Fabio Podesta', Andrea Spiro
A simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold $G^C/P$ is provided in terms of symplectic data.
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For Fano homogeneous toric bundles, we obtain a formula of the greatest lower bound on Ricci curvature. We also give a criteria for the ampleness of a kind of line bundles over general homogeneous toric bundles.
It is proved that an homogeneous toric bundles over a flag manifold G^\C/P admits a Kaehler-Ricci solitonic metric if and only if it is Fano. In particular, an homogeneous toric bundle of this kind is Kaehler-Einstein if and only if it is Fano and its Futaki invariant vanishes identically.
We study a generalized Abreu Equation in $n$-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles.
In this paper, we study the generalized Abreu equation on a Delzant ploytope $Δ\subset \mathbb{R}^2$ and prove the existence of the constant scalar metrics of homogeneous toric bundles under the assumption of an appropriate stability.
In this note, we prove that on an $n$-dimensional compact toric manifold with positive first Chern class, the Kähler-Ricci flow with any initial $(S^1)^n$-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, we give another proof for the existence of Kähler-Ricci solitons on a compact toric manifold with positive first Chern class by using the Kähler-Ricci flow.
In this paper, we prove the Yau-Tian-Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles.
This is the first of a sequence of two papers. Here, a simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G^C/P is provided in terms of symplectic data. The result of this paper is used in the second paper, where it is proved that an homogeneous toric bundle over a flag manifold admits a Kaehler-Ricci solitonic metric if and only if it is Fano.
We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic spinors isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms between other types of harmonic spinors and Bott-Chern, Aeppli and twisted cohomology.
Starting from the data of a nonsingular complex projective toric variety, we define an associated notion of toric co-Higgs bundle. We provide a Lie-theoretic classification of these objects by studying the interaction between Klyachko's fan filtration and the fiber of the co-Higgs bundle at a closed point in the open orbit of the torus action. This can be interpreted, under certain conditions, as the construction of a coarse moduli scheme of toric co-Higgs bundles of any rank and with any total equivariant Chern class.
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.
Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.
Equivariant vector bundles over topological toric manifolds
该研究证明2n维拓扑环面流形上的拓扑/光滑T=(C*)^n等变向量丛均为arXiv:2504.02205定义的拓扑/光滑Klyachko向量丛，建立了等变向量丛与Klyachko向量丛的等价性。
In this paper, we give a simple proof of the Gauss-Bonnet-Chern theorem for a real oriented Finsler vector bundle with rank equal to the dimension of the base manifold. As an application, a Gauss-Bonnet-Chern formula for any metric-compatible connection is established on Finsler manifolds.
Klyachko vector bundles over topological toric manifolds
本文给出了在不变仿射图上等变平凡的拓扑/光滑/全纯 $(\mathbb{C}^{*})^n$-等变向量丛的 Klyachko 型分类，推广了 Klyachko 对环面向量丛的分类。
We find sufficient conditions for a principal toric bundle over compact Kähler manifolds to admit Calabi-Yau connections with torsion. With the aids of a topological classification, we construct such geometry on $n(S^2\times S^4)#(n+1)(S^3\times S^3)$
We prove a GAGA-style result for toric vector bundles with smooth base and give an algebraic construction of the Frölicher approximating vector bundle that has recently been introduced by Dan Popovici using analytic techniques.
A principal toric bundle $M$ is a complex manifold equipped with a free holomorphic action of a compact complex torus $T$. Such a manifold is fibered over $M/T$, with fiber $T$. We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety $X\subset M$ of a positive principal toric bundle, we show that either $X$ is $T$-invariant, or it lies in an orbit of $T$-action. For principal elliptic bundles, this theorem is known (math.AG/0403430). As follows from Borel-Remmert-Tits theorem, any compact simply connected homogeneous compl…
We determine the greatest lower bounds on the transverse Ricci curvature of compact toric Sasaki manifolds with positive basic first Chern class and with the first Chern class of the contact bundle being trivial. This is based on Wang-Zhu's and Futaki-Ono-Wang's works, and is an analogue of C. Li's work on toric Fano manifolds.
We study the contact equivalence problem for toric contact structures on $S^3$-bundles over $S^2$. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To find inequivalent toric contact structures that are contact equivalent, we show that the corresponding 3-tori belong to distinct conjugacy classes in the contactomorphism group. To show that two toric contact structures with the same first Chern class are contact inequivalent, we …
Assume that $X$ is a homogeneous toric bundle of the form $G^{\mathbb{C}}\times_{P,τ} F$ and is Fano, where $G$ is a compact semisimple Lie group with complexification $G^\mathbb{C}$, $P$ a parabolic subgroup of $G^\mathbb{C}$, $τ:P\rightarrow (T^m)^\mathbb{C}$ is a surjective homomorphism from $P$ to the algebraic torus $(T^m)^\mathbb{C}$, and $F$ is a compact toric manifold of complex dimension $m$. In this note we show that the normalized Kähler-Ricci flow on $X$ with a $G\times T^m$-invariant initial Kähler form in $c_1(X)$ converges, modulo the algebraic torus action, to a Kähler-Ricci so…
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 $f_1$ 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 {\it Sasaki-Ricci soliton}) on compact toric Sasaki manifolds whose basic first Chern form of the nor…
3-Manifolds admitting toric integrable geodesic flows
本文利用余球面丛在环面可积时具有的接触环面流形结构，通过比较Betti数，得到紧连通3维流形上测地流环面可积的必要条件。
Slope stability of tangent bundles of smooth toric Fano varieties
本文对三维至六维的8630个光滑环面Fano簇的切丛反典范斜率稳定性分类，引入根扭环面dP-纤维丛构造，明确稳定与凯勒-爱因斯坦性质的关系，完成相关分类与证明。
The mirror of a projective toric manifold $X_Σ$ is given by a Landau-Ginzburg model $(Y,W)$. We introduce a class of Lagrangian submanifolds in $(Y,W)$ and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over $X_Σ$. Through this geometric correspondence, we also identify the mirrors of Hermitian-Einstein metrics, which are given by distinguished Lagrangian sections whose potentials satisfy certain Laplace-type equations.
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 imm…
Toric Fano manifolds that do not admit extremal Kähler metrics
本文构造了维数为 10 的环面 Fano 流形，使其第一陈类中无极值 Kähler 度量，并通过乘积推广到所有 n≥11，回答了 Mabuchi 的问题。
We present two formulas for Chern classes of the tensor product of two vector bundles. In the first formula we consider a matrix containing Chern classes of the first bundle and we take a polynomial of this matrix with Chern classes of the second bundle as coefficients. The determinant of this expression equals the Chern polynomial of the tensor product. In the second formula we express the total Chern class of the tensor product as resultant of two explicit polynomials with coefficient involving Chern classes of each vector bundles. This approach leads to determinantal formulas for the total …
Toric vector bundles, non-abelianization, and spectral networks
本文基于镜像对称，通过源自拉格朗日多截面的谱网络与非阿贝尔化在完全环面曲面上构造环面向量丛，并证明秩2环面向量丛模空间具有$A$型$\mathcal{X}$-cluster结构。
We introduce the fibred toric varieties as equivariant $\mathbb{C}P^r$ bundles over lower dimensional toric varieties. An equivalent characterization is that the natural morphisms on them degenerate to bundle projections in the context of variation of toric varieties as GIT quotients. Our main observation is that these fibred toric varieties also arise naturally in the variation of hyperkähler varieties, namely, the fibred toric varieties are contained in the exceptional sets of the hyperkähler natural morphisms and the Mukai flops.
We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi-Yau threefolds.
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