Fabrizio Catanese, Meng Chen, De-Qi Zhang
Let X be a smooth projective minimal 3-fold of general type. We prove the sharp inequality K^3_X >= (2 /3)(2p_g(X) - 5), an analogue of the classical Noether inequality for algebraic surfaces of general type
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We prove the Conjecture of Catenese--Chen--Zhang: the inequality $K_X^3\geq \frac{4}{3}p_g(X)-\frac{10}{3}$ holds for all projective Gorenstein minimal 3-folds $X$ of general type.
If $X$ is a smooth complex projective 3-fold with ample canonical divisor $K$, then the inequality $K^3\ge {2/3}(2p_g-7)$ holds, where $p_g$ denotes the geometric genus. This inequality is nearly sharp. We also give similar, but more complicated, inequalities for general minimal 3-folds of general type. (A noether type of inequality lies, by all means, on the other side of the Miyaoka-Yau inequality from the geographical point of view.)
1) We give a 3-dimensional analogue of M. Noether's inequality for canonically polarized threefolds: $K^3\ge 2(2p_g-5)/3$. This inequality is sharp by known examples of M. Kobayashi. 2) Given a minimal 3-fold $X$ of general type with canonical singularities, if $K^3< (3p_g-5)/2$, we show that $X$ must be birationally fibred by curves of genus 2.
Let $X$ be a Gorenstein minimal $3$-fold of general type. We prove the optimal inequality: $$K_X^{3}\geq \frac{4}{3}χ(ω_X)-2,$$ where $χ(ω_X)$ is the Euler-Poincar$\acute{\text{e}}$ characteristic of the dualizing sheaf $ø_{X}$.
Moduli spaces of stable bundles on ruled 3-folds and Brill-Noether problems
本文重新定义墙与室理论，用于研究高维代数簇上秩2稳定向量丛的模空间，并应用于规则三维流形，证明某些Brill-Noether轨迹非空。
We prove effective upper bounds on the global sections of nef line bundles of small generic degree over a fibered surface over a field of any characteristic. It can be viewed as a relative version of the classical Noether inequality for surfaces. As a consequence, we give a new proof of the slope inequality for fibered surface without using any stability method. The treatment is essentially different from these of Xiao, Cornalba--Harris and Moriwaki. We also study the geography problem of surfaces in positive characteristics and show that the Severi inequality is true for surfaces of general t…
We study canonically polarized Gorenstein $3$-folds with at most terminal singularities and satisfying $K_X^3=\frac 43p_g(X)-\frac {10}3$ and $p_g(X) \ge 7$. We characterize the canonical maps of such $3$-folds, describe a structure theorem for the locally factorial ones and completely classify the smooth ones. New examples of canonically polarized smooth $3$-folds with $K_X^3=\frac 43p_g(X)-\frac {10}3$ and $p_g(X) \ge 7$ are constructed. These examples are natural extensions of those constructed by M.~Kobayashi.
A Brill-Noether locus is a subscheme of the moduli of bundles E over a curve C defined by requiring E to have a given number of sections, or homomorphisms from another bundle. There are a number of different types, that can be treated by determinantal methods, with symmetry or skewsymmetry arising from Serre duality. They have many beautiful applications to curves, K3 surfaces and Fano 3-folds.
In this short note, we give a refinement of our previous work (arXiv:1803.05553) stating that for a projective $3$-fold $X$ of general type with either $p_g(X)\leq 4$ or $p_g(X)\geq 11$, $$\text{vol}(X)\geq \frac{4}{3}p_g(X)-{\frac{10}{3}}.$$
We prove the optimal Noether-Severi inequality that $\mathrm{vol}(X) \ge \frac{4}{3} χ(ω_{X})$ for all smooth and irregular $3$-folds $X$ of general type over $\mathbb{C}$. For those $3$-folds $X$ attaining the equality, we completely describe their canonical models and show that the topological fundamental group $π_1(X) \simeq \mathbb{Z}^2$. As a corollary, we obtain for the same $X$ another optimal inequality that $\mathrm{vol}(X) \ge \frac{4}{3}h^0_a(X, K_X)$ where $h^0_a(X, K_X)$ stands for the continuous rank of $K_X$, and we show that $X$ attains this equality if and only if $\mathrm{vol…
Noether inequality for irregular threefolds of general type
本文研究一般型光滑不规则三维簇的最优 Noether 不等式，证明在几何亏格至少为16或存在 Gorenstein 极小模型时体积下界成立，并刻画等号情形的典范模型。
We establish the Noether inequality for projective $3$-folds. More precisely, we prove that the inequality $${\rm vol}(X)\geq \tfrac{4}{3}p_g(X)-{\tfrac{10}{3}}$$ holds for all projective $3$-folds $X$ of general type with either $p_g(X)\leq 4$ or $p_g(X)\geq 21$, where $p_g(X)$ is the geometric genus and ${\rm vol}(X)$ is the canonical volume. This inequality is optimal due to known examples found by M. Kobayashi in 1992.
Sharp Noether-type inequalities for $3$-folds of intermediate Kodaira dimensions
本文针对中间小平维数的光滑射影三维流形，建立了尖锐的 Noether 型不等式，给出了伊塔卡体积与几何亏格的下界关系，并刻画了等号成立的条件。
Stability conditions on Calabi-Yau threefolds via Brill-Noether theory of curves
通过Brill-Noether理论研究Calabi-Yau三重曲面的稳定性条件，证明了相关不等式并确保稳定性条件的存在。
The Noether inequality for threefolds and three moduli spaces with minimal volumes
本文建立了几何亏格5≤p_g≤10的一般型射影三维簇的诺特不等式，解决三维簇诺特不等式所有剩余情形，并证明三个特定极小体积典范三维簇的粗模空间是不可约单有理簇且成员单连通。
Asymptotic Brill-Noether Existence at the Half-Canonical Degree: Energy Pairing, Cheeger Inequality and Covering Radii
利用数论几何技术研究图上布里尔 - 诺特存在性猜想渐近版本，在半典范度及附近证实多个图族的情况，关键工具是切赫型不等式，还应用于动力系统相关图，最后给出处理一般情况的建议。
The conjectural Bogomolov-Gieseker (BG) type inequality for tilt semistable objects on projective 3-folds was proposed by Bayer, Macri and the author. In this note, we prove our conjecture for slope stable sheaves with the smallest first Chern class on certain Calabi-Yau 3-folds, e.g. quintic 3-folds.
In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3-folds of general type. We generalize the classical Noether-Castelnuovo inequalities for irregular surfaces to irregular 3-folds according to the Albanese dimension.
We prove a Bogomolov-Gieseker type inequality for the third Chern characters of stable sheaves on Calabi-Yau 3-folds and a large class of Fano 3-folds with given rank and first and second Chern classes. The proof uses the spreading-out technique, vanishings from the tilt-stability conditions, and Langer's estimation theorem of the global sections of torsion free sheaves. In particular, the result implies that the conjectural sufficient conditions on the Chern numbers for the existence of stable sheaves on a Calabi-Yau 3-fold by Douglas-Reinbacher-Yau needs to be modified.
We consider the Noether-Lefschetz problem for surfaces in Q-factorial normal 3-folds with rational singularities. We show the existence of components of the Noether-Lefschetz locus of maximal codimension, and that there are indeed infinitely many of them. Moreover, we show that their union is dense in the natural topology.
We propose a conjectural stronger version of Bogomolov-Gieseker inequality for stable sheaves on quintic 3-folds. Our conjecture is derived from an attempt to construct a Bridgeland stability condition on graded matrix factorizations, which should correspond to the Gepner point via mirror symmetry and Orlov equivalence. We prove our conjecture in the rank two case.
It is known that the optimal Noether inequality $\mathrm{vol}(X) \ge \frac{4}{3}p_g(X) - \frac{10}{3}$ holds for every $3$-fold $X$ of general type with $p_g(X) \ge 11$. In this paper, we give a complete classification of $3$-folds $X$ of general type with $p_g(X) \ge 11$ satisfying the above equality by giving the explicit structure of a relative canonical model of $X$. This model coincides with the canonical model of $X$ when $p_g(X) \ge 23$. We also establish the second and third optimal Noether inequalities for $3$-folds $X$ of general type with $p_g(X) \ge 11$. These results answer two op…
Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda (such as $\mathbb P^3$, the quintic threefold or an abelian threefold). Let $L$ be a line bundle supported on a very positive surface in $X$. If $c_1(L)$ is a primitive cohomology class then we show it has very negative square.
We develop a framework to modify the Bogomolov-Gieseker type inequality conjecture introduced by Bayer-Macri-Toda, in order to construct a family of geometric Bridgeland stability conditions on any smooth projective 3-fold. We show that it is enough to check these modified inequalities on a small class of tilt stable objects. We extend some of the techniques in the works by Li and Bernardara-Macri-Schmidt-Zhao to formulate a strong form of Bogomolov-Gieseker inequality for tilt stable objects on Fano 3-folds. Consequently, we establish our modified Bogomolov-Gieseker type inequality conjecture…
We show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are proper algebraic stacks of finite type, if they satisfy the Bogomolov-Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson-Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi-Yau 3-folds satisfying the BG inequality conjecture, for example on…
On minimal varieties growing from quasismooth weighted hypersurfaces
本文研究小典范体积极小簇的构造，通过建立加权超曲面加权爆破的有效nef准则构造了多族极小3-折叠，并证明了一般型极小n-折叠典范体积的有效下界，证明了其在低维最优、高维近似最优。
We modify the conjectural Bogomolov-Gieseker type inequality introduced by Bayer, Macri and Toda to construct a family of geometric Bridgeland stability conditions on smooth projective 3-folds. We give an equivalent conjecture which needs to check these inequalities for a small class of tilt stable objects. We extend some of the techniques in Li's work for Fano 3-folds of Picard rank one to establish our modified Bogomolov-Gieseker type inequality conjecture for general Fano 3-folds.
Generalize Kobayashi's example for the Noether inequality in dimension three, we provide examples of n-folds of general type with small volumes.
Let $S$ be a minimal surface of general type with $p_g(S)=2$ and $K^2_S=1$, so called by a minimal $(1,2)$-surface. Then we obtain that the global log canonical threshold of the surface $S$ via $K_S$ is greater than equal to $\frac{1}{2}$. As an application we have \[ {\rm{vol}}(X)\ge\frac{4}{3}p_g(X)-\frac{10}{3} \] for all projective $3$-folds $X$ of general type which answers Question 1.4 of [J. A. Chen, M. Chen, C. Jiang, "The Noether inequality for algebraic threefolds", arXiv:1803.05553] about Noether inequality for $X$ with $5\le p_g(X)\le 26$.
We establish Noether's inequality for surfaces of general type in positive characteristic.Then we extend Enriques' and Horikawa's classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.
We establish the canonical class inequality for families of higher dimensional projective manifolds. As an application, we get a new inequality between the Chern numbers of 3-folds with smooth families of minimal surfaces of general type over a curve, $c_1^3<18c_3$.
Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors
本文研究带nef反典范除子的终端射影3-簇的Miyaoka型不等式，给出陈类乘积的下界及紧性结果，通过部分分类与例子完成证明，还拓展了终端弱Fano簇的相关非零性与Miyaoka–Kawamata型不等式研究。
Construction of minimal varieties from quasi-smooth weighted complete intersections
本文推广了极小簇的构造，为加权完全交集的加权吹胀典范除子建立了nef准则，并据此构造了大量新的极小3-簇。
In this paper, we first construct varieties of any dimension $n>2$ fibered over curves with low slopes. These examples violate the conjectural slope inequality of Barja and Stoppino [BS14b]. Led by their conjecture, we focus on finding the lowest possible slope when $n=3$. Based on a characteristic $p > 0$ method, we prove that the sharp lower bound of the slope of fibered $3$-folds over curves is $4/3$, and it occurs only when the general fiber is a $(1, 2)$-surface. Otherwise, the sharp lower bound is $2$. We also obtain a Cornalba-Harris-Xiao type slope inequality for families of surfaces o…
Let $X$ be a minimal projective Gorenstein 3-fold of general type. We give two applications of an inequality between $χ(ω_X)$ and $p_g(X)$: 1) Assume that the canonical map $Φ_{|K_X|}$ is of fiber type. Let $F$ be a smooth model of a generic irreducible component in the general fiber of $Φ_{|K_X|}$. Then the birational invariants of $F$ are bounded from above. 2) If $X$ is nonsingular, then $c_1^3\leq {1/27} c_1c_2+{10/3}$ where $c_1$, $c_2$ are Chern invariants of $X$.
Let $f:\, X\to Y$ be a semistable non-isotrivial family of $n$-folds over a smooth projective curve with discriminant locus $S \subseteq Y$ and with general fibre $F$ of general type. We show the strict Arakelov inequality \[\frac{\mathrm{deg}\, f_*ω_{X/Y}^ν}{\mathrm{rank}\, f_*ω_{X/Y}^ν} < {nν\over 2}\cdot\mathrm{deg}\,Ω^1_Y(\log S),\] for all $ν\in \mathbb N$ such that the $ν$-th pluricanonical linear system $|ω^ν_F|$ is birational. This answers a question asked by Möller, Viehweg and the third named author.
We study a conjectural relationship among Donaldson-Thomas type invariants on Calabi-Yau 3-folds counting torsion sheaves supported on ample divisors, ideal sheaves of curves and Pandharipande-Thomas's stable pairs. The conjecture is a mathematical formulation of Denef-Moore's formula derived in the study of Ooguri-Strominger-Vafa's conjecture relating black hole entropy and topological string. The main result of this paper is to prove our conjecture assuming a conjectural Bogomolov-Gieseker type inequality proposed by Bayer, Macri and the author.
Let $\Cal E$ be a very ample vector bundle of rank two on a smooth complex projective threefold $X$. An inequality about the third Segre class of $\Cal E$ is provided when $K_X+\det \Cal E$ is nef but not big, and when a suitable positive multiple of $K_X+\det \Cal E$ defines a morphism $X\to B$ with connected fibers onto a smooth projective curve $B$, where $K_X$ is the canonical bundle of $X$. As an application, the case where the genus of $B$ is positive and $\Cal E$ has a global section whose zero locus is a smooth hyperelliptic curve of genus $\geq 2$ is investigated, and our previous res…
A Kawamata--Miyaoka type inequality for Fano varieties of arbitrary Picard number
针对维数不小于2的Q-可分典范弱Fano簇，证明当Q-Fano指数不小于3时其满足Kawamata–Miyaoka型不等式，并给出Gorenstein典范Fano三维簇的Q-Fano指数取值范围。
On the Feyzbakhsh-Thomas programme for Fano $3$-folds
研究关于满足特定条件的法诺三维簇上唐纳森 - 托马斯不变量的计算，核心方法是借助\(K\)理论唐纳森 - 托马斯理论及顶点代数组合性质，实现将秩\(r>0\)的吉塞克半稳定层不变量用特定秩\(0\)层不变量表示。
Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical map of $X$ and so $F$ is a smooth curve or a smooth surface. The main result of the paper is that there is a computable constant $K$ (independent of $X$) such that $g(F)\leq 647$ or $p_g(F)\leq 38$ whenever $p_g(X)\geq K$. The method heavily relies on both a Noether type of inequality and a Miyaoka-Yau inequality and that is the reason we only treat a Goren…
Given an open subset U of a projective curve Y and a smooth family f:V-->U of curves, with semi-stable reduction over Y, we show that for a sub variation of Hodge structures of rank >2 the Arakelov inequality must be strict. For families of n-folds we prove a similar result under the assumption that the (n,0) component of the Higgs bundle defines fibrewise a birational map.
Let $X$ be a smooth projective threefold of Picard number one for which the generalized Bogomlov-Gieseker inequality holds. We characterize the limit Bridgeland semistable objects at large volume in the vertical region of the geometric stability conditions associated to $X$ in complete generality and provide examples of asymptotically semistable objects. In the case of the projective space and $ch^β(E)=(-R,0,D,0)$, we prove that there are only a finite number of nested walls in the $(α,s)$-plane. Moreover, when $R=0$ the only semistable objects in the outermost chamber are the 1-dimensional Gi…
On general type varieties admitting global holomorphic forms
本文证明了非奇异射影一般型n-簇的Noether型不等式，并以此证明了特定3-簇的最小体积猜想，揭示了一般型簇典范稳定指数的提升原理及强提升原理。
We prove that there exists a positive number $C_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over {\bf C}, the automorphism group $Aut(X)$ satisfies the inequality $\sharp{Aut}(X)\leq C_{n}\cdotμ(X,K_{X})$, where $μ(X,K_{X})$ is the volume of $X$ with respect to $K_{X}$.
On generic finiteness of pluricanonical maps of threefolds of general type
本文证明了对满足P2(X)≥2的一般型极小三维流形X，|6K_X|定义通用有限映射，且该结果最优；同时证明对所有一般型极小三维流形X，当n≥38时|nK_X|定义通用有限映射。关键技术是曲面的新通用有限性准则和特殊解消下的有效比较不等式。
In this paper, we prove that the group $\mathrm{Aut}_\mathbb{Q}(X)$ of numerically trivial automorphisms are uniformly bounded for smooth projective threefolds $X$ of general type which either satisfy $q(X)\geq 3$ or have a Gorenstein minimal model. If $X$ is furthermore of maximal Albanese dimension, then $|\mathrm{Aut}_\mathbb{Q}(X)|\leq 4$, and equality can be achieved by an unbounded family of threefolds previously constructed by the third author. Along the way we prove a Noether type inequality for log canonical pairs of general type with the coefficients of the boundary divisor from a gi…
On divisorial (i) classes
本文针对若干点处爆破的射影空间，引入并研究i=-1、0、1时的除子（i）类，推广Noether不等式，证明其与Weyl群作用轨道双射对应，还将其不可约性条件替换为正相交的数值条件。
Degenerations of shifted symplectic moduli stacks of sheaves
本文通过推广 Brav--Dyckerhoff 的构造，证明了 Calabi--Yau 族上凝聚层的导出模叠具有相对平移辛结构，并识别其为伪完美对象模的子叠，为枚举几何提供了潜在应用。
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