Canonical Line Bundle Research Articles

On the zero set of the holomorphic sectional curvature

AbstractA notable example due to Heier, Lu, Wong, and Zheng shows that there exist compact complex Kähler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi‐definite and vanishes along high‐dimensional linear subspaces in every tangent space. The main result of this note is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real‐valued bihomogeneous polynomial of bidegree (2,2) on every tangent space, the proof is based on making a connection with the work of D'Angelo on complex subvarieties of real algebraic varieties and the decomposition of polynomials into differences of squares. Our bound involves an invariant that we call the holomorphic sectional curvature square decomposition length, and our arguments work as long as the holomorphic sectional curvature is semi‐definite, be it negative or positive.

The moduli space of stable supercurves and its canonical line bundle

Abstract: We prove that the moduli of stable supercurves with punctures is a smooth proper DM stack and study an analog of the Mumford's isomorphism for its canonical line bundle.

Non-degeneracy of Cohomological Traces for General Landau–Ginzburg Models

We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is a non-compact complex manifold with trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$, assuming only that the critical locus of $W$ is compact (but may not consist of isolated points). These results can be viewed as certain "deformed" versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of $W$, while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the $\mathbb{Z}_2$-graded category of topological D-branes of such models.

Inequalities of Chern classes on nonsingular projective $n$-folds with ample canonical or anti-canonical line bundles

Let $X$ be a nonsingular projective $n$-fold $(n \geq 2)$ which is either Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $\kappa$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, \dotsc , c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $\kappa$ is $0$, then the Chern ratios $\left( \dfrac{c_{2,1^{n-2}}}{c_{1^n}} , \dfrac{c_{2,2,1^{n-4}}}{c_{1^n}} , \dotsc , \frac{c_n}{c_{1^n}} \right)$ are contained in a convex polyhedron depending on the dimension of $X$ only. So we give an affirmative answer to a generalized open question, that whether the region described by the Chern ratios is bounded, posted by Hunt [Hun] to all dimensions. As a corollary, we can get that there exist constants $d_1$, $d_2$, $d_3$ and $d_4$ depending only on $n$ such that $d_1 K^n_X \leq \chi_\mathrm{top} (X) \leq d_2 K^n_X$ and $d_3 K^n_X \leq \chi (X, \mathcal{O}_X) \leq d_4 K^n_X$ . If the characteristic of $\kappa$ is positive, $K_X$ (or $-K_X$) is ample and $\mathcal{O}_X (K_X) (\mathcal{O}_X(-K_X) \textrm{, respectively})$ is globally generated, then the same results hold.

On Almost Nonpositive k-Ricci Curvature

Motivated by the recent work of Chu–Lee–Tam on the nefness of canonical line bundle for compact Kähler manifolds with nonpositive k-Ricci curvature, we consider a natural notion of almost nonpositive k-Ricci curvature, which is weaker than the existence of a Kähler metric with nonpositive k-Ricci curvature. When \(k=1\), this is just the almost nonpositive holomorphic sectional curvature introduced by Zhang. We firstly give a lower bound for the existence time of the twisted Kähler-Ricci flow when there exists a Kähler metric with k-Ricci curvature bounded from above by a positive constant. As an application, we prove that a compact Kähler manifold of almost nonpositive k-Ricci curvature must have nef canonical line bundle.

Kähler manifolds and mixed curvature

In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a “convex combination” of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni [Comm. Pure Appl. Math. 74 (2021), pp. 1100–1126] concerning manifolds with negative k k -Ricci curvature and generalize a result of Wu-Yau [Comm. Anal. Geom. 24 (2016), pp. 901–912] and Diverio-Trapani [J. Differential Geom. 111 (2019), pp. 303–314] to the conformally Kähler case. We also show that the compact Kähler manifold is projective and simply connected if the mixed curvature is positive.

Curvature estimates for the continuity method

We obtain curvature estimates for long-time solutions of the continuity method on compact Kähler manifolds with semi-ample canonical line bundles. In this setting, initiated in [G. La Nave and G. Tian, A continuity method to construct canonical metrics, Math. Ann. 365(3) (2016) 911–921; Y. A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math. 218(5) (2008) 1526–1565], we adapt arguments from [F. T.-H. Fong and Y. Zhang, Local curvature estimates of long-time solutions to the Kähler–Ricci flow, Adv. Math. 375 (2020) 107416] for the Kähler–Ricci flow to this setup. As an application, we derive curvature bounds for general metrics on product manifolds.

Sheaves on non-reduced curves in a projective surface

Sheaves on non-reduced curves can appear in moduli spaces of 1-dimensional semistable sheaves over a surface and moduli spaces of Higgs bundles as well. We estimate the dimension of the stack Mx (nC, χ) of pure sheaves supported at the non-reduced curve nC (n ≽ 2) with C an integral curve on X. We prove that the Hilbert-Chow morphism $${h_{L,\chi}}:{\cal M}_X^H\left({L,\chi} \right) \to \left| L \right|$$ sending each semistable 1-dimensional sheaf to its support has all its fibers of the same dimension for X Fano or with the trivial canonical line bundle and |L| contains integral curves.

On the positivity of the dimension of the global sections of adjoint bundle for quasi-polarized manifold with numerically trivial canonical bundle

Let $(X, L)$ denote a quasi-polarized manifold of dimension $n \geq 5$ defined over the field of complex numbers such that the canonical line bundle $K_{X}$ of $X$ is numerically equivalent to zero. In this paper, we consider the dimension of the global sections of $K_{X} + mL$ in this case, and we prove that $h^{0}(K_{X} + mL) > 0$ for every positive integer $m$ with $m \geq n - 3$. In particular, a Beltrametti–Sommese conjecture is true for quasi-polarized manifolds with numerically trivial canonical divisors.

On the Hölder Estimate of Kähler–Ricci Flow

AbstractIt is conjectured by Song–Tian that the normalized Kähler–Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle is uniformly smooth away from the singular fibers. In this work, we adapt the method in the work of Hein–Tosatti on collapsing Calabi–Yau metrics and obtain a uniform spatial Hölder estimate along the normalized Kähler–Ricci flow for all time away from the singular fiber.

Analytic approach to a generalization of Chern classes in supergeometry

Some cohomology elements, called [Formula: see text]-classes, as a supergeneralization of universal Chern classes, are introduced for canonical super line bundles over [Formula: see text]-projective spaces, a novel supergeometric generalization of projective spaces. It is shown that these classes may be described by analytic representatives of elements of generalized de Rham cohomology.

Flat conical Laplacian in the square of the canonical bundle and its regularized determinants

Let $X$ be a compact Riemann surface of genus $g\geq 2$ equipped with flat conical metric $|\Omega|$, where $\Omega$ be a holomorphic quadratic differential on $X$ with $4g-4$ simple zeroes. Let $K$ be the canonical line bundle on $X$. Introduce the Cauchy-Riemann operators $\bar \partial$ and $\partial$ acting on sections of holomorphic line bundles over $X$ ($K^2$ in the definition of $\Delta^{(2)}_{|\Omega|}$ below) and, respectively, anti-holomorphic line bundles ($\bar { K}^{-1}$ below). Consider the Laplace operator $\Delta^{(2)}_{|\Omega|}:=|\Omega| \partial |\Omega|^{-2}\bar\partial$ acting in the Hilbert space of square integrable sections of the bundle $K^2$ equipped with inner product $ _{K^2}=\int_X\frac {Q_1\bar Q_2}{|\Omega|}$. We discuss two natural definitions of the determinant of the operator $\Delta^{(2)}_{|\Omega|}$. The first one uses the zeta-function of some special self-adjoint extension of the operator (initially defined on smooth sections of $K^2$ vanishing near the zeroes of $\Omega$), the second one is an analog of Eskin-Kontsevich-Zorich (EKZ) regularization of the determinant of the conical Laplacian acting in the trivial bundle. In contrast to the situation of operators acting in the trivial bundle, for operators acting in $K^2$ these two regularizations turn out to be essentially different. Considering the regularized determinant of $\Delta^{(2)}_{|\Omega|}$ as a functional on the moduli space $Q_g(1, \dots, 1)$ of quadratic differentials with simple zeroes on compact Riemann surfaces of genus $g$, we derive explicit expressions for this functional for the both regularizations. The expression for the EKZ regularization is closely related to the well-known explicit expressions for the Mumford measure on the moduli space of compact Riemann surfaces of genus $g$.

Convergence of Narasimhan–Simha measures on degenerating families of Riemann surfaces

Given a compact Riemann surface $Y$ and a positive integer $m$, Narasimhan and Simha defined a measure on $Y$ associated to the $m$-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense of Amini and Baker. We also study the limit of the measure induced by the Hermitian pairing defined by the Narasimhan-Simha measure. For $m = 1$, both these measures coincide with the Bergman measure on $Y$. We also extend the definition of the Narasimhan-Simha measure to the singular curves on the boundary of $\overline{\mathcal{M}_g}$ in such a way that these measures form a continuous family of measures on the universal curve over $\overline{\mathcal{M}_g}$.

Higher-order estimates of long-time solutions to the Kähler-Ricci flow

In this article, we study the higher-order regularity of the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. By developing sharp new parabolic Schauder estimates on cylinders, we proved that when the generic fibers of the Iitaka fibration are biholomorphic to each other, the flow converges in Cloc∞-topology away from singular fibers to a negative Kähler-Einstein metric on the base manifold. In particular, we proved that the Ricci curvature of the flow is uniformly bounded on any compact subsets away from singular fibers when the generic fibers are biholomorphic to each other.

Extension of the curvature form of the relative canonical line bundle on families of Calabi–Yau manifolds and applications

Given a proper, open, holomorphic map of Kahler manifolds, whose general fibers are Calabi-Yau manifolds, the volume forms for the Ricci-flat metrics induce a hermitian metric on the relative canonical bundle over the regular locus of the family. We show that the curvature form extends as a closed positive current. Consequently the Weil-Petersson metric extends as a positive current. In the projective case, the Weil-Petersson form is known to be the curvature of a certain determinant line bundle, equipped with a Quillen metric. As an application we get that after blowing up the singular locus, the determinant line bundle extends, and the Quillen metric extends as singular hermitian metric, whose curvature is a positive current.

On the Negativity of Moduli Spaces for Polarized Manifolds

Given a log base space (Y, S), parameterizing a smooth family of complex projective varieties with semi-ample canonical line bundle, we briefly recall the construction of the deformation Higgs sheaf and the comparison map on (Y, S) made in the work by Viehweg–Zuo. While almost all hyperbolicities in the sense of complex analysis such as Brody, Kobayashi, big Picard and Viehweg hyperbolicities of the base U = Y ∖ S (under some technical assumptions) follow from the negativity of the kernel of the deformation Higgs bundle we pose a conjecture on the topological hyperbolicity on U. In order to study the rigidity problem we then introduce the notions of the length and characteristic varieties of a family f : X → Y, which provide an infinitesimal characterization of products of sub log pairs in (Y, S) and an upper bound for the number of subvarieties appearing as factors in such a product. We formulate a conjecture on a characterization of non-rigid families of canonically polarized varieties.

An application of a C2-estimate for a complex Monge–Ampère equation

By studying a complex Monge–Ampère equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact Kähler manifold [Formula: see text] with [Formula: see text] for some integer [Formula: see text] with [Formula: see text], and the ampleness of the canonical line bundle [Formula: see text].

Irregular Eguchi–Hanson type metrics and their soliton analogues

We verify the extension to the zero section of momentum construction of Kaehler-Einstein metrics and Kaehler-Ricci solitons on the total space Y of positive rational powers of the canonical line bundle of toric Fano manifolds with possibly irregular Sasaki-Einstein metrics. More precisely, we show that the extended metric along the zero section has an expression which can be extended to Y, restricts to the associated unit circle bundle as a transversely Kaehler-Einstein (Sasakian eta-Einstein) metric scaled in the Reeb flow direction, and that there is a Riemannian submersion from the scaled Sasakian eta-Einstein metric to the induced metric of the zero section.

Holomorphic sectional curvature, nefness and Miyaoka–Yau type inequality

On a compact Kahler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kahler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kahler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle, contains no rational curves and satisfies some Miyaoka-Yau type inequalities. In the course of the discussions, we attach a real value to any fixed Kahler class which, up to a constant factor depending only on the dimension of manifold, turns out to be an upper bound for the nef threshold.

Local curvature estimates of long-time solutions to the Kähler-Ricci flow

We study the local curvature estimates of long-time solutions to the normalized Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at time-infinity is independent of the choice of the initial Kähler metric. Moreover, when a regular fiber of the semi-ample fibration is not a finite quotient of a torus, we determine the exact curvature blow-up rate of the Kähler-Ricci flow near the regular fiber.