Algebraic theory of characteristic classes of bundles with connection

Universal degeneracy classes for vector bundles on $\mathbb{P}^1$ bundles

Given a vector bundle on a $\mathbb{P}^1$ bundle, the base is stratified by degeneracy loci measuring the spitting type of the vector bundle restricted to each fiber. The classes of these degeneracy loci in the Chow ring or cohomology ring of the base are natural invariants characterizing the degenerations of the vector bundle. When these degeneracy loci occur in the expected codimension, we find their classes. This yields universal formulas for degeneracy classes in terms of naturally arising vector bundles on the base. Our results hold over arbitrary fields of any characteristic.

结点代数曲线上主$\Bbb C^*$-丛的模栈的上同调

Cohomology of Moduli Stacks of Principal $\Bbb C^*$-Bundles Over Nodal Algebraic Curves

研究结点曲线上主$\Bbb C^*$-丛模栈，用陈类确定其有理上同调代数。

Algebraic vector bundles on the 2-sphere and smooth rational varieties with infinitely many real forms

We construct smooth rational real algebraic varieties of every dimension $\ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.

Stability of syzygy bundles on an algebraic surface

Given a very ample line bundle L on a projective variety X, the syzygy bundle M_L associated to L is the kernel of the evaluation map on sections of L. Our main result is that if X is a smooth projective surface defined over an algebraically closed field, then M_L is slope-stable for any sufficiently positive L.

Stable real algebraic vector bundles over a Klein bottle

Let X be a geometrically connected smooth projective curve of genus one, defined over the field of real numbers, such that X does not have any real points. We classify the isomorphism classes of all stable real algebraic vector bundles over X.

A construction of Chern classes of parabolic vector 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.

An introduction to moduli stacks, with a view towards Higgs bundles on algebraic curves

This article is based in part on lecture notes prepared for the summer school "The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles" at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a "bootstrap" method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a…

Algebraic cobordism of varieties with G-bundles

Lee and Pandharipande studied a "double point" algebraic cobordism theory of varieties equipped with vector bundles, and speculated that some features of that story might extend to the case of varieties with principal G-bundles. This note shows that this expectation holds rationally, and more generally after inverting the torsion index of the group, for reductive G. We show that (after inverting the torsion index) the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism, that the theory for a point is dual to Omega^*(BG), and describe how the theory f…

Parabolic bundles in positive characteristic

Algebraic parabolic bundles on smooth projective curves over algebraically closed field of positive characteristic is defined. It is shown that the category of algebraic parabolic bundles is equivalent to the category of orbifold bundles defined in \cite{KP}. Tensor, dual, pullback and pushforward operations are also defined for parabolic bundles.

Characterization of generic projective space bundles and algebraicity of foliations

In this paper we consider various notions of positivity for distributions on complex projective manifolds. We start by analyzing distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. We classify foliations attaining this bound, and describe those whose algebraic rank slightly exceeds this bound.

The moduli space of stable vector bundles over a real algebraic curve

We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension Z/2 by the fundamental group. By comparison with the space of real or quaternionic connections, some of the basic topological invariants of these spaces are calculated.

Some Remarks on H-stability of syzygy bundle on algebraic surface

Let $L$ be a globally generated line bundle over a smooth irreducible complex projective surface $X$. The syzygy bundle $M_{L}$ is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_X\to L$. We prove the $L$-stability of $M_L$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. The $(-K_X)$-stability of syzygy bundles $M_L$ over del Pezzo surfaces is also obtained.

Maximally Frobenius-destabilized vector bundles over smooth algebraic curves

Vector bundles in positive characteristics have a tendency to be destabilized after pulling back by the Frobenius morphism. In this paper, we closely examine vector bundles over curves that are, in an appropriate sense, maximally destabilized by the Frobenius morphism. Then we prove that such bundles of rank 2 exist over any curve in characteristic 3, and are unique up to twisting by a line bundle. We also give an application of such bundles to the study of ample vector bundles, which is valid in all characteristics.

曲线上向量丛的代数算法

Algebraic algorithms for vector bundles over curves

本文提出用函数域极大序上的格对表示曲线上的向量丛，并给出计算行列式、同态核像、同构、上同调、扩张及不可分解直和分解的多项式时间代数算法，同时实现为SageMath包并应用于椭圆曲线向量丛分类与代数几何码构造。

Essentially Finite Vector Bundles on Normal Pseudo-proper Algebraic Stacks

Let $X$ be a normal, connected and projective variety over an algebraically closed field $k$. It is known that a vector bundle $V$ on $X$ is essentially finite if and only if it is trivialized by a proper surjective morphism $f:Y\to X$. In this paper we introduce a different approach to this problem which allows to extend the results to normal, connected and strongly pseudo-proper algebraic stack of finite type over an arbitrary field $k$.

关于代数曲线上主丛模栈的 Frobenius 态射作用

On actions of Frobenius morphisms for moduli stacks of principal bundles over algebraic curves

本文研究代数曲线上主丛模栈的算术与几何 Frobenius 态射，并利用 Chern 类明确其在 ℓ-adic 上同调上的作用。

Characteristic classes of flat bundles and determinant of the Gauss-Manin connection

The purpose of this note is to give a survey on recent progress on characteristic classes of flat bundles, and how they behave in a family.

光滑代数簇上半稳定丛的逼近

Approximation of semistable bundles on smooth algebraic varieties

本文证明了具有消失数值Chern类的强半稳定丛可由商为相似斜率线丛的过滤逼近，推广了曲线和曲面情形的早期结果并证实了相关猜想。

On algebraic Chern classes of flat vector bundles

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.

On higher Chern classes of vector bundles of conformal blocks

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 $\overline{\operatorname{M}}_{0,n}$.

簇簇集簇上的代数向量丛

Algebraic Vector Bundles on Cluster Varieties

本文研究簇簇集上的代数向量丛，证明特征不为2的完美域上特定簇上的代数向量丛平凡，构造了含不可分解秩2向量丛的光滑簇，给出稳定自由判据并计算相关上同调与闭链类映射像。

Semistable Higgs bundles and representations of algebraic fundamental groups: Positive characteristic case

Let $k$ be an algebraic closure of finite fields with odd characteristic $p$ and a smooth projective scheme $\mathbf{X}/W(k)$. Let $\mathbf{X}^0$ be its generic fiber and $X$ the closed fiber. For $\mathbf{X}^0$ a curve Faltings conjectured that semistable Higgs bundles of slope zero over $\mathbf{X}^0_{\mathbb{C}_p}$ correspond to genuine representations of the algebraic fundamental group of $\mathbf{X}^0_{\mathbb{C}_p}$ in his $p$-adic Simpson correspondence. This paper intends to study the conjecture in the characteristic $p$ setting. Among other results, we show that isomorphism classes of…

Geometry of Moduli Stacks of $(k,l)$-stable vector bundles over an algebraic curve

We study the geometry of the moduli stack of vector bundles of fixed rank and degree over an algebraic curve by introducing a filtration made of open substacks build from $(k, l)$-stable vector bundles. $(k, l)$-stability was introduced by Narasimhan and Ramanan to study the geometry of the coarse moduli space of stable bundles. We will exhibit the stacky picture and analyse the geometric and cohomological properties of the moduli stacks of $(k, l)$-stable vector bundles. For particular pairs $(k, l) $ of integers we also show that these moduli stacks admit coarse moduli spaces and we discuss …

Chern classes of rank two globally generated vector bundles on P^2

We determine the Chern classes of globally generated rank two vector bundles on P^2.

Comparison of stratified-algebraic and topological K-theory

Stratified-algebraic vector bundles on real algebraic varieties have many desirable features of algebraic vector bundles but are more flexible. We give a characterization of the compact real algebraic varieties having the following property: There exists a positive integer r such that for any topological vector bundle E on X, the direct sum of r copies of E is isomorphic to a stratified-algebraic vector bundle. In particular, each compact real algebraic variety of dimension at most 8 has this property. Our results are expressed in terms of K-theory.

Lagrangian subbundles of orthogonal bundles of odd rank over an algebraic curve

An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a Lagrangian subbundle. This invariant, and the induced stratifications on moduli spaces of orthogonal bundles, were studied for bundles of even rank in [4]. In this paper, we obtain analogous results for bundles of odd rank. We obtain a sharp upper bound on the isotropic Segre invariant. We show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions. As a key ingredient of the proofs, we study the correspondence between L…

Higgs bundles and characteristic classes

We calculate the characteristic classes for flat SL(n,R) and Sp(2m,R)-bundles over a compact surface as functions of the spectral data in the Higgs bundle description, which consists of the points of order 2 in an abelian variety. Using similar methods we produce a conjectural mirror for the moduli space of flat U(m,m)-bundles with a fixed characteristic class. This is a hyperholomorphic bundle over the Sp(2m,C)-moduli space.

Real algebraic morphisms on 2-dimensional conic bundles

Given two nonsingular real algebraic varieties V and W, we consider the problem of deciding whether a smooth map f: V -> W can be approximated by regular maps in the space of smooth maps from V to W. Our main result is a complete solution to this problem in case W is the usual 2-dimensional sphere and V is a real algebraic surface of negative Kodaira dimension.

On $(k,l)$-stable vector bundles over algebraic curves

In this paper, we study the $(k,l)$-stable vector bundles over non-singular projective curve $X$ of genus $g\geq 2,$ its relation with stability and Segre invariants. For rank 2 and 3, we give an explicit description and relation of $(k,l)$-stability and Brill-Noether loci.

On the algebraic holonomy of stable principal bundles

Apart from math.AG/0608569, it contains the following applications of it. Let M be a simply connected, irreducible smooth complex projective variety of dimension $n$ such that the Picard number of $M$ is one. If the canonical line bundle $K_M$ is ample, then the algebraic holonomy of $TM$ is $\text{GL}(n, {\mathbb C})$. If $K^{-1}_M$ is ample, $\text{rank}(\text{NS}(M)) = 1$, the biholomorphic automorphism group of $M$ is finite, and $M$ admits a Kähler--Einstein metric, then the algebraic holonomy of $TM$ is ${\rm GL}(n, {\mathbb C})$.

丛模空间上的泛陈类

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。

Divisor classes and the virtual canonical bundle for genus 0 maps

Some divisor class relations are proved and used to compute the Cartier divisor class of the virtual canonical bundle for genus 0 maps to a smooth target.

Bundles of rank 2 with small Clifford index on algebraic curves

In this paper, we remove a technical hypothesis from a previous paper. This allows us to construct stable bundles of rank 2 with Clifford index significantly smaller than the classical Clifford index on suitably chosen curves of any genus $\ge13$.

Exact sequence of semistable vector bundles on algebraic curves

Let X be a smooth complex projective curve of genus g bigger or equal to 1. If g is bigger than 1 assume further that X is either bielliptic or with general moduli. Under a natural condition on slopes, we prove that there exists a short exact sequence of semistable vector bundles with given ranks and degrees. The importance of the paper consists of the new method adopted for the proves.

Regulous vector bundles

Among recently introduced new notions in real algebraic geometry is that of regulous functions. Such functions form a foundation for the development of regulous geometry. Several interesting results on regulous varieties and regulous sheaves are already available. In this paper, we define and investigate regulous vector bundles. We establish algebraic and geometric properties of such vector bundles, and identify them with stratified-algebraic vector bundles. Furthermore, using new results on curve-rational functions, we characterize regulous vector bundles among families of vector spaces param…

Principal bundles, quasi-abelian varieties and structure of algebraic groups

We classify principal bundles over anti-affine schemes with affine and commutative structural group. We show that this yields the classification of quasi-abelian varieties over a field k (i.e., group k-schemes with no non constant global functions). The interest of this result is given by the fact that the classification of smooth group k-schemes is reduced to the classification of quasi-abelian varieties and of certain affine group schemes.

Chern classes and Gromov--Witten theory of projective bundles

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.

Chern classes of Gauß-Manin bundles of weight 1 vanish

We show that the Chern character of a variation of polarized Hodge structures of weight one with nilpotent residues at $\infty$ dies up to torsion in the Chow ring, except in codimension 0.

数值平坦 Higgs 丛的消失 Chern 类

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。

A Riemann-Roch theorem for flat bundles, with values in the algebraic Chern-Simons theory

Let $f: X \to S$ be flat morphism over an algebraically closed field $k$ with a relative normal crossings divisor $Y\subset X$, $(E, \nabla)$ be a bundle with a connection with log poles along $Y$ and curvature with values in $f^*Ω^2_{k(S)}$. Then the Gauß-Manin sheaf $R^if_*(Ω^*_{X/S}({\rm log} Y)\otimes E)$ carries a Gauß-Manin connection $GM^i(\nabla)$. We establish a Riemann-Roch formula relating the algebraic Chern-Simons invariants of $\nabla$, $GM^i(\nabla)$ and the top Chern class of $Ω^1_{X/S}({\rm log}Y)$.

Comparison of motivic Chern classes and stable envelopes for cotangent bundles

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.

自守向量丛的Chern类，II

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。

Rational curves and ampleness properties of the tangent bundle of algebraic varieties

The purpose of this paper is to translate positivity properties of the tangent bundle (and the anti-canonical bundle) of an algebraic manifold into existence and movability properties of rational curves and to investigate the impact on the global geometry of the manifold $X$. Among the results we prove are these: \quad If $X$ is a projective manifold, and ${\cal E} \subset T_X$ is an ample locally free sheaf with $n-2\ge rk {\cal E}\ge n$, then $X \simeq \EP_n$. \quad Let $X$ be a projective manifold. If $X$ is rationally connected, then there exists a free $T_X$-ample family of (rational) cur…

Cohomology of line bundles on horospherical varieties

A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties.

模栈的动机类及其在Higgs丛栈中的应用

Motivic classes of stacks in finite characteristic and applications to stacks of Higgs bundles

本文在有限特征下定义了栈的动机类环，并计算了半稳定Higgs丛的模栈的动机类，恢复了特征零和有限域上的相关结果，并给出了更简单的公式。

Real Algebraic Threefolds III: Conic Bundles

This is the third of a series of papers studying real algebraic threefolds, but the methods are mostly independent from the previous two. Let $f:X\to S$ be a map of a smooth projective real algebraic 3-fold to a surface $S$ whose general fibers are rational curves. Assume that the set of real points of $X$ is an orientable 3-manifold $M$. The aim of the paper is to give a topological description of $M$. It is shown that $M$ is either Seifert fibered or a connected sum of lens spaces. Much stronger results hold if $S$ is rational.

Holomorphic bundles on O(-k) are algebraic

We show that holomorphic bundles on O(-k) for k > 0 are algebraic. We also show holomorphic bundles on O(-1) are trivial outside the zero section.

Globally generated vector bundles on $\mathbf{P}^1 \times \mathbf{P}^1 \times \mathbf{P}^1$ with low first Chern classes

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.

On the positivity of high-degree Schur classes of an ample vector bundle

Let $X$ be a smooth projective variety of dimension $n$, and let $E$ be an ample vector bundle over $X$. We show that any non-zero Schur class of $E$, lying in the cohomology group of bidegree $(n-1, n-1)$, has a representative which is strictly positive in the sense of smooth forms. This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level, and thus strengthens the celebrated positivity results of Fulton-Lazarsfeld for certain degrees.

典范丛公式与Schnell关于某类代数纤维空间的猜想

Canonical bundle formula and a conjecture on certain algebraic fiber spaces by Schnell

本文利用典范丛公式，在两种独立假设下证明了Schnell关于代数纤维空间的猜想，其中一种假设涉及刚性流表示的典范类。

Equivariant principal bundles and logarithmic connections on toric varieties

Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$--orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic group. We prove that an algebraic principal $G$--bundle $E_G\to M$ admits a $T$--equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$--bundle, where $H$ is any complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection si…

Segre classes of tautological bundles on Hilbert schemes of surfaces

We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande on top Segre classes of the tautological bundles on Hilbert schemes of $K3$ surfaces equipped with a line bundle. We then turn to the blow-up of $K3$ surface at one point and establish vanishing results for the corresponding top Segre classes in a certain range. This determines, at least theoretically, all top Segre classes of tautological bundles for any pair $(Σ,H),\,H\in {\rm Pic}\,Σ$.

曲线上主丛模栈的线丛

The line bundles on moduli stacks of principal bundles on a curve

该研究针对代数闭域上仿射约化代数群的情形，确定了光滑射影曲线上主G-丛模栈的Picard群，推进了代数几何中模栈线丛的相关研究。

Algebraic Nahm transform for Parabolic Higgs Bundles on P^1

We study Nahm transformation for parabolic Higgs bundles on the projective line \PP^1, with logarithmic singularities on a finite set P. Such a Higgs bundle can be given by its spectral data: a Hirzebruch surface Z together with a coherent sheaf M on Z, supported in dimension 1 and away from infinity. We describe the transform in terms of these data. The main technical tool is the notion of proper transform of a coherent sheaf with respect to a blow-up. Finally, we prove the main properties of the induced map on moduli spaces: involutibility and preservation of the hyper-Kaehler structure.