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.)
We prove a more general and precise version of the Noether-Fano inequalities for birational maps between Mori fiber spaces. This is applied to give descriptions of global canonical thresholds on Fano varieties of Picard number one.
I give a survey of Noether-Fano inequalities in birational geometry, starting with the original Noether inequality and up to the modern approach of Log Minimal Model program. The paper is based on my talk at the Fano conference in Torino in October 2002. This is the revised version: an erroneous reference in my paper published in the Proceedings of Fano conference is corrected.
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.
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 $ø_{X}$.
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.
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
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.
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.
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…
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 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.
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…
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.
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 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.
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 study the geography of Gorenstein stable log surfaces and prove two inequalities for their invariants: the stable Noether inequality and the $P_2$-inequality. By constructing examples we show that all invariants are realised except possibly some cases where the inequalities become equalities.
Based on work of R. Lazarsfeld and M. Popa, we use the derivative complex associated to the bundle of the holomorphic p-forms to provide inequalities for all the Hodge numbers of a special class of irregular compact Kaehler manifolds. For 3-folds and 4-folds we give an asymptotic bound for all the Hodge numbers in terms of the irregularity. As a byproduct, via the BGG correspondence, we also bound the regularity of the exterior cohomology modules of bundles of holomorphic p-forms.
For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus W^r(L,X):={η\in Pic^0(X)|h^0(L+η)\geq r+1}: we describe its natural scheme structure and compute the Zariski tangent space. If X is a smooth surface of maximal Albanese dimension and C is a curve on X, we define a Brill-Noether number ρ(C, r) and we prove, under some mild additional assumptions, that if ρ(C, r) is non negative then W^r(C,X) is nonempty of dimension bigger or equal to ρ(C,r). As an application, we derive lower bounds for h^0(K_D) for a divisor D that moves linearly o…
Let $X$ be a nonsingular projective $n$-fold $(n\ge 2)$ of Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $κ$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, \cdots, c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $κ$ is $0$, then the Chern ratios $(\frac{c_{2,1^{n-2}}}{c_{1^n}}, \frac{c_{2,2,1^{n-4}}}{c_{1^n}}, \cdots, \frac{c_{n}}{c_{1^n}})$ are contained in a convex polyhedron for al…
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 $5\le p_g(X)\le 26$.
Local inequalities for $cD$ and $cE$ singularities
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所交叉理论与数学科学中心）；National Central University（中央大学）；Kyushu University（九州大学）
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 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…
