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Kefeng Liu

Publications and source records attributed to Kefeng Liu.

At least 73 records · Page 4Linked to original sources
arXiv2014

Quasi-isometry and deformations of Calabi-Yau manifolds

We prove several formulas related to Hodge theory and the Kodaira-Spencer-Kuranishi deformation theory of Kähler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi-Yau manifolds and also a construction of global canonical family of holomorphic $(n,0)$-forms on the deformation spaces of Calabi-Yau manifolds. Similar constructions are also applied to the deformation spaces of compact Kähler manifolds.

math.DG↗
arXiv2013

Curvatures of direct image sheaves of vector bundles and applications I

Let $p:\sXS$ be a proper Kähler fibration and $\sE\sX$ a Hermitian holomorphic vector bundle. As motivated by the work of Berndtsson(\cite{Berndtsson09a}), by using basic Hodge theory, we derive several general curvature formulas for the direct image $p_*(K_{\sX/S}\ts \sE)$ for general Hermitian holomorphic vector bundle $\sE$ in a simple way. A straightforward application is that, if the family $\sXS$ is infinitesimally trivial and Hermitian vector bundle $\sE$ is Nakano-negative along the base $S$, then the direct image $p_*(K_{\sX/S}\ts \sE)$ is Nakano-negative. We also use these curvature formulas to study the moduli space of projectively flat vector bundles with positive first Chern classes and obtain that, if the Chern curvature of direct image $p_*(K_{X}\ts E)$--of a positive projectively flat family $(E,h(t))_{t\in \mathbb D}X$--vanishes, then the curvature forms of this family are connected by holomorphic automorphisms of the pair $(X,E)$.

math.CV↗
arXiv2013

New structure for orthogonal quantum group invariants

Based on the orthogonal Labastida-Mari{ñ}o-Ooguri-Vafa conjecture made by L. Chen & Q. Chen [5], we derive an infinite product formula for Chern-Simons partition functions, which generalizes the Liu-Peng's [19] recent results to the orthogonal case. Symmetry property of this new infinite product structure is also discussed.

math.QA↗
arXiv2013

$E_8$ Bundles and Rigidity

In this paper, we establish rigidity and vanishing theorems for Dirac operators twisted by $E_8$ bundles.

math.DG↗
arXiv2013

Boltje-Maisch resolutions of Specht modules

In \cite{21}, Boltje and Maisch found a permutation complex of Specht modules in representation theory of Hecke algebras, which is the same as the Boltje-Hartmann complex appeared in the representation theory of symmetric groups and general linear groups. In this paper we prove the exactness of Boltje-Maisch complex in the dominant weight case.

math.RT↗
arXiv2013

A Basis of the $q$-Schur Module

In this paper, we construct the $q$-Schur modules as left principle ideals of the cyclotomic $q$-Schur algebras, and prove that they are isomorphic to those cell modules defined in \cite{8} and \cite{15} at any level $r$. Then we prove that these $q$-Schur modules are free modules and construct their bases. This result gives us new versions of several results about the standard basis and the branching theorem. With the help of such realizations and the new bases, we re-prove the Branch rule of Weyl modules which was first discovered and proved by Wada in \cite{23}.

math.RT↗
arXiv2013

New proofs of the Torelli theorems for Riemann surfaces

In this paper, by using the Kuranishi coordinates on the Teichmüller space and the explicit deformation formula of holomorphic one-forms on Riemann surface, we give an explicit expression of the period map and derive new differential geometric proofs of the Torelli theorems, both local and global, for Riemann surfaces.

math.DG↗
arXiv2012

Anomaly Cancellation and Modularity. II: $E_8\times E_8$ case

In this paper we show that both of the Green-Schwarz anomaly factorization formula for the gauge group $E_8\times E_8$ and the Hořava-Witten anomaly factorization formula for the gauge group $E_8$ can be derived through modular forms of weight 14. This answers a question of J. H. Schwarz. We also establish generalizations of these factorization formulas and obtain a new Hořava-Witten type factorization formula.

hep-th↗
arXiv2012

Anomaly Cancellation and Modularity

It has been shown that the Alvarez-Gaum$\mathrm{\acute{e}}$-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula in type I superstring theory can also be derived from the modularity of those characteristic forms and thus unify the Alvarez-Gaum$\mathrm{\acute{e}}$-Witten formula, the Green-Schwarz formula as well as the Schwarz-Witten formula in the same framework. Various generalizations of these remarkable formulas are also established.

math.DG↗
arXiv2012

Deforming submanifolds of arbitrary codimension in a sphere

In this paper, we prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere $\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in $\mathbb{S}^{n+d}$.

math.DG↗
arXiv2012

Mean curvature flow of higher codimension in Riemannian manifolds

We investigate the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry. We prove that if the initial submanifold satisfies a pinching condition, then along the mean curvature flow the submanifold contracts smoothly to a round point in finite time. As a consequence we obtain a differentiable sphere theorem for submanifolds in a Riemannian manifold.

math.DG↗
arXiv2011

Formal pseudodifferential operators and Witten's r-spin numbers

We derive an effective recursion for Witten's r-spin intersection numbers, using Witten's conjecture relating r-spin numbers to the Gel'fand-Dikii hierarchy (Theorem 4.1). Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions: Propositions 5.2 and 5.4, Corollary 5.5). We use these closed-form descriptions to prove Harer-Zagier's formula for the Euler characteristic of M_{g,1}. Finally in Section 6, we extend Witten's series expansion formula for the Landau-Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brezin and Hikami.

math.AG↗
arXiv2011

On an algebraic formula and applications to group action on manifolds

We consider a purely algebraic result. Then given a circle or cyclic group of prime order action on a manifold, we will use it to estimate the lower bound of the number of fixed points. We also give an obstruction to the existence of $\mathbb{Z}_p$ action on manifolds with isolated fixed points when $p$ is a prime.

math.AT↗
arXiv2011

Modular Forms and Generalized Anomaly Cancellation Formulas

In this paper, we generalize the anomaly cancellation formulas in \cite{AW, Liu1, HZ2} to the cases that an auxiliary bundle $W$ as well as a complex line bundle $ξ$ are involved with no conditions on the first Pontryagin forms being assumed.

math.DG↗
arXiv2011

A remark on Mirzakhani's asymptotic formulae

In this note, we answer a question of Mirzakhani on asymptotic behavior of the one-point volume polynomial of moduli spaces of curves. We also present some applications of Mirzakhani's asymptotic formulae of Weil-Petersson volumes.

math.AG↗