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Qiang Tan

Publications and source records attributed to Qiang Tan.

15 recordsLinked to original sources
arXiv2026

Bochner-Yano Type Theorems for Conformal Killing Vector Fields under Curvature Pinching Conditions

In this article, we investigate conformal Killing vector fields on closed Riemannian manifolds under a curvature pinching condition. By establishing a new Bochner-type identity for the $1$-form dual to a conformal Killing vector field, we derive a sharp gradient estimate via a Moser iteration procedure. Based on this estimate, we prove that, under a suitable upper bound on the Ricci curvature, every nontrivial conformal Killing vector field must be nowhere vanishing. Consequently, on even-dimensional manifolds with non-zero Euler characteristic, every conformal Killing vector field vanishes identically, which in turn implies that the conformal transformation group of such a manifold is finite. Our results extend the classical rigidity theorems of Yano and Bochner from the setting of non-positive Ricci curvature to that of small positive Ricci curvature, and generalize the recent results of Chen and Han from Killing vector fields to conformal Killing vector fields.

math.DG↗
arXiv2026

The existence criterion of holomorphic discs for higher $A_\infty$ operations via minimal discs

The main theorem of the paper provides an existence criterion of holomorphic discs for higher $A_\infty$ operations. The key step is to show that if a minimal disc in a K\"ahler manifold with boundary in a sequence of Lagrangian submanifolds intersecting transversely such that its partial Maslov indices are either all no less than $1$ or all no larger than $-1$, then there is a holomorphic disc with the same image as this minimal disc. As a by-product, we show that all minimal discs in $\C\mathrm{P}^m$ with boundary on $\R\mathrm{P}^m$ are holomorphic.

math.SG↗
arXiv2025

Dolbeault-Morse-Novikov Cohomology on Complex manifolds and its applications

In this article, we investigate the topological properties of complex manifolds by studying Dolbeault-Morse-Novikov cohomology. By establishing an integral inequality, we obtain two main results: (1) When a closed complex manifold admits a nonzero parallel $(0,1)$-form, the Dolbeault-Morse-Novikov cohomology must be trivial, which implies that the Hirzebruch $\chi_{y}$-genus vanishes. (2) When a closed complex manifold admits a nowhere vanishing $(0,1)$-form, we establish a vanishing theorem for a certain class of twisted Dirac operators, which also forces the Hirzebruch $\chi_{y}$-genus to be zero. In particular, we prove that the Hirzebruch $\chi_{y}$-genus of a closed complex manifold vanishes if and only if the manifold admits a nowhere vanishing real vector field. These results generalize some classical theorems from Riemannian manifolds to the complex setting. As a culminating application, we prove that the Hirzebruch $\chi_{y}$-genus must vanish on closed Gauduchon manifolds admitting positive holomorphic scalar curvature.

math.DG↗
arXiv2025

On the $\mathcal{D}^+_J$ operator on higher-dimensional almost K\"{a}hler manifolds

In this paper, we introduce $\mathcal{D}^+_J$, a generalization of $\partial\bar{\partial}$ operator on higher dimensional almost K\"{a}hler manifolds. Using the $\mathcal{D}^+_J$ operator, we investigate the $\bar{\partial}$-problem in almost K\"{a}hler geometry and explore the generalized Monge-Amp\`{e}re equation on almost K\"{a}hler manifolds. We establish a uniqueness up to the addition of a constant and local existence theorem for this equation. At last, we find an elliptical system for $\mathcal{D}^+_J$ operator. As an application, we reorganize the result of Tosatti-Weinkove-Yau in \cite{TWY}.

math.DG↗
arXiv2023

The Calabi-Yau Equation on Symplectic Manifolds

By using the global deformation of almost complex structures which are compatible with a symplectic form off a Lebesgue measure zero subset, we construct a (measurable) Lipschitz Kahler metric such that the one-form type Calabi-Yau equation on an open dense submanifold is reduced to the complex Monge-Ampere equation with respect to the measurable Kahler metric. We give an existence theorem for solutions to the one-form type Calabi-Yau equation on closed symplectic manifolds.

math.DG↗
arXiv2023

$L^{2}$-Hodge theory on Complete Almost K\"{a}hler Manifolds and the Hopf Conjecture

In this article, we develop an $L^{2}$-Hodge theory on complete $2n$-dimensional almost K\"{a}hler manifolds $(X,\omega)$. In the first part, we establish several identities for various Laplacians, generalized Hodge and Serre dualities, a generalized Hard Lefschetz duality, and a Lefschetz decomposition, all restricted to the space $\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}$ of forms of pure bidegree. In the second part, as applications of these identities, we prove vanishing theorems for $L^{2}$-harmonic $(p,q)$-forms on $X$ under some growth assumptions on the K\"{a}her form $\omega$. We also provide refined $L^{2}$-estimates to sharpen the vanishing theorems in three specific settings. As a final application, the topology of compact almost K\"ahler manifolds with negative sectional curvature is studied. Under a smallness condition on the Nijenhuis tensor depending on the curvature, the authors prove that the Hirzebruch $\chi_{y}$-genus satisfies $(-1)^{n-p}\chi_{p}(X)\geq1$ for all $p=0,1,\cdots,n$, which in particular implies the Hopf conjecture for the Euler number $(-1)^{n}\chi(X)\geq n+1$. This extends a classical result of Gromov [J. Differential Geom., 1991] from the K\"ahler to the almost K\"ahler setting.

math.DG↗
arXiv2020

Remarks on some compact symplectic solvmanifolds

We study the hard Lefschetz property on compact symplectic solvmanifolds, i.e., compact quotients $M=\Gamma\backslash G$ of a simply-connected solvable Lie group $G$ by a lattice $\Gamma$, admitting a symplectic structure.

math.DG↗
arXiv2020

$L^{2}$-hard Lefschetz complete symplectic manifolds

For a complete symplectic manifold $M^{2n}$, we define the $L^{2}$-hard Lefschetz property on $M^{2n}$. We also prove that the complete symplectic manifold $M^{2n}$ satisfies $L^{2}$-hard Lefschetz property if and only if every class of $L^{2}$-harmonic forms contains a $L^{2}$ symplectic harmonic form. As an application, we get if $M^{2n}$ is a closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler characteristic satisfies the inequality $(-1)^{n}\chi(M^{2n})\geq0$.

math.DG↗
arXiv2019

Vanishing theorems on complete Riemannian manifold with a parallel $1$-form

In this article, we first consider the $L^{2}$ \textit{Morse-Novikov cohomology} on a complete Riemannian manifold $M$ equipped with a parallel $1$-form which includes Vaisman manifold. Based on a vanishing theorem of $L^{2}$ \textit{Morse-Novikov cohomology}, we prove that the $L^{2}$-harmonic forms on $M$ are identically zero.

math.DG↗
arXiv2017

On tamed almost complex four manifolds

This paper proves that on any tamed closed almost complex four-manifold $(M,J)$ whose dimension of $J$-anti-invariant cohomology is equal to the self-dual second Betti number minus one, there exists a new symplectic form compatible with the given almost complex structure $J$. In particular, if the self-dual second Betti number is one, we give an affirmative answer to a question of Donaldson for tamed closed almost complex four-manifolds. Our approach is along the lines used by Buchdahl to give a unified proof of the Kodaira conjecture.

math.DG↗
arXiv2016

Symplectic Parabolicity and L^2 Symplectic Harmonic Forms

In this paper, we study the symplectic cohomologies and symplectic harmonic forms which introduced by Tseng and Yau. Based on this, we get if $(M^{2n},\omega)$ is a compact symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler number satisfies the inequality $(-1)^n\chi(M)\geq 0$.

math.SG↗
arXiv2014

Symplectic Cohomology and the Stability of J-Anti-Invariant Cohomology

In this paper, we investigate the relationship between J-anti-invariant cohomology of a closed symplectic 4-manifold introduced by T.-J. Li and W. Zhang and new symplectic cohomologies introduced by L.-S. Tseng and S.-T. Yau. We also prove that the dimension of J-anti-invariant cohomology is constant for almost structures J which are compatible with a fixed symplectic form.

math.SG↗
arXiv2014

Primitive cohomology of degree 2 on compact symplectic manifolds

In this paper, we define the generalized Lejmi's $P_J$ operator on a compact almost Kähler $2n$-manifold. We get that $J$ is $C^\infty$-pure and full if $\dim\ker P_J=b^2-1$. Additionally, we investigate the relationship between $J$-anti-invariant cohomology introduced by T.-J. Li and W. Zhang and new symplectic cohomologies introduced by L.-S. Tseng and S.-T. Yau on a closed symplectic $4$-manifold.

math.SG↗
arXiv2013

On cohomology of almost complex 4-manifolds

Based on recent work of T. Draghici, T.-J. Li and W. Zhang, we further investigate properties of the dimension h_J of the J-anti-invariant cohomology subgroup H_J of a closed almost Hermitian 4-manifold (M, g, J, F) using metric compatible and symplectic 2-form compatible almost complex structures. We prove that h_J = 0 for generic almost complex structures J on M.

math.SG↗