Let $M$ be an oriented even-dimensional Riemannian manifold on which a discrete group $Γ$ of orientation-preserving isometries acts freely, so that the quotient $X=M/Γ$ is compact. We prove a vanishing theorem for a half-kernel of a $Γ$-invariant Dirac operator on a $Γ$-equivariant Clifford module over $M$, twisted by a sufficiently large power of a $Γ$-equivariant line bundle, whose curvature is non-degenerate at any point of $M$. This generalizes our previous vanishing theorems for Dirac operators on a compact manifold.
In particular, if $M$ is an almost complex manifold we prove a vanishing theorem for the half-kernel of a $\spin^c$ Dirac operator, twisted by a line bundle with curvature of a mixed sign. In this case we also relax the assumption of non-degeneracy of the curvature. When $M$ is a complex manifold our results imply analogues of Kodaira and Andreotti-Grauert vanishing theorems for covering manifolds.
As another application, we show that semiclassically the $\spin^c$ quantization of an almost complex covering manifold gives an "honest" Hilbert space. This generalizes a result of Borthwick and Uribe, who considered quantization of compact manifolds.
Application of our results to homogeneous manifolds of a real semisimple Lie group leads to new proofs of Griffiths-Schmidt and Atiyah-Schmidt vanishing theorems.
We establish cosection localization and vanishing results for virtual fundamental classes of derived manifolds, combining the theory of derived differential geometry by Joyce with the theory of cosection localization by Kiem-Li. As an application, we show that the stable pair invariants of hyperkähler fourfolds, defined by Cao-Maulik-Toda, are zero.
In this paper we will first show some Kollar-Enoki type injectivity theorems on compact Kahler manifolds, by using the Hodge theory, the Bochner- Kodaira-Nakano identity and the analytic method provided by O. Fujino and S. Matsumura in [15, 25, 36, 39]. We have some straightforward corollaries. In particular, we will show that our main injectivity theorem implies several Nadel type vanishing theorems on smooth projective manifolds. Second, by applying the transcendental method, especially the Demailly-Peternell-Schneider equisingular approximation theorem and the Hormander L2 estimates, we wil…
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci boundedcovering geometry.
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.
In this work, we (partially) generalize two classical tools in study of collapsed manifolds with bounded sectional curvature: a (singular) fibration theorem by Fukaya (1987) and Cheeger-Fukaya-Gromov (1992), and the stability for isometric compact Lie group actions on manifolds by Palais (1961) and Grove-Karcher (1973), to manifolds with local bounded Ricci covering geometry. Our two generalized results have been used in a recent work of Xiaochun Rong in generalizing Gromov's almost flat manifolds theorem to maximally collapsed manifolds with local bounded Ricci covering geometry.
For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant.
In this paper, we consider orthogonal Ricci curvature $Ric^{\perp}$ for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate their subtle relationship.
We show that if a bounded domain in complex Euclidean space with $\mathcal{C}^{1,1}$ boundary covers a compact manifold, then the domain is biholomorphic to the unit ball.
In this paper, we show several rigidity results for harmonic $(p,q)$-forms in complete Kähler manifolds. We also give several applications to study non-compact Kähler manifolds with parallel Bochner tensor or quaternion Kähler manifolds. Our results are natural extensions of Petersen and Wink's results in \cite{PW21, PW} in the setting of complete, non-compact Kähler manifolds.
We prove the convergence of the Bergman kernels and the $L^2$-Hodge numbers on a tower of Galois coverings $\{X_j\}$ of a compact Kähler manifold $X$ converging to an infinite Galois (not necessarily universal) covering $\widetilde{X}$. We also show that, as an application, sections of canonical line bundle $K_{X_j}$ for sufficiently large $j$ give rise to an immersion into some projective space, if so do sections of $K_{\widetilde{X}}$.
In this paper we consider an Einstein-type equation which generalizes important geometric equations, like static and critical point equations. We prove that a complete Einstein-type manifold with fourth-order divergence-free Weyl tensor and zero radial Weyl curvature is locally a warped product with $(n-1)$-dimensional Einstein fibers, provided that the potential function is proper. As a consequence, we prove a result about the nonexistence of multiple black holes in static spacetimes.
We prove a Bochner type vanishing theorem for compact complex manifolds $Y$ in Fujiki class $\mathcal C$, with vanishing first Chern class, that admit a cohomology class $[α] \in H^{1,1}(Y,\mathbb R)$ which is numerically effective (nef) and has positive self-intersection (meaning $\int_Y α^n \,>\, 0$, where $n\,=\,\dim_{\mathbb C} Y$). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold $Y$ are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group o…
Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle $\mathscr{O}_{T_M^*}(1)$ is not pseudo-effective and $\mathscr{O}_{T_N^*}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature …
In this paper we prove: if a bounded domain with $C^2$ boundary covers a manifold which has finite volume with respect to either the Bergman volume, the Kähler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is biholomorphic to the unit ball. This answers an old question of Yau. Further, when the domain is convex we can assume that the boundary only has $C^{1,ε}$ regularity.
We study the property of spectral-tightness of Riemannian manifolds, which means that the bottom of the spectrum of the Laplacian separates the universal covering space from any other normal covering space of a Riemannian manifold. We prove that spectral-tightness of a closed Riemannian manifold is a topological property characterized by its fundamental group. As an application, we show that a non-positively curved, closed Riemannian manifold is spectrally-tight if and only if the dimension of its Euclidean local de Rham factor is zero. In their general form, our results extend the state of th…
