Canonical bundle
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An extension of a theorem of Wu–Yau Open
We show that a compact Kähler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjectur…
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On real bisectional curvature for Hermitian manifolds Open
Motivated by the recent work of Wu and Yau on the ampleness of a canonical line bundle for projective manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called real bisectional curvature for Hermit…
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A remark on our paper “Negative holomorphic curvature and positive canonical bundle” Open
This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to pro…
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On Gauduchon connections with Kähler-like curvature Open
We study Hermitian metrics with a Gauduchon connection being “Kähler-like”, namely, satisfying the same symmetries for curvature as the Levi–Civita and Chern connections. In particular, we investigate dimensional solvmanifolds with invaria…
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On a generalized canonical bundle formula for generically finite morphisms Open
We prove a canonical bundle formula for generically finite morphisms in the setting of generalized pairs (with -coefficients). This complements Filipazzi’s canonical bundle formula for morphisms with connected fibres. It is then applied to…
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Attractor invariants, brane tilings and crystals Open
Supersymmetric D-brane bound states on a Calabi-Yau threefold $X$ are counted by generalized Donaldsdon-Thomas invariants $Ω_Z(γ)$, depending on a Chern character (or electromagnetic charge) $γ\in H^*(X)$ and a stability condition (or cent…
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Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow Open
We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded…
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Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle Open
We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.
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A Weitzenböck formula for canonical metrics on four-manifolds Open
We first provide an alternative proof of the classical Weitzenböck formula for Einstein four-manifolds using Berger curvature decomposition, motivated by which we establish a unified framework for a Weitzenböck formula for a large class of…
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Singular spaces with trivial canonical class Open
The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected C…
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Towards a splitting of the $K(2)$-local string bordism spectrum Open
We show that $K(2)$-locally, the smash product of the string bordism spectrum and the spectrum $T_2$ splits into copies of Morava $E$-theories. Here, $T_2$ is related to the Thom spectrum of the canonical bundle over $ΩSU(4)$.
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An index obstruction to positive scalar curvature on fiber bundles over aspherical manifolds Open
We exhibit geometric situations where higher indices of the spinor Dirac operator on a spin manifold [math] are obstructions to positive scalar curvature on an ambient manifold [math] that contains [math] as a submanifold. In the main resu…
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Convergence of scalar curvature of Kahler-Ricci flow on manifolds of positive Kodaira dimension Open
In this paper, we consider Kahler-Ricci flow on n-dimensional Kahler manifold with semi-ample canonical line bundle and 0< m:= Kod(X)
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Boundedness of elliptic Calabi–Yau varieties with a rational section Open
We show that for each fixed dimension $d\geq 2$, the set of $d$-dimensional klt elliptic varieties with numerically trivial canonical bundle is bounded up to isomorphism in codimension one, provided that the torsion index of the canonical …
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Effective algebraic integration in bounded genus Open
We introduce and study birational invariants for foliations on projective surfaces built from the adjoint linear series of positive powers of the canonical bundle of the foliation. We apply the results in order to investigate the effective…
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Shokurov’s conjecture on conic bundles with canonical singularities Open
A conic bundle is a contraction $X\to Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov that predicts that if $X\to Z$ is a conic bundle such that X has canonical …
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Vafa-Witten invariants for projective surfaces I: stable case Open
On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a action with compact fixed …
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Positivity of holomorphic vector bundles in terms of $L^p$-conditions of $\bar\partial$ Open
We study the positivity properties of Hermitian (or even Finsler) holomorphic vector bundles in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic objects. To this end, we introduce four conditions, called the o…
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Rational Curves on Fibered Calabi-Yau Manifolds Open
We show that a smooth projective complex manifold of dimension greater than two endowed with an elliptic fiber space structure and with finite fundamental group always contains a rational curve, provided its canonical bundle is relatively …
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On the image of MRC fibrations of projective manifolds with\n semi-positive holomorphic sectional curvature Open
In this paper, we pose several conjectures on structures and images of\nmaximal rationally connected fibrations of smooth projective varieties\nadmitting semi-positive holomorphic sectional curvature. Toward these\nconjectures, we prove th…
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Quasiprojective manifolds with negative holomorphic sectional curvature Open
Let $(M,\\omega)$ be a compact K\\"ahler manifold with negative holomorphic\nsectional curvature. It was proved by Wu-Yau and Tosatti-Yang that $M$ is\nnecessarily projective and has ample canonical bundle. In this paper, we show\nthat any…
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Localized Donaldson-Thomas theory of surfaces Open
Let $S$ be a projective simply connected complex surface and $\mathcal{L}$ be a line bundle on $S$. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of $\mathcal{L}$. The moduli space admits…
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On real bisectional curvature and Kähler-Ricci flow Open
In their recent work, X. Yang and F. Zheng proved the positivity of a canonical line bundle for compact Hermitian manifolds with negative real bisectional curvature, a curvature condition they introduced that generalizes the holomorphic se…
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Existence of cscK metrics on smooth minimal models Open
Given a compact Kähler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature Kähler (cscK) metric. In this short note we show that there always exist cscK metrics on compact Kähler manifolds with nef canonical…
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Boundedness of elliptic Calabi-Yau varieties with a rational section Open
We show that for each fixed dimension $d\geq 2$, the set of $d$-dimensional klt elliptic varieties with numerically trivial canonical bundle is bounded up to isomorphism in codimension one, provided that the torsion index of the canonical …
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Complex Manifolds With Negative Curvature Operator Open
We prove that compact complex manifolds admitting metrics with negative Chern curvature operator either admit a $d d^c$-exact positive $(1,1)$ current or are Kähler with ample canonical bundle. In the case of complex surfaces we obtain a c…
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On the Effective Freeness of the Direct Images of Pluricanonical Bundles Open
We give effective bounds on the number of twists by ample line bundles, for global generations of pushforwards of log-pluricanonical bundles on klt pairs. This gives a partial answer to a conjecture proposed by Popa and Schnell. We prove t…
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Strong $(\delta,n)$-Complements for Semi-Stable Morphisms Open
We prove the boundedness of global strong $(\delta,n)$-complements for generalized $\epsilon$-log canonical pairs of Fano-type. We also prove some partial results towards boundedness of local strong $(\delta,n)$-complements for semi-stable…
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Theta functions on varieties with effective anti-canonical class Open
We show that a large class of maximally degenerating families of n-dimensional polarized varieties come with a canonical basis of sections of powers of the ample line bundle. The families considered are obtained by smoothing a reducible un…
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Canonical bundle formula and degenerating families of volume forms Open
Canonical bundle formula due to Kawamata and others has played fundamental roles in algebraic geometry. We show that the canonical bundle formula has analytic characterization in terms of fiberwise integration, which confirms a folklore co…