Bogomolov-Miyaoka-Yau Inequality Research Articles

Miyaoka-Yau type inequalities of complete intersection threefolds in products of projective

Geography of projective varieties is one of the fundamental problems in algebraic geometry. There are many researches toward the characteristics of Chern number of some projective spaces, for example Noethers inequalities, the theorem of Chang-Lopez, and the Miyaoka-Yau inequality. In this paper, we compute the Chern numbers of any smooth complete intersection threefold in the product of projective spaces via the standard exact sequences of cotangent bundles. Then we obtain linear Chern number inequalities for (c_1 (X)c_2 (X))/(c_1^3 (X)) and (c_3 (X))/(c_1^3 (X)) on such threefolds under conditions of d_ij4 and d_ij6 respectively. They can be considered as a generalization of the Miyaoka-Yau inequality and an improvement of Yaus inequality for such threefolds.

The Miyaoka–Yau inequality on smooth minimal models

AbstractIn this short note, we offer an observation that the Miyaoka–Yau inequality holds for any compact Kähler manifold with nef canonical bundle, that is, a smooth minimal model. It follows directly from the existence of cscK metrics in a neighborhood of the canonical class which was confirmed both by the work of Dyrefelt and Song using different approaches.

An Explicit Bound for the Log-Canonical Degree of Curves on Open Surfaces

Let $X$, $D$ be a smooth projective surface and a simple normal crossing divisor on $X$, respectively. Suppose $\kappa (X, K\_X + D)\ge 0$, let $C$ be an irreducible curve on $X$ whose support is not contained in $D$ and $\alpha$ a rational number in $\[ 0, 1 ]$. Following Miyaoka, we define an orbibundle $\mathcal{E}\alpha$ as a suitable free subsheaf of log differentials on a Galois cover of $X$. Making use of $\mathcal{E}\alpha$ we prove a Bogomolov–Miyaoka–Yau inequality for the couple $(X, D+\alpha C)$. Suppose moreover that $K\_X+D$ is big and nef and $(K\_X+D)^2$ is greater than $e\_{X\setminus D}$, namely the topological Euler number of the open surface $X\setminus D$. As a consequence of the inequality, by varying $\alpha$, we deduce a bound for $(K\_X+D)\cdot C$ by an explicit function of the invariants: $(K\_X+D)^2$, $e\_{X\setminus D}$ and $e\_{C \setminus D}$ , namely the topological Euler number of the normalization of $C$ minus the points in the set-theoretic counterimage of $D$. We finally deduce that on such surfaces curves, with $- e\_{C\setminus D}$ bounded, form a bounded family, in particular there are only a finite number of curves $C$ on $X$ such that $- e\_{C\setminus D}\le 0$.

Orbifold stability and Miyaoka–Yau inequality for minimal pairs

After establishing suitable notions of stability and Chern classes for singular pairs, we use K\"ahler-Einstein metrics with conical and cuspidal singularities to prove the slope semistability of orbifold tangent sheaves of minimal log-canonical pairs of log general type. We then proceed to prove the Miyaoka-Yau inequality for all minimal pairs with standard coefficients. Our result in particular provides an alternative proof of the Abundance theorem for threefolds that is independent of positivity results for tangent sheaves.

Equality in the Bogomolov–Miyaoka–Yau inequality in the non-general type case

Abstract We classify all minimal models X of dimension n, Kodaira dimension n - 1 {n-1} and with vanishing Chern number c 1 n - 2 ⁢ c 2 ⁢ ( X ) = 0 {c_{1}^{n-2}c_{2}(X)=0} . This solves a problem of Kollár. Completing previous work of Kollár and Grassi, we also show that there is a universal constant ϵ > 0 {\epsilon>0} such that any minimal threefold satisfies either c 1 ⁢ c 2 = 0 {c_{1}c_{2}=0} or - c 1 ⁢ c 2 > ϵ {-c_{1}c_{2}>\epsilon} . This settles completely a conjecture of Kollár.

Miyaoka–Yau Inequality for Compact Kähler Manifolds with Semi-positive Canonical Bundles

In this paper, we prove the Miyaoka–Yau inequality for compact Kahler manifolds with semi-positive canonical bundles. The key point of the proof is the estimate for the $$L^2$$ -norm of the scalar curvature along the Kahler–Ricci flow.

On generalized Kummer surfaces and the orbifold Bogomolov-Miyaoka-Yau inequality

A generalized Kummer surface X = Km(T, G) is the resolution of a quotient of a torus T by a finite group of symplectic automorphisms G. We complete the classification of generalized Kummer surfaces by studying the two last groups which have not been yet studied. For these surfaces we compute the associated Kummer lattice K G , which is the minimal primitive sub-lattice containing the exceptional curves of the resolution X → T /G. We then prove that a K3 surface is a generalized Kummer surface of type Km(T, G) if and only if its Neron-Severi group contains K G. For smooth-orbifold surfaces X of Kodaira dimension ≥ 0, Kobayashi proved the orbifold Bogomolov Miyaoka Yau inequality c 2 1 (X) ≤ 3c 2 (X). For Kodaira dimension 2, the case of equality is characterized as X being uniformized by the complex 2-ball B 2. For smooth-orbifold K3 and Enriques surfaces we characterize the case of equality as being uniformized by C 2 .

The Miyaoka-Yau inequality and uniformisation of canonical models

We establish the Miyaoka-Yau inequality in terms of orbifold Chern classes for the tangent sheaf of any complex projective variety of general type with klt singularities and nef canonical divisor. In case equality is attained for a variety with at worst terminal singularities, we prove that the associated canonical model is the quotient of the unit ball by a discrete group action.

Xiao's conjecture on canonically fibered surfaces

A canonically fibered surface is a surface whose canonical series maps it to a curve. Using Miyaoka–Yau inequality, A. Beauville proved that a canonically fibered surface has relative genus at most 5 when its geometric genus is sufficiently large. G. Xiao further conjectured that the relative genus cannot exceed 4. We give a proof of this conjecture.

The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic

We generalize Bogomolov’s inequality for Higgs sheaves and the Bogomolov– Miyaoka–Yau inequality in positive characteristic to the logarithmic case. We also generalize Shepherd-Barron’s results on Bogomolov’s inequality on surfaces of special type from rank 2 to the higher-rank case. We use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo p2. These examples contradict some claims by Xie.

Bogomolov’s inequality for Higgs sheaves in positive characteristic

We prove Bogomolov’s inequality for Higgs sheaves on varieties in positive characteristic $$p$$ that can be lifted modulo $$p^2$$ . This implies the Miyaoka–Yau inequality on surfaces of non-negative Kodaira dimension liftable modulo $$p^2$$ . This result is a strong version of Shepherd-Barron’s conjecture. Our inequality also gives the first algebraic proof of Bogomolov’s inequality for Higgs sheaves in characteristic zero, solving the problem posed by Narasimhan.

Arrangements of rational sections over curves and the varieties they define

We introduce arrangements of rational sections over curves. They generalize line arrangements on $\mathbb P^2$. Each arrangement of $d$ sections defines a single curve in $\mathbb P^{d-2}$ through the Kapranov's construction of $\overline{M}{0,d+1}$. We show a one-to-one correspondence between arrangements of $d$ sections and irreducible curves in $M{0,d+1}$, giving also correspondences for two distinguished subclasses: transversal and simple crossing. Then, we associate to each arrangement $\mathcal A$ (and so to each irreducible curve in $M\_{0,d+1}$) several families of nonsingular projective surfaces $X$ of general type with Chern numbers asymptotically proportional to various log Chern numbers defined by $\mathcal A$. For example, for the main families and over $\mathbb C$, any such $X$ is of positive index and $\pi\_1(X) \simeq \pi\_1(\overline{A})$, where $\overline{A}$ is the normalization of $A$. In this way, any rational curve in $M\_{0,d+1}$ produces simply connected surfaces with Chern numbers ratio bigger than $2$. Inequalities like these come from log Chern inequalities, which are in general connected to geometric height inequalities (see Appendix). Along the way, we show examples of étale simply connected surfaces of general type in any characteristic violating any sort of Miyaoka-Yau inequality.

Miyaoka-Yau inequality for minimal projective manifolds of general type

In this short paper, we prove the Miyaoka-Yau inequality for minimal projective n-manifolds of general type by using Kahler-Ricci flow.

Families of $K3$ surfaces over curves reaching the Arakelov-Yau type upper bounds and Modularity

Let f : X → C be a family of semi-stable curves of genus g over a smooth projective C of genus q, and S ⊂ C the degeneration locus of f. The so-called Arakelov inequality states that deg f∗ωX/C ≤ g2 deg Ω 1 C(log S) = g 2 (2q − 2 + #S). When g ≥ 2 and #S = 0, the Miyaoka-Yau inequality for surfaces implies a much stronger inequality deg f∗ωX/C ≤ g − 1 6 deg Ω 1 C . In general, Tan [28] proved that the Arakelov inequality for a family f : X → C of semi-stable curves of genus ≥ 2 holds strictly. If g = 1, then deg f∗ωX/C can reach the upper bound in the inequality. Beauville has classified such families over C = P with #S = 4. More precisely, there are exactly 6 non-isotrivial families of semi-stable elliptic curves over P with 4 singular fibres. All of them are modular families of elliptic curves [2]. In this paper, we will consider the similar question for families of higher dimensional varieties. The Arakelov inequality is a special case of some more general inequalities for Hodge bundles. To state them, let V denote a polarized real variation of Hodge structure on a smooth projective curve C S such that the local monodromies around S are all unipotent, let (⊕p+q=kE, θ) denote the corresponding Hodge bundles. In [8] the following Arakelov-Yau type inequality was proven (also see [18] for a similar inequality): If k = 2l + 1, then

THE BOGOMOLOV–MIYAOKA–YAU INEQUALITY FOR LOG CANONICAL SURFACES

Let X be a smooth projective surface of non-negative Kodaira dimension. Bogomolov [1, Theorem 5] proved that c21 [les ] 4c2. This was improved to c21 [les ] 3c2 by Miyaoka [12, Theorem 4] and Yau [19, Theorem 4]. Equality c21 [les ] 3c2 is attained, for example, if the universal cover of X is a ball (if κ(X) = 2 then this is the only possibility). Further generalizations of inequalities for Chern classes for some singular surfaces with (fractional) boundary were obtained by Sakai [16, Theorem 7.6], Miyaoka [13, Theorem 1.1], Kobayashi [6, Theorem 2; 7, Theorem 12], Wahl [18, Main Theorem] and Megyesi [10, Theorem 10.14; 11, Theorem 0.1].In [8] we introduced Chern classes of reflexive sheaves, using Wahl's local Chern classes of vector bundles on resolutions of surface singularities. Here we apply them to obtain the following generalization of the Bogomolov–Miyaoka–Yau inequality.

Configurations of conics with many tacnodes

We investigate configurations of conics in the projective plane which have the property that the number of tacnodes is equal or close to the upper bound obtained from the Miyaoka-Yau inequality. We show that for 5 conics there are exactly 3 configurations, including 2 new ones, achieving the maximum 17 tacnodes, and for 6 conics the maximum number of tacnodes is 22, which together with previous results implies that the Miyaoka-Yau bound can never be achieved.

On Veys' conjecture

We show that W. Veys' conjecture which was proved by A.J. de Jong and J.H.M. Steenbrink easily follows from S. litaka's virtual singularity theorem and the log Miyaoka-Yau inequality.

Canonical bundles of Deligne-Lusztig varieties

In this paper we consider Deligne–Lusztig varieties. We explicitly describe the canonical bundles of their smooth compactifications in terms of homogeneous line bundles pulled back from G/B. Using this description we show that the members (one member in each dimension) of a special family of Deligne–Lusztig varieties have ample canonical bundles. A consequence is that, unlike the closely related Schubert varieties, Deligne–Lusztig varieties are not in general Frobenius split. Several examples are given. Among these we exhibit (Example 4) an infinite family of counter-examples to the Miyaoka–Yau inequality (one for each prime power).

Bounding codimension-one subvarieties and a general inequality between Chern numbers

We extend the Miyaoka-Yau inequality for a surface to an arbitrary nonuniruled normal complex projective variety, eliminating the hypothesis that the variety must be minimal. The inequality is sharp in dimension three and is also sharp among minimal varieties. For nonminimal varieties in dimension four or higher, an error term is picked up which can be controlled. As a consequence, we bound codimension one subvarieties in a variety of general type linearly in terms of their Chern numbers. In particular, we show that there are only a finite number of smooth Fano, Abelian and Calabi-Yau subvarieties of codimension one in any variety of general type.

Einstein metrics and Mostow rigidity

Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness theorem is proved for Einstein metrics on compact quotients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian version of the Miyaoka-Yau inequality.