Angela Gibney, Swarnava Mukhopadhyay
Here we consider higher Chern classes of vector bundles of conformal blocks on $\overline{\operatorname{M}}_{0,n}$, giving explicit formulas for them, and extending various results that hold for first Chern classes to them. We use these classes to form a full dimensional subcone of the Pliant cone on $\overline{\operatorname{M}}_{0,n}$.
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We describe nef vector bundles on a projective space with first Chern class three and second Chern class eight over an algebraically closed field of characteristic zero by giving them a minimal resolution in terms of a full strong exceptional collection of line bundles.
Given a parabolic vector bundle, we construct for it a projectivization and tautological line bundle. These are analogs of the projectivization and tautological line bundle for an usual vector bundle. Using these we give a construction of the parabolic Chern classes.
Chern character of Schur bundles
本文提出一种方法，通过E的Chern类显式计算Schur丛S^αE的Chern类。
If $ D \in X$ is a curve with multiple points in a surface, a parabolic bundle defined on $(X;D)$ away from the singularities can be extended in several ways to a parabolic bundle on a resolution of singularities. We investigate the possible parabolic Chern classes for these extensions.
Chern invariants of some flat bundles in the arithmetic Deligne cohomology
本文研究了Green-Griffiths，Asakura-Saito提出的算术Deligne上同调中平坦丛的Chern类。我们证明了在某些情况下陈类的非平凡性，证明也表明一般平坦丛（在其模空间）可以预期有非平凡类，这是在某些Gauss-Manin系统。
Nef vector bundles on a quadric surface with first Chern class $(2,2)$
本研究针对特征零代数闭域上光滑二次曲面，完成了第一陈类为$(2,2)$的nef向量丛的分类工作，填补了相关代数几何研究的内容。
Nef vector bundles on a hyperquadric with first Chern class two
本研究在特征零的代数闭域下，完成了维数不小于4的光滑超二次曲面上第一陈类为2的nef向量丛的分类工作。
We determine the Chern classes of globally generated rank two vector bundles on P^2.
We show that under some assumptions on the monodromy group some combinations of higher Chern classes of flat vector bundles are torsion in the Chow group. Similar results hold for flat vector bundles that deform to such flat vector bundles (also in case of quasi-projective varieties). The results are motivated by Bloch's conjecture on Chern classes of flat vector bundles on smooth complex projective varities but in some cases they give a more precise information. We also study Higgs version of Bloch's conjecture and analogous problems in the positive characteristic case.
We classify nef vector bundles on a projective space with first Chern class three over an algebraically closed field of characteristic zero; we see, in particular, that these nef vector bundles are globally generated if the second Chern class is less than eight, and that there exist nef but non-globally generated vector bundles with second Chern class eight and nine on a projective plane.
Chern classes of Gauß-Manin bundles of weight 1 vanish
我们表明，陈特征的变化的极化霍奇结构的重量与幂零残留在$\infty$死亡扭转的Chow环，除了在余维0。
The order of the top Chern class of the Hodge bundle on the moduli space of abelian varieties
给出了主极化阿贝尔簇模空间上Hodge丛的顶Chern类的阶的上、下界。
We prove an explicit formula for the total Chern character of the Verlinde bundle over the moduli space of pointed stable curves in terms of tautological classes. The Chern characters of the Verlinde bundles define a semisimple CohFT (the ranks, given by the Verlinde formula, determine a semisimple fusion algebra). According to Teleman's classification of semisimple CohFTs, there exists an element of Givental's group transforming the fusion algebra into the CohFT. We determine the element using the first Chern class of the Verlinde bundle on the moduli space of nonsingular curves and the proje…
The top Chern class of the Hodge bundle on the moduli space of abelian varieties
给出了主极化阿贝尔簇模空间上Hodge丛的顶Chern类的阶的上、下界。我们也给出了一个推广到更高的属的著名公式$12 λ_1=δ$的亏格1。
Nef vector bundles on a quadric threefold with first Chern class two
本文分类了特征零代数闭域上三维光滑超二次簇上第一陈类为二的 nef 向量丛，并证明它们都是整体生成的。
The Chern character of a parabolic bundle, and a parabolic Reznikov theorem in the case of finite order at infinity
本文给出了局部交换抛物丛的Chern特征标关于其组成丛的一个显式公式。抛物型结构的几个特点和变种进行了讨论。由对数联络产生的抛物丛是一类重要的例子。作为应用，我们考虑了局部单值点在无穷远处是半单的有穷级的情形。在这种情况下，相应的局部交换抛物丛的抛物陈类在$\geq 2$度的有理Deligne上同调中被导出为零。
Calculating the parabolic Chern character of a locally abelian parabolic bundle
利用Deligne-Mumford叠加的定义，计算了光滑严格正规交叉因子上具有局部交换抛物结构丛的抛物Chern特征标.我们得到了$ch_1$，$ch_2$和$ch_3$的显式公式，并验证了这些公式与Borne给出的$ch_1$公式和Mochizuki给出的$ch_2$公式相对应.
We classify globally generated vector bundles on $\mathbf{P}^1 \times \mathbf{P}^1 \times \mathbf{P}^1$ with small first Chern class, i.e. $c_1= (a_1, a_2, a_3)$, $a_i \le 2$. Our main method is to investigate the associated smooth curves to globally generated vector bundles via the Hartshorne-Serre correspondence.
We prove that the Gromov--Witten theory (GWT) of a projective bundle can be determined by the Chern classes and the GWT of the base. It completely answers a question raised in a previous paper (arXiv:1607.00740). Its consequences include that the GWT of the blow-up of X at a smooth subvariety Z is uniquely determined by GWT of X, Z plus some topological data.
A formula for the first Chern class of the Verlinde bundle over the moduli space of smooth genus g curves is given. A finite-dimensional argument is presented in rank 2 using geometric symmetries obtained from strange duality, relative Serre duality, and Wirtinger duality together with the projective flatness of the Hitchin connection. A derivation using conformal-block methods is presented in higher rank. An expression for the first Chern class over the compact moduli space of curves is obtained.
On Chern ratios for surfaces with ample cotangent bundle
本文研究了具有充裕余切丛的曲面的Chern比在（1，3）中的稠密性问题。特别地，我们通过在具有大余切丛的簇的乘积中构造一族完全交曲面来证明（1，2）中的稠密性。我们还分析了完全相交的情况下，在产品的曲线亏格至少2。
Vanishing Chern classes for numerically flat Higgs bundles
本文研究了 Higgs 丛的 ample 性，证明了 H-ample 和数值有效 Higgs 丛 Chern 类的非负性，并据此确立了数值平坦 Higgs 丛 Chern 类的消失性。
We consider a complex smooth projective variety equipped with an action of an algebraic torus with a finite number of fixed points. We compare the motivic Chern classes of Białynicki-Birula cells with the $K$-theoretic stable envelopes of cotangent bundle. We prove that under certain geometric assumptions satisfied e.g. by homogenous spaces these two notions coincide up to normalization.
Universal Chern classes on the moduli of bundles
本文针对曲线稳定丛模空间非精细（秩与次数不互素）的情况，构造泛上同调类，证明曲线与丛模栈乘积上的泛丛的部分陈类可提升至曲线与稳定丛模空间的乘积上。
A relation between the parabolic Chern characters of the de Rham bundles
本文考虑光滑簇上的权$i$ de Rham-Gauss-Manin丛，它由$i\geq 0$的光滑射影态射$f:X\_U\lrar U$产生.我们将每个权$i$ de Rham丛与$S$上的某个抛物丛相关联，并考虑它们在有理Chow群中的抛物Chern特征标，从而得到$U$的良好紧化$S$.我们证明了这些抛物丛的交替和的平凡性（正度）理性周群。这消除了由于Esnault和Viehweg的这种原始结果中的半稳定约化假设。
On the Chern filtration for the moduli of bundles on curves
该研究引入并研究曲线丛模空间上同调的陈滤过，完成秩2情形下稳定丛模空间及Harder--Narasimhan分层中间栈的滤过计算，发现奇特对称性并构造范畴化该对称性的$\frak{sl}_2$-作用，受相关几何中$P=C$现象启发。
Toric vector bundles with trivial Chern class and flag decorations
本文用热带方法研究完备环面簇上Chern类平凡的环面向量丛，证明其可经特征标扭曲后由低维簇拉回，推广了Payne定理，并构造了无此类非平凡丛的例子，引入旗装饰作为关键工具。
Cycles Representing The Top Chern Class of the Hodge Bundle on the Moduli Space of Abelian Varieties
我们将著名的亏格1的公式$12 λ=δ$推广到更高的亏格。我们还计算类的某些阶层的佐竹紧化的重言式环的下推的元素。
Chern classes of automorphic vector bundles, II
本文证明Hodge型Shimura簇环面紧化上自守向量丛典范扩张的ℓ-adic Chern类可下降至最小紧化上同调，且在Galois群作用下不变。
Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane
光滑射影曲面上点的Hilbert格式的上同调可以用顶点代数工具和等变工具来逼近。利用第一个工具，我们研究了射影曲面上点的Hilbert概型上的切丛的Chern类的普适公式的存在性和结构。第二个工具，然后导致很好的生成公式在特定情况下的希尔伯特计划的点在仿射平面。
Computations of Bott-Chern classes on P(E)
计算了复流形上全纯厄米向量丛（E，h）的超平面簇P（E）的相对切丛的度量欧拉序列的Bott-Chern类.给出了算术向量丛的算术特征类的构造和P（E）关于重言式商丛O_E（1）的高度的计算.
Chern Numbers of Ample Vector Bundles on Toric Surfaces
设$\sE$是光滑复曲面$S$上的充分秩$r$丛，其拓扑欧拉示性数是$e(S)$ .在本文中，我们证明了$c_1(\sE)^2$和$c_2(\sE)$的一些令人惊讶的强下界。我们还列举了不等式$c_1(\sE)^2\ge 4e(S)$或不等式$c_2(\sE)\ge e(S)$成立的例外。
We give a new proof of the classification due to Peternell-Szurek-Wiśniewski of nef vector bundles on a projective space with the first Chern class less than three and on a smooth hyperquadric with the first Chern class less than two over an algebraically closed field of characteristic zero.
A Riemann-Roch theorem for flat bundles, with values in the algebraic Chern-Simons theory
设$f: X \to S$是代数闭域$k$上的平坦态射，其相对法交叉因子为$Y\subset X$，$(E, \nabla)$是一个丛，其对数极点沿$Y$连接，曲率值在$f^*Ω^2_{k(S)}$中.然后Gaueline-Manin层$R^if_*(Ω^*_{X/S}({\rm log} Y)\otimes E)$携带Gaueline-Manin连接$GM^i(\nabla)$ .建立了联系$\nabla$、$GM^i(\nabla)$和$Ω^1_{X/S}({\rm log}Y)$的顶Chern类的代数Chern-Simons不变量的Riemann-Roch公式.
We derive a formula for the Chern classes of the bundles of conformal blocks on \bar{M}_{0,n} associated to simple finite dimensional Lie algebras and explore its consequences in more detail for sl_2 and in general for level 1. We also give a method for computing the first Chern class of such bundles on \bar{M}_{g,n} for g>0.
Equivariant Chern Classes of Toric Vector Bundles over a DVR and Bruhat--Tits Buildings
本文定义 DVR 上正常环面格式中环面向量丛的等变陈类，借助多面体复形上的分段多项式函数及扩展 Bruhat–Tits 建筑给出组合刻画，并联系非阿基米德 Arakelov 理论。
Vanishing of the top Chern classes of the moduli of vector bundles
设$Y$是亏格为$g\ge 2$的光滑射影曲线，$M_{r,d}(Y)$是$Y$上秩为$r$、度为$d$的稳定向量丛的模空间. Newstead和Ramanan的一个经典猜想指出，$i>2(g-1)$的$ c_i(M_{2,1}(Y))=0$，即顶级$2g-1$ Chern类消失。本文的目的是通过推广Gieseker退化方法，将这个消失结果推广到秩为3的情形。更准确地说，我们证明了$c_i(M_{3,1}(Y))=0$为$i>6g-5$。换句话说，顶级$3g-3$ Chern类消失了。请注意，我们还有$i>6g-5$的$c_i(M_{3,2}(Y))=0$。
We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.
Let $S \subset \mathbb P^3$ be a very general sextic surface over complex numbers. Let $\mathcal{M}(H, c_2)$ be the moduli space of rank $2$ stable bundles on $S$ with fixed first Chern class $H$ and second Chern class $c_2$. In this article we study the configuration of points of certain reduced zero dimensional subschemes on $S$ satisfying Cayley-Bacharach property, which leads to the existence of non-trivial sections of a general memeber of the moduli space for small $c_2$. Using this study we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components …
We prove that Chern classes in continuous $\ell$-adic cohomology of automorphic bundles associated to representations of $G$ on a projective Shimura variety with data $(G,X)$ are trivial rationally. It is a consequence of Beilinson's conjectures which predict that the Chern classes in the Chow groups vanish rationally.
Chern Classes of Tautological Sheaves on Hilbert Schemes
我们给出了一个算法描述的行动的陈类重言丛上的Hilbert计划的上同调的点表面内的Nakajima的振子代数的框架。这导致识别的上同调环的希尔伯特计划的仿射平面与一个环的微分算子的Fock空间。最后计算了Hilb^n上直到n=7的线丛的重言式丛的顶Segre类，并给出了生成级数的一个猜想。
We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern numbers of such a holomorphic vector bundle are nonnegative and can be bounded below and above respectively by two special Chern numbers. As applications, we obtain a family of new results on compact connected complex manifolds which are homogeneous or can be holomorphically imm…
We prove a canonical identifications of the spaces of generalized theta functions on the moduli spaces of anti-invariant vector bundles in the ramified case and the conformal blocks associated to twisted Kac-Moody affine algebras.
For any simple Lie algebra, a positive integer, and tuple of compatible weights, the conformal blocks bundle is a globally generated vector bundle on the moduli space of pointed rational curves. We classify all $S_n$-invariant vector bundles of conformal blocks for $\mathfrak{sl}_n$ which have rank one. We show that the cone generated by their base point free first Chern classes is polyhedral, generated by level one divisors.
We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum of two terms, namely the secondary Bott Chern character class of the sequence and its Chern character with supports on the finite fibers. Next, we compute these classes in the situation encountered by the second author when proving a "Kodaira vanishing theorem" for arithmetic surfaces.
We define Chern and Segre forms, or rather currents, associated with a Griffiths positive singular hermitian metric $h$ with analytic singularities on a holomorphic vector bundle $E$. The currents are constructed as pushforwards of generalized Monge-Ampère products on the projectivization of $E$. The Chern and Segre currents represent the Chern and Segre classes of $E$, respectively, and coincide with the Chern and Segre forms of $E$ and $h$, where $h$ is smooth. Moreover, our currents coincide with the Chern and Segre forms constructed by the first three authors and Ruppenthal in the cases wh…
We classify globally generated vector bundles on the projective n-space with first Chern class = 4. This extends previous results for first Chern class at most 3, namely for 2 of Sierra and Ugaglia [J. Pure Appl. Algebra 213 (2009), 2141-2146] and for 3 of Anghel and Manolache [arXiv:1202.6261] and, independently, of Sierra and Ugaglia [arXiv:1203.0185]. It turns out that the case first Chern class = 4 is much more involved than the previous cases, especially on the projective 3-space. Among the bundles appearing in our classification one can find the Sasakura rank 3 vector bundle on the proje…
We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an algebraic $K$-theory and depends on the canonical $\mathbb{Z}$-torsor of a locally linearly compact $k$-vector space. Analogs of certain auxiliary results for the case of an arithmetic surface are also discussed.
We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi-Yau threefolds.
For a stable curve of genus $g\geq 2$ and simple Lie algebra of type A or C, we show that the conformal blocks algebra $\mathcal{A}$ on $\overline{\mathcal{M}}_g$ is finitely generated and establish an explicit connection to Schmitt and Muñoz-Castañeda's compactification of the moduli space of $G$-bundles.
We study the positivity of the first Chern class of a rank r Ulrich vector bundle E on a smooth n-dimensional variety $X \subseteq \mathbb P^N$. We prove that $c_1(E)$ is very positive on every subvariety not contained in the union of lines in X. In particular if X is not covered by lines, then E is big and $c_1(E)^n \ge r^n$. Moreover we classify rank r Ulrich vector bundles E with $c_1(E)^2=0$ on surfaces and with $c_1(E)^2=0$ or $c_1(E)^3=0$ on threefolds (with some exceptions).
Chern classes for representations of reductive groups
设G是复连通约化群.表示环R（G）允许一个用正则结构定义的正则滤子.我们计算了相关的分次环gr R（G）（在Q上）和一个表示的Chern类-一个简单的练习，我们一直无法找到在文献中。作为应用，我们给出了gr K（B）中主丛P --> B的特征类的简单定义，以及计算相应向量丛的Chern类的简单方法.
This article is concerned with Chern class and Chern number inequalities on polarized manifolds and nef vector bundles. For a polarized pair $(M,L)$ with $L$ very ample, our first main result is a family of sharp Chern class inequalities. Among them the first one is a variant of a classical result and the equality case of the second one is a characterization of hypersurfaces. The second main result is a Chern number inequality on it, which includes a reverse Miyaoka-Yau type inequality. The third main result is that the Chern numbers of a nef vector bundle over a compact Kähler manifold are bo…
Chern Classes of Bundles over Rational Surfaces
考虑有理曲面$X$在一点处的爆破$π: \widetilde{X} \to X$。设$\widetilde{V}$是$\widetilde{X}$上的一个全纯丛，其例外因子的限制等于${\cal{O}(j) \oplus {\cal O}(-j)$，并定义$V =(π_*\widetilde{V})^{\vee \vee}.$ Friedman和Morgan给出了第二类$j \leq c_2(\widetilde{V}) - c_2(V) \leq j^2.$的如下界，我们证明了这些界是尖锐的。
Chern Classes of Toric Variety Bundles
本文解决了Khovanskii--Monin猜想，给出环簇丛切丛全陈类用基空间与环簇纤维陈类表示的公式，并推广至稳定复拟环流形，应用于环面螺旋球簇及Masuda公式的新证明。
We prove a result of Chern-Weil type for canonically metrized line bundles on one-parameter families of smooth complex curves. Our result generalizes a result due to J.I. Burgos Gil, J. Kramer and U. Kühn that deals with a line bundle of Jacobi forms on the universal elliptic curve over the modular curve with full level structure, equipped with the Petersson metric. Our main tool, as in the work by Burgos Gil, Kramer and Kühn, is the notion of a b-divisor.
Chern numbers on positive vector bundles and combinatorics
本文利用组合方法研究丰富与 nef 向量丛的陈数，给出有效下界并证明反向优势序，还提出并验证半正切丛流形上的同时正性猜想。
Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of Aluffi and Wu-Zhou. The first application of our formula is a geometric description of Chern-Mather classes of an arbitrary very affine variety, generalizing earlier results of Huh which held under the smooth and schon assumptions. As the second application, we confirm an involution formula relating sectional maximum likelihood (ML) degrees and ML bidegrees, whi…
We compute an explicit formula for the first Chern class of the Hodge Bundle over the space of admissible cyclic $\mathbb{Z}/3\mathbb{Z}$ covers of $n$-pointed rational stable curves as a linear combination of boundary strata. We then apply this formula to give a recursive formula for calculating certain Hodge integrals containing $λ_1$. We also consider covers with a ${\mathbb{Z}}/{2\mathbb{Z}}$ action for which we compute $λ_2$ as a linear combination of codimension two boundary strata.
We construct Chern classes of vector bundles in the universal precobordism theory of Annala--Yokura over an arbitrary Noetherian base ring of finite Krull dimension. As an immediate corollary of this, we show that the Grothendieck ring of vector bundles can be recovered from the universal precobordism ring, and that we can construct candidates for Chow rings satisfying an analogue of the classical Grothendieck--Riemann--Roch theorem. We also strengthen the weak projective bundle formula of Annala--Yokura to work for arbitrary projective bundles.
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