平凡向量丛上带联络的不变量

Invariants of stably trivial vector bundles with connection

本文定义了稳定平凡向量丛上联络的Chern-Simons不变量，证明其可加性，并由此得到共形浸入的障碍。

Higgs bundles, the Toledo invariant and the Cayley correspondence

We define the Toledo invariant of a G-Higgs bundle on a Riemann surface, where G is a real semisimple group of Hermitian type, and we prove a Milnor-Wood type bound for this invariant when the bundle is semistable. We prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley correspondence when the symmetric space defined by G is of tube type. This gives a new proof of the Milnor-Wood inequality of Burger-Iozzi-Wienhard for representations of the fundamental group of a Riemann surface into G. Compared to previous results using Higgs bundles, it uses gener…

紧秩一对称空间切球丛上的所有不变切触度量结构

All invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces

本文完整分类了紧秩一对称空间切球丛上的所有不变切触度量结构，证明仅复射影空间情形存在3-Sasakian度量，并给出球面与实射影空间上唯一的Einstein切触度量，同时将这些结构延拓到穿孔切丛上的相应近Kähler、Kähler及hyperKähler结构。

An Approach to Differential Invariants of $G$-Structures

In a natural way, the local diffeomorphisms of a manifold onto itself act on the reference frame bundles of any order and on the bundles associated with them. Due to the transitivity, the invariants by diffeomorphisms of an associated bundle correspond to the real functions on the orbit space of the action of the jet group on the typical fiber, which enables to construct the said associated bundle. Scalar differential invariants are functions on bundles of jets of sections of associated bundles, which are invariant by the standard action of local diffeomorphisms. We show that these jet bundles…

紧致对称空间的切球丛上的不变接触度量结构

Invariant contact metric structures on tangent sphere bundles of compact symmetric spaces

本文刻画了紧致秩一对称空间，指出其球丛上的标准向量场关于某不变黎曼度量是Killing的，并确定了相关度量及接触结构；对秩≥2情形，构造了包含标准结构的不变接触度量结构族。

Invariant forms, associated bundles and Calabi-Yau metrics

We develop a method, initially due to Salamon, to compute the space of ``invariant'' forms on an associated bundle X=P\times_G V, with a suitable notion of invariance. We determine sufficient conditions for this space to be d-closed. We apply our method to the construction of Calabi-Yau metrics on TCP^1 and TCP^2.

Geometric Invariant Theory, holomorphic vector bundles and the Harder-Narasimhan filtration

This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed.

圆包上Fano流形的eta不变量

The eta invariant of a circle bundle on a Fano manifold

研究正线丛上圆包的spin-c狄拉克算子，计算其eta不变量并扩展了adiabatic参数的范围。

On Rho invariants of fiber bundles

Using adiabatic limits of Eta invariants, Rho invariants of the total space of a fiber bundle are investigated. One concern is to formulate the aspects of local index theory for families of Dirac operator in terms of the odd signature operator, and place known results in a context which permits the treatment of Rho invariants. The main focus is, however, to use this to compute Rho invariants for explicit classes of fibered 3-manifolds. More precisely, we consider $\U(1)$-Rho invariants of principal $S^1$-bundles over closed, oriented surfaces as well as mapping tori with torus fiber. Hyperboli…

Gerbes, simplicial forms and invariants for families of foliated bundles

The notion of a gerbe with connection is conveniently reformulated in terms of the simplicial deRham complex. In particular the usual Chern-Weil and Chern-Simons theory is well adapted to this framework and rather easily gives rise to `characteristic gerbes' associated to families of bundles and connections. In turn this gives invariants for families of foliated bundles. A special case is the Quillen line bundle associated to families of flat SU(2)-bundles

Circle-invariant fat bundles and symplectic Fano 6-manifolds

We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4 and CP^2-bar are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Kliener's theorem that only S^4 and CP^2 admit circle-invariant metrics of positive sectional curvature.

Eta-invariants, Torsion forms and Flat vector bundles

We present a new proof, as well as a ${\bf C/Q}$ extension, of the Riemann-Roch-Grothendieck theorem of Bismut-Lott for flat vector bundles. The main techniques used are the computations of the adiabatic limits of $η$-invariants associated to the so-called sub-signature operators. We further show that the Bismut-Lott analytic torsion form can be derived naturally from the transgression of the $η$-forms appearing in the adiabatic limit computations.

Boundaries, eta invariant and the determinant bundle

Cobordism invariance shows that the index, in K-theory, of a family of pseudodifferential operators on the boundary of a fibration vanishes if the symbol family extends to be elliptic across the whole fibration. For Dirac operators with spectral boundary condition, Dai and Freed \cite{dai-freed1} gave an explicit version of this at the level of the determinant bundle. Their result, that the eta invariant of the interior family trivializes the determinant bundle of the boundary family, is extended here to the wider context of pseudodifferential families of cusp type.

Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence.

Separating Vector Bundle Sections by Invariant Means

We sharpen the construction of representation space in the paper "Principal Series Representations of Infinite Dimensional Lie Groups II: Construction of Induced Representations". We show that the principal series representation spaces constructed there, are completions of spaces of sections of Hilbert bundles rather than completions of quotient spaces of sections.