A note on Fano manifolds whose second Chern character is positive

Bogomolov-Gieseker Type Inequality on Calabi-Yau and Fano 3-folds

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.

Classification of 2-Fano manifolds with high index

In this paper we classify n-dimensional Fano manifolds with index >=n-2 and positive second Chern character.

Classification of Toric 2-Fano 4-folds

In this notes we classify toric Fano 4-folds having positive second Chern Character.

Extremal contractions of 2-Fano fourfolds

We consider extremal contractions on smooth Fano fourfolds whose second Chern character is positive. We show that such contractions can neither be of fiber type nor contract a divisor to a point.

Inequalities of Chern classes on nonsingular projective $n$-folds of Fano and general type with ample canonical bundle

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…

Polarized minimal families of rational curves and higher Fano manifolds

In this paper we investigate Fano manifolds $X$ whose Chern characters $ch_k(X)$ satisfy some positivity conditions. Our approach is via the study of polarized minimal families of rational curves $(H_x,L_x)$ through a general point $x\in X$. First we translate positivity properties of the Chern characters of $X$ into properties of the pair $(H_x,L_x)$. This allows us to classify polarized minimal families of rational curves associated to Fano manifolds $X$ satisfying $ch_2(X)\geq0$ and $ch_3(X)\geq0$. As a first application, we provide sufficient conditions for these manifolds to be covered by…

Computing Gromov-Witten invariants of some Fano varieties

We present a recursive algorithm computing all the genus-zero Gromov-Witten invariants from a finite number of initial ones, for Fano varieties with generically tame semi-simple quantum (and small quantum) (p, p)- type cohomology, whose first Chern class is a strictly positive combination of effective integral basic divisors.

不允许存在极值 Kähler 度量的环面 Fano 流形

Toric Fano manifolds that do not admit extremal Kähler metrics

本文构造了维数为 10 的环面 Fano 流形，使其第一陈类中无极值 Kähler 度量，并通过乘积推广到所有 n≥11，回答了 Mabuchi 的问题。

Compactification of the moduli space of instanton sheaves on the Fano threefold $V_5$

We study semistable sheaves of rank $2$ with Chern classes $c_1=0$, $c_2=2$ and $c_3=0$ on the Fano 3-fold $V_5$ of Picard number $1$, degree $5$ and index $2$. We show that the moduli space of such sheaves has a component that is isomorphic to $\mathbb{P}^5$ by identifying it with the moduli space of semistable quiver representations. This provides a natural smooth compactification of the moduli space of minimal instantons, as well as Ulrich bundles of rank $2$ on $V_5$.

On a sufficient condition for a Fano manifold to be covered by rational $N$-folds

In this paper, we prove a conjecture by T. Suzuki, which says if a smooth Fano manifold satisfies some positivity condition on its Chern character, then it can be covered by rational $N$-folds. We prove this conjecture by using purely combinatorial properties of Bernoulli numbers.

On the quantum cohomology of Fano bundles over projective spaces

In their paper "Quantum cohomology of projective bundles over $P^n$" (Trans. Am. Math. Soc. (1998)350:9 3615-3638) Z.Qin and Y.Ruan introduce interesting techniques for the computation of the quantum ring of manifolds which are projectivized bundles over projective spaces; in particular, in the case of splitting bundles they prove under some restrictions the formula of Batyrev about the quantum ring of toric manifolds. Here we prove the formula of Batyrev on the quantum Chern-Leray equation for a large class of splitting bundles.

Espace des modules des faisceaux semi-stables de rang 2 et de classes de Chern $c_{1}=0$, $c_{2}=2$ et $c_{3}=0$ sur une hypersurface cubique lisse de $\mathbb{P}^{4}$

In this note we study the moduli space of rank two semistable sheaves on a smooth cubic hypersurface $X\subset\mathbb{P}^{4}$ with $c_{1}=0$, $c_{2}=2$ and $c_{3}=0$. We show that it is isomorphic to the blow-up of the intermediate jacobian $J(X)$ of $X$ along the Fano surface of $X$. We complete previous results of Iliev-Markushevich AG/9910058 and Markushevich-Tikhomirov AG/9910063.

正特征中的Fano三维流形 III

Fano threefolds in positive characteristic III

本文分类了正特征下Picard数至少为二以及Picard秩为二的原始Fano三维流形。

正特征下的Fano三重曲面IV

Fano threefolds in positive characteristic IV

本文在正特征下对Fano三重曲面进行了分类，延续了前几部分的研究工作。

Fano Hypersurfaces in Positive Characteristic

We prove that a general Fano hypersurface in a projective space over an algebraically closed field of arbitrary characteristic is separably rationally connected.

正特征下的105族Fano三维簇

105 families of Fano threefolds in positive characteristic

在正特征代数闭域上，证明光滑Fano三维簇共有105个不可约形变族。

Some pathologies of Fano manifolds in positive characteristic

We construct examples of Fano manifolds, which are defined over a field of positive characteristic, but not over $\com$.

Remarks on Kawamata's effective non-vanishing conjecture for manifolds with trivial first Chern classes

Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial global sections. We verify this conjecture for several cases, including (i) all hyperkähler varieties of dimension $\leq 6$; (ii) all known hyperkähler varieties except for O'Grady's 10-dimensional example; (iii) general complete intersection Calabi-Yau varieties in certain Fano manifolds (e.g. toric ones). Moreover, we investigate the effectivity of Todd classes of hyperkähler varieties and Calabi-Yau varieties. We prove that the fourth Todd…

特征为正的素 Fano 三维簇

Prime Fano threefolds in positive characteristic

该研究证明特征为正的域上亏格≥7的素 Fano 三维簇均为 Mukai 簇的线性截影，完善了代数几何中相关分类结论。

正特征下的Fano三维流形I

Fano threefolds in positive characteristic I

本文在正特征下研究了 Picard 数为一的 Fano 三维流形，证明了高 genus 的 Fano 流形是二次曲面的交集。

Fano manifolds with big tangent bundle: a characterisation of $V_5$

Let X be a Fano manifold with Picard number one such that the tangent bundle T_X is big. If X admits a rational curve with trivial normal bundle, we show that X is isomorphic to the del Pezzo threefold of degree five.

关于toric 2-Fano manifold分类：通用P^2-包络

On the classification of toric $2$-Fano manifolds: generic $\mathbb{P}^2$-bundles

本文通过研究最小投影包络维度m(X)扩展了toric 2-Fano manifold的分类，证明只有当m(X)=2时，X同构于P^2。

素数Fano三种折面的 genus 8 在正特征下的情况

Prime Fano threefolds of genus 8 in positive characteristic

研究证明 genus 8 的素数Fano三种折面在正特征下与Grassmannian的线性截面同构，并指出其为非理性且在特征大于2时全局F-正则。

正特征域上纤维化Fano三维簇的补集有界性

Boundedness of complements for fibered Fano threefolds in positive characteristic

本文在正特征大特征情形下证明了Fano型纤维化的典范丛公式，并解决了具有纤维化结构的Fano型三维配对的Shokurov补集有界性猜想，特别覆盖了$-(K_X+B)$为nef且非big的情形。

$c(\mathrm{Pol}^d(\mathbb{C}^n))$的斯特林系数的多项式性与Fano概形

Polynomiality of the Striling coefficients of $c(\mathrm{Pol}^d(\mathbb{C}^n))$ and Fano schemes

该研究引入斯特林系数概念，证明了$n$元$d$次齐次多项式的$\mathrm{GL}(n)$表示的陈类对$d$的多项式依赖关系及渐近性质，解决了Manivel的相关猜想，并给出一般超曲面的直线Fano概形的次数与欧拉示性数新公式。

Fano varieties with large Seshadri constants in positive characteristic

We prove that a Fano variety (with arbitrary singularities) of dimension $n$ in positive characteristic is isomorphic to $\mathbb{P}^n$ if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than $n$ and classify Fano varieties whose anti-canonical divisors have Seshadri constants $n$. In characteristic $p>5$ and dimension $3$, we also show that Fano varieties $X$ with Seshadri constants $ε(-K_X,x)>2+ε$ at some smooth point $x\in X$ (for some fixed $ε>0$) have bounded anti-canonical degrees.

A characterization of some Fano 4-folds through conic fibrations

We find a characterization for Fano 4-folds $X$ with Lefschetz defect $δ_{X}=3$: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure $f\colon X\to Y$ with $ρ_{X}-ρ_{Y}=3$. Moreover, we observe that all of these varieties are rational. We give the list of all possible targets of such contractions. Combining our results with the classification of toric Fano $4$-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric $4$-folds with $δ_{X}=3$.

Mirror Symmetry and String Vacua from a Special Class of Fano Varieties

Because of the existence of rigid Calabi--Yau manifolds, mirror symmetry cannot be understood as an operation on the space of manifolds with vanishing first Chern class. In this article I continue to investigate a particular type of Kähler manifolds with positive first Chern class which generalize Calabi--Yau manifolds in a natural way and which provide a framework for mirrors of rigid string vacua. This class is comprised of a special type of Fano manifolds which encode crucial information about ground states of the superstring. It is shown in particular that the massless spectra of $(2,2)$--…

A Fano compactification of the $\mathrm{SL}_2(\mathbb{C})$ free group character variety

We show that a certain compactification $\mathfrak{X}_g$ of the $\mathrm{SL}_2(\mathbb{C})$ free group character variety $\mathcal{X}(F_g, \mathrm{SL}_2(\mathbb{C}))$ is Fano. This compactification has been studied previously by the second author, and separately by Biswas, Lawton, and Ramras. Part of the proof of this result involves the construction of a large family of integral reflexive polytopes.

正特征下的 Fano 三维簇 II

Fano threefolds in positive characteristic II

本文研究正特征域上指数和 Picard 数均为一且反典范线性系极丰富的光滑 Fano 三维簇，确定其亏格范围并排除沿光滑曲线爆破仍为 Fano 的情形。

正特征下素法诺三维簇的向井（Mukai）丛与布里尔-诺特（Brill-Noether）一般性

Mukai bundles and Brill-Noether generality for prime Fano threefolds in positive characteristic

针对正特征下亏格≥8的素法诺三维簇，证明其存在对应亏格非平凡分解的唯一向井丛，核心方法是证明其光滑超平面截面为布里尔-诺特一般，丰富了代数几何中相关簇的结构理论。

正特征中一个不规则光滑 Fano 四维簇

An irregular smooth Fano fourfold in positive characteristic

在特征二的代数闭域上构造了一个光滑射影 Fano 四维簇，具有非零的 $H^1$ 且无自由有理曲线，从而不是可分有理连通的，该例子借助 ChatGPT 构建。

正特征下具有固定Gorenstein指数的弱Fano三维簇的有界性

Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic

该研究在特征p>5的不可数代数闭域上，通过引入Gorenstein指数、补与Kodaira型零化条件，部分证明了三维簇的BAB猜想，同时给出了射影三维簇上nef大可逆层Seshadri常数的一致下界。

Fano manifolds whose elementary contractions are smooth $\mathbb P^1$-fibrations: A geometric characterization of flag varieties

The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if $X$ is a Fano manifold whose all elementary contractions are $\mathbb P^1$-fibrations then $X$ is isomorphic to the complete flag manifold $G/B$ where $G$ is a semi-simple Lie algebraic group and $B$ is a Borel subgroup of $G$.

正特征Fano三维流形的可提升性与消失定理I

Liftability and vanishing theorems for Fano threefolds in positive characteristic I

本文证明正特征光滑Fano三维流形可提升到Witt向量环，并满足Akizuki-Nakano消失、Hodge到de Rham谱序列E1退化及晶体上同调无挠性，针对$|-K_X||$非常丰沛且Picard群由$\omega_X$生成的情形。

正特征域上 Fano 三维簇的提升性与消失定理 II

Liftability and vanishing theorems for Fano threefolds in positive characteristic II

本文研究正特征域上光滑 Fano 三维簇的提升性与上同调性质，证明当 Picard 数或 Fano 指数大于 1 时其具有拟 F-分裂性，从而在除特定极丰沛情形外建立 Akizuki–Nakano 消失、Hodge–de Rham 谱序列 E1-退化及晶体上同调无挠性。

A note on Fano surfaces of nodal cubic threefolds

We study the Picard variety of the Fano surface of nodal and mildly cuspidal cubic threefolds in arbitrary characteristic by relating divisors on the Fano surface to divisors on the symmetric product of a curve of genus 4.

Weak approximation for Fano complete intersections in positive characteristic

For a smooth curve $B$ over an algebraically closed field $k$, for every $B$-flat complete intersection $X_B$ in $B\times_{\text{Spec}\ k} \mathbb{P}^n_k$ of type $(d_1,\dots,d_c)$, if the Fano index is $\geq 2$ and if $\text{char}(k)>\max(d_1,\dots,d_c)$, we prove weak approximation of $\widehat{\mathcal{O}}_{B,b}$-points of $X_B$ by $k(B)$-points at all places of (strong) potentially good reduction, including all places of good reduction. The key step is the proof that such complete intersections are \emph{separably uniruled by lines}, and even \emph{separably rationally connected}, whenever…

On some Fano 4-folds with Lefschetz defect 3

We show that Fano 4-folds with Picard number 5 have Lefschetz defect 3 if and only if they are toric of combinatorial type K. We also find a characterization for such varieties in terms of Picard number of prime divisors. Moreover, we discuss classification results for 4-dimensional complex smooth projective varieties admitting some particular fiber type contractions.

Examples of Fano manifolds with non-pseudoeffective tangent bundle

Let $X$ be a Fano manifold. While the properties of the anticanonical divisor $-K_X$ and its multiples have been studied by many authors, the positivity of the tangent bundle $T_X$ is much more elusive. We give a complete characterisation of the pseudoeffectivity of $T_X$ for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds.

Higher Codimensional Alpha Invariants and Characterization of Projective Spaces

We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable Q-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.

关于特征 $p>0$ 下 Fano 簇的典范丛公式与有效双有理性

On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$

本文研究了正特征下 Fano 型簇的双有理几何与有界性问题，证明了有界性在正规化下不变，给出了更易用的典范丛公式，并证实了具有良好奇点的弱 Fano 簇的有效双有理性。

Fano纤维与扭曲的Kähler-爱因斯坦度量I

Fano Fibrations and Twisted Kähler-Einstein Metrics I

本文研究Fano纤维的几何结构及其在Kähler-Ricci流中的应用，构造了与Weil-Petersson度量相关的(1,1)形式，并证明了扭曲Kähler-爱因斯坦方程的解。

秩为二的半稳定sheaves在主系列有理Fano三维流形上的模空间

Moduli of rank two semistable sheaves on rational Fano threefolds of the main series

本文研究了在有理Fano三维流形上秩为二的半稳定sheaves的模空间，证明了其第三陈类的有界性，并构造了新的有理和无理成分的无限系列。

Some results on rational surfaces and Fano varieties

The goal of this article is to study the equations and syzygies of embeddings of rational surfaces and certain Fano varieties. Given a rational surface X and an ample and base-point-free line bundle L on X, we give an optimal numerical criterion for L to satisfy property Np. This criterion turns out to be a characterization of property Np if X is anticanonical. We also prove syzygy results for adjunction bundles and a Reider type theorem for higher syzygies. For certain Fano varieties we also prove results on very ampleness and higher syzygies.

Remarks on toric manifolds whose Chern characters are positive

We show that the second Chern character of any projective toric manifold of Picard number three is not ample. In connection with this result, we give various examples of the positivity of higher Chern characters of projective toric manifolds.

Notes on the Chern-character

Notes for some talks given at the seminar on characteristic classes at NTNU in autumn 2006. In the paper a proof of the existence of a Chern-character from complex K-theory to any cohomology theory with values in graded Q-algebras is given. The Chern-character respects the Adams and Steenrod operations.

A vanishing theorem for Fano varieties in positive characteristic

We prove a Kodaira-type vanishing theorem for the Witt vector sheaf on a Fano variety over a perfect field of characteristic p. As a corollary, we deduce that the number of rational points on a Fano variety over a finite field with q=p^n elements is congruent to 1 mod q. This refines a result of Esnault which say that the number is congruent to 1 mod p.

Fano generalized Bott manifolds

We give a necessary and sufficient condition for a generalized Bott manifold to be Fano or weak Fano. As a consequence we characterize Fano Bott manifolds.

Fano $4$-folds with nef tangent bundle in positive characteristic

In characteristic $0$, the Campana-Peternell conjecture claims that the only smooth Fano variety with nef tangent bundle should be homogeneous. In this paper, we study the positive characteristic version of the Campana-Peternell conjecture. In particular, we give an affirmative answer for Fano $4$-folds with nef tangent bundle and Picard number greater than one.

Toric Fano varieties associated to building sets

We characterize building sets whose associated nonsingular projective toric varieties are Fano. Furthermore, we show that all such toric Fano varieties are obtained from smooth Fano polytopes associated to finite directed graphs.

Instanton bundles on $\mathbb{P}^1\times\mathbb{F}_1$

In this paper we deal with a particular class of rank two vector bundles (\emph{instanton} bundles) on the Fano threefold of index one $F:=\mathbb{F}_1 \times \mathbb{P}^1$. We show that every instanton bundle on $F$ can be described as the cohomology of a monad whose terms are free sheaves. Furthermore we prove the existence of instanton bundles for any admissible second Chern class and we construct a nice component of the moduli space where they sit. Finally we show that minimal instanton bundles (i.e. with the least possible degree of the second Chern class) are aCM and we describe their mo…

Einstein-Kaehler Metrics on Symmetric Toric Fano Manifolds

Let $X$ be a complex toric Fano $n$-fold and ${\cal N}(T)$ the normalizer of a maximal torus $T$ in the group of biholomorphic authomorphisms $Aut(X)$. We call $X$ {\em symmetric} if the trivial character is a single ${\cal N}(T)$-invariant algebraic character of $T$. Using an invariant $α_G(X)$ introduced by Tian, we show that all symmetric toric Fano $n$-folds admit an Einstein-Kähler metric. We remark that so far one doesn't know any example of a toric Fano $n$-fold $X$ such that $Aut(X)$ is reductive, the Futaki character of $X$ vanishes, but $X$ is not symmetric.

A Chern-Weil formula for the Chern character of a perfect curved module

Let $k$ be a field of characteristic 0 and $\mathcal{A}$ a curved $k$-algebra. We obtain a Chern-Weil-type formula for the Chern character of a perfect $\mathcal{A}$-module taking values in $HN_0^{II}(\mathcal{A})$, the negative cyclic homology of the second kind associated to $\mathcal{A}$, when $\mathcal{A}$ satisfies a certain smoothness condition.