Vector bundles on a three dimensional neighborhood of a ruled surface

直纹曲面上的高秩优先丛及其整体截面

Higher rank prioritary bundles on ruled surfaces and their global sections

本文研究亏格非负曲线的直纹曲面，构造任意秩的简单优先向量丛，并给出其整体截面模空间维数的有效界。

Moduli spaces of vector bundles on a singular rational ruled surface

We study moduli spaces $M_X(r,c_1,c_2)$ parametrizing slope semistable vector bundles of rank $r$ and fixed Chern classes $c_1, c_2$ on a ruled surface whose base is a rational nodal curve. We show that under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases. We show that for a rational ruled surface defined over real numbers, the moduli space $M_X(r,c_1,c_2)$ is rational as a variety defined over $\mathbb R$.

直纹曲面爆破的切丛的丰沛性

Bigness of tangent bundles of blow-ups of ruled surfaces

研究直纹曲面\(S = \mathbb{P}_C(\mathcal{E})\)在点爆破下切丛丰沛性的保持情况，通过第一塞格雷不变量\(s_1(\mathcal{E})\)给出爆破点数的精确界。

Stable Higgs Bundles on ruled surfaces

Let $π: X = \mathbb{P}_C(E) \longrightarrow C$ be a ruled surface over an algebraically closed field $k$ of characteristic 0, with a fixed polarization $L$ on $X$. In this paper, we show that pullback of a (semi)stable Higgs bundle on $C$ under $π$ is a $L$-(semi)stable Higgs bundle. Conversely, if $(V,θ)$ is a $L$-(semi)stable Higgs bundle on $X$ with $c_1(V)= π^*(\bf d \rm)$ for some divisor $\bf d \rm $ of degree $d$ on $C$ and $c_2(V)=0$, then there exists a (semi)stable Higgs bundle $(W,ψ)$ of degree $d$ on $C$ whose pullback under $π$ is isomorphic to $(V,θ)$. As a consequence, we get an…

Elliptic ruled surfaces over arbitrary characteristic fields

Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically closed field of any characteristic. On the other hand, a rank $2$ vector bundle on $E$ defines a surface $S$ with a $\mathbb{P}^1$-bundle structure on $E$. We study when $S$ has an elliptic fibration according to the Atiyah's classification, and what kinds of singular fibers appear.

$\mathbb{P}^1$-束 over 几何射影曲面的自同构群

Automorphism groups of $\mathbb{P}^1$-bundles over geometrically ruled surfaces

本文研究了在非理几何射影曲面上的$\mathbb{P}^1$-束$(X,π)$的自同构群$\mathrm{Aut}^\circ(X)$，并证明其在包含于$\mathrm{Bir}(X/S)$中的情况下是最大化的。

非极小直纹曲面上的秩二向量丛

Rank-two vector bundles on non-minimal ruled surfaces

本文通过引入两个数值不变量将稳定秩二向量丛表示为自然扩张，研究非极小直纹曲面上模空间的双有理几何，并证明有理曲面上模空间的不可约分支均为有理或稳定有理。

Second Infintesimal Neighborhoods of Projective Bundle Sections

This short paper shows that a projective bundle over a smooth variety with a section can be uniquely recovered from the embedding of the section in its second infinitesimal neighborhood within the bundle.

Remarks on Ruled surfaces and rank two bundles with canonical determinant and 4 sections

Let $C$ be a smooth irreducible complex projective curve of genus $g$ and let $B^k(2,K_C)$ be the Brill-Noether loci parametrizing classes of (semi)-stable vector bundles $E$ of rank two with canonical determinant over $C$ with $h^0(C,E)\geq k$. We show that $B^4(2,K_C)$ it has an irreducible component $\mathcal B$ of dimension $3g-13$ on a general curve $C$ of genus $g\geq 8$. Moreover, we show that for the general element $[E]$ of $\mathcal B$, $E$ fits in a exact sequence $0\to\mathcal O_C(D)\to E\to K_C(-D)\to 0$, with $D$ a general effective divisor of degree three, and the corresponding …

On the existence of Ulrich bundles on geometrically ruled surfaces

Let $S$ be a geometrically ruled surface with invariant $e$ on a curve $C$. We deal with Ulrich line bundles and $μ$-stable special Ulrich bundles of rank $2$ on $S$ when $e\ge0$, slightly extending a recent result due to M. Aprodu, L. Costa and R.M. Miró-Roig. If $C$ is elliptic, we also prove that $S$ always supports Ulrich bundles of rank at most $2$, without any restriction on $e$. Finally, we show that in many cases $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$.

关于椭圆挠面稳定sheaves模空间的注记

A note on the moduli of stable sheaves on elliptic ruled surfaces

研究椭圆挠面带有 nef 抗canonical 包含的稳定sheaves模空间非空性，采用模空间理论与稳定性条件分析。

具有伪有效切丛的光滑射影曲面

Smooth projective surfaces with pseudo-effective tangent bundles

该数学代数几何研究刻画了非单直纹光滑射影曲面切丛伪有效的等价条件，并探讨了非有理直纹曲面吹胀的相关性质。

Irreducibility of Moduli Spaces of Vector Bundles on Birationally Ruled Surfaces

Let $S$ be a birationally ruled surface. We show that the moduli schemes $M_S(r,c_1,c_2)$ of semistable sheaves on $S$ of rank $r$ and Chern classes $c_1$ and $c_2$ are irreducible for all $(r,c_1,c_2)$ provided the polarization of $S$ used satisfies a simple numerical condition. This is accomplished by proving that the stacks of prioritary sheaves on $S$ of fixed rank and Chern classes are smooth and irreducible.

Extremal Metrics On Ruled Manifolds

In this paper, we consider a compact Kahler manifold with extremal Kahler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kahler metrics on the projectivization of the dual vector bundle.

直纹曲面上稳定向量丛的 Brill-Noether 理论

Brill-Noether Theory of stable vector bundles on ruled surfaces

本文研究直纹曲面上 H-稳定秩 2 向量丛模空间的 Brill-Noether 轨迹，给出其非空时 k 值在多数情形下的精确界。

Stable vector bundles on algebraic surfaces

We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of certain moduli space of stable bundles on a rational ruled surface.

On Higher Syzygies of ruled varieties over a curve

For a vector bundle $\mathcal{E}$ of rank $n+1$ over a smooth projective curve $C$ of genus $g$, let $X=¶_C (\mathcal{E})$ with projection map $π:X\to C$. In this paper we investigate the minimal free resolution of homogeneous coordinate rings of $X$. We first clarify the relations between higher syzygies of very ample line bundles on $X$ and higher syzygies of Veronese embedding of fibres of $π$ by the same line bundle. More precisely, letting $H = \mathcal{O}_{¶_C (\mathcal{E})} (1)$ be the tautological line bundle, we prove that if $(¶^n,\mathcal{O}_{¶^n} (a))$ satisfies Property $N_p$, the…

Vector Bundles on a Neighborhood of an Exceptional Curve and Elementary Transformations

Let W be the germ of a smooth complex surface around an exceptional curve and let E be a rank 2 vector bundle on W. We study the cohomological properties of a finite sequence $E_i, 1 \leq i \leq t$ of rank 2 vector bundles canonically associated to E. We calculate numerical invariants of E in terms of the $E_i.$ If S is a compact complex smooth surface and E is a rank 2 bundle on the blow- up of S at a point, we show that all values of $c_2(E)-c_2(π_* E^{\vee\vee})$ that are numerically possible are actually attained.

Obstruction bundles, semiregularity, and Seiberg-Witten invariants

We compare the deformation theory and the analytic structure of the Seiberg-Witten moduli spaces of a Kähler surface to the corresponding components of the Hilbert scheme, and show that they are isomorphic. Next we show how to compute the invariant in case the moduli space is smooth but not of the expected dimension, and apply this study to elliptic surfaces. Finally we discuss ruled surfaces, both products and more general ruled surfaces. For product ruled surfaces we relate the infinitesimal structure of the moduli spaces to Brill-Noether theory and compute the invariant in special cases. Fo…

Formal classication of two-dimensional neighborhoods of genus g $\ge$ 2 curves with trivial normal bundle

In this paper we study the formal classication of two-dimensional neighborhoods of genus g $\ge$ 2 curves with trivial normal bundle. We first construct formal foliations on such neighborhoods with holonomy vanishing along many loops, then give the formal/analytic classication of neighborhoods equipped with two foliations, and finally put this together to obtain a description of the space of neighborhoods up to formal equivalence.

On higher syzygies of ruled surfaces

We study higher syzygies of a ruled surface $X$ over a curve of genus $g$ with the numerical invariant $e$. Let $L \in {Pic}X$ be a line bundle in the numerical class of $aC_0 +bf$. We prove that for $0 \leq e \leq g-3$, $L$ satisfies property $N_p$ if $a \geq p+2$ and $b-ae \geq 3g-1-e+p$ and for $e \geq g-2$, $L$ satisfies property $N_p$ if $a \geq p+2$ and $b-ae\geq 2g+1+p$. By using these facts, we obtain Mukai type results. For ample line bundles $A_i$, we show that $K_X + A_1 + ... + A_q$ satisfies property $N_p$ when $0 \leq e < \frac{g-3}{2}$ and $q \geq g-2e+1 +p$ or when $e \geq \fra…

Obstructions of extension of vector bundles

In the holomorphic or algebraic setting we consider a vector bundle E on a smooth subvariety X in a smooth variety Y over a field of characteristic zero. Assuming E extends to the l-th neighborhood of X in Y, we study cohomological obstructions to extending it further to the k-th neighborhood, for k > l.

An elliptic conic bundle in P^4 arising from a stable rank-3 vector bundle

In this note we show the existence of a family of elliptic conic bundles in P^4 of degree 8. This family has been overlooked and in fact falsely ruled out in a series of classification papers. Our surfaces provide a counterexample to a conjecture of Ellingsrud and Peskine. According to this conjecture there should be no irregular m-ruled surface in P^4 for m at least 2.

Multi-point Seshadri constants on ruled surfaces

Let $X$ be a surface and let $L$ be an ample line bundle on $X$. We first obtain a lower bound for the Seshadri constant $\varepsilon(X,L,r)$, when $r \ge 3$. We then assume that $X$ is a ruled surface and study Seshadri constants on $X$ in greater detail. We also make precise computations of Seshadri constants on $X$ in some cases.

直纹曲面爆炸上的 Seshadri 常数与负曲线

Seshadri constants and negative curves on blowups of ruled surfaces

本文研究 Hirzebruch 曲面及可分解直纹曲面在非常一般点爆炸后的 Seshadri 常数，并给出相关有界负性结果。

Topological rigidity of algebraic $¶_3$-bundles over curves

A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus $g\ge 1$ is itself a ruled surface over a curve of genus $g$. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case $g\ge 2$.

Higher Syzygies of Elliptic Ruled Surfaces

Let L be a normally generated line bundle on X; we say L satisfies property N_p (notation after Mark Green) if the matrices in the free resolution of R (the homogeneous coordinate ring of X) over S (the homogeneous coordinate ring of the projective space corresponding to the complete linear series |L|) have linear entries until the p-th stage. In this article we prove the following result: Let X be an elliptic ruled surface and let L be a product of p+1 base point free and ample line bundles on X. Then L satisfies property N_p. In particular we prove that numerical classes of all divisors whic…

On metrics with minimal singularities of line bundles whose stable base loci admit holomorphic tubular neighborhoods

We investigate the minimal singularities of metrics on a big line bundle $L$ over a projective manifold when the stable base locus $Y$ of $L$ is a submanifold of codimension $r\geq 1$. Under some assumptions on the normal bundle and a neighborhood of $Y$, we give a explicit description of the minimal singularity of metrics on $L$. We apply this result to study a higher (co-)dimensional analogue of Zariski's example, in which the line bundle $L$ is not semi-ample, however it is nef and big.

On the singularities of effective loci of line bundles

We prove that every irreducible component of semi-regular loci of effective line bundles in the Picard scheme of a smooth projective variety has at worst rational singularities. This generalizes Kempf's result on rational singularities of $W^0_d$ for smooth curves. We also work out an example of such loci for a ruled surface.

关于紧致凯勒流形上解析子集邻域上全纯线丛的形式原理

Formal principle for line bundles on neighborhoods of an analytic subset of a compact Kähler manifold

本文研究了紧致凯勒流形上解析子集邻域全纯线丛的形式原理，通过上同调准则和构造反例揭示了形式原理在族中的不稳定性。

Moduli spaces of framed sheaves on certain ruled surfaces over elliptic curves

Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the group ${\mathbf C}^*$, and we determine its fixed-point set, which leads in some special cases to explicit formulas for the rational homology of the moduli space.

Two-dimensional moduli spaces of irregular Higgs bundles

We give a complete description of the two-dimensional moduli spaces of stable Higgs bundles of rank 2 over complex projective line with one irregular singular point, having a regular leading-order term, and endowed with a generic compatible parabolic structure such that the parabolic degree of the Higgs bundle is 0. Our method relies on elliptic fibrations of the rational elliptic surface, an equivalence of categories between irregular Higgs bundles and some sheaves on a ruled surface, and an analysis of stability conditions.

椭圆曲线上的直纹曲面的典范扩张

Canonical extensions of ruled surfaces over elliptic curves

本文研究椭圆曲线上负次数线丛对应的直纹曲面，发现其切丛的对称张量环非有限生成，并构造全纯函数证明其典范扩张非 Stein 流形。

Blowups of surfaces and moduli of holomorphic vector bundles

We examine the moduli of framed holomorphic bundles over the blowup of a complex surface, by studying a filtration induced by the behavior of the bundles on a neighborhood of the exceptional divisor.

直纹曲面爆破上 Seshadri 常数的有理性

Rationality of Seshadri constants on blow-ups of ruled surfaces

本文研究直纹曲面爆破上的 Seshadri 常数，提出分类负自交曲线的猜想，并在假设该猜想成立时构造出具有无理 Seshadri 常数的丰富线丛。

Nef cones of projective bundles over surfaces and Seshadri constants

In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over some special surfaces $X$ and for some special bundles on $X$. As an application, we also calculate the Seshadri constants of semistable ample vector bundles with vanishing discriminant on some special ruled surfaces at special points.

关于基本形式与密切丛

On fundamental forms and osculating bundles

本文利用主部层定义射影代数簇的高阶基本形式与密切空间，证明第m个基本形式是第(m-1)个高斯映射的微分，并借助第二基本形式和富比尼三次式讨论曲面为直纹曲面的条件。

ADE-bundles over rational surfaces, configuration of lines and rulings

To each del Pezzo surface (resp. ruled surface, ruled surface with a section), we describe a natural Lie algebra bundle of type E_n (resp. D_n, A_n) over it. Using lines and rulings on any such surface, we describe various representation bundles corresponding to fundamental representations of the corresponding Lie algebra. When we specify a geometric structure on the surface to reduce the Lie algebra to a smaller one, then the classical geometry of the configuration of lines and rulings is encoded beautifully by the branching rules in Lie theory. We discuss this relationship in details. When w…

Poisson surfaces and algebraically completely integrable systems

One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there is one case of a ruled surface that does not seem to be covered. This is the case of projective bundle associated to the first jet bundle of a topologically nontrivial line bundle. We give the integrable system corresponding to this surface; it turns out to be a deformation of the Hitchin system.

An explicit construction of ruled surfaces

The main goal of this paper is to give a general method to compute (via computer algebra systems) an explicit set of generators of the ideals of the projective embeddings of some ruled surfaces, namely projective line bundles over curves such that the fibres are embedded as smooth rational curves. Indeed, although the existence of the embeddings that we consider is well known, often in literature there are no explicit descriptions of the corresponding projective ideals. Such an explicit description allows to compute, besides all the syzygies, some of the important algebraic invariants of the s…

Non-commutative Mori contractions and $\PP^1$-bundles

We give a method for constructing maps from a non-commutative scheme to a commutative projective curve. With the aid of Artin-Zhang's abstract Hilbert schemes, this is used to construct analogues of the extremal contraction of a $K$-negative curve with self-intersection zero on a smooth projective surface. This result will hopefully be useful in studying Artin's conjecture on the birational classification of non-commutative surfaces. As a non-trivial example of the theory developed, we look at non-commutative ruled surfaces and, more generally, at non-commutative $\PP^1$-bundles. We show in pa…

Characterization of Ulrich bundles on Hirzebruch surfaces

In this work we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over $\mathbb{P}^1$. We show that every Ulrich bundle admits a resolution in terms of line bundles. Conversely, given an injective map between suitable totally decomposed vector bundles, we show that its cokernel is Ulrich if it satisfies a vanishing in cohomology. As a consequence we obtain, once we fix a polarization, the existence of Ulrich bundles for any admissible rank and first Chern class. Moreover we show the existence of stable Ulrich bundles for certain pairs $(\textrm{rk}(E),c_1(E))$ and wi…

On neighborhoods of embedded complex tori

The goal of the article is to show that an n-dimensional complex torus embedded in a complex manifold of dimensional n+d, with a split tangent bundle, has neighborhood biholomorphic a neighborhood of the zero section in its normal bundle, provided the latter has (locally constant) Hermitian transition functions and satisfies a non-resonant Diophantine condition.

Infinitesimal extensions of rank two vector bundles on submanifolds of small codimension

Let $X$ be a submanifold of dimension $n$ of the complex projective space $\mathbb P^N$ ($n<N$), and let $E$ be a vector bundle of rank two on $X$ . If $n\geq\frac{N+3}{2}\geq 4$ we prove a geometric criterion for the existence of an extension of $E$ to a vector bundle on the first order infinitesimal neighborhood of $X$ in $\mathbb P^N$ in terms of the splitting of the normal bundle sequence of $Y\subset X\subset\mathbb P^N$, where $Y$ is the zero locus of a general section of a high twist of $E$. In the last section we show that the universal quotient vector bundle on the Grassmann variety $…

Secant varieties and Hirschowitz bound on vector bundles over a curve

For a vector bundle V over a curve X of rank n and for each integer r in the range 1 \le r \le n-1, the Segre invariant s_r is defined by generalizing the minimal self-intersection number of the sections on a ruled surface. In this paper we generalize Lange and Narasimhan's results on rank 2 bundles which related the invariant s_1 to the secant varieties of the curve inside certain extension spaces. For any n and r, we find a way to get information on the invariant s_r from the secant varieties of certain subvariety of a scroll over X. Using this geometric picture, we obtain a new proof of the…

Minimal sections of conic bundles

Let the threefold X be a general smooth conic bundle over the projective plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this paper we prove the existence of two natural families C(+) and C(-) of curves on X, such that the Abel-Jacobi map F sends one of these families onto a copy of the theta divisor (Theta), and the other -- onto the jacobian J(X). The general curve C of any of these two families is a section of the conic bundle projection, and our approach relates such C to a maximal subbundle of a rank 2 vector bundle E(C) on C, or -- to a minimal section of the rule…

Relative K-polystability of projective bundles over a curve

Let $P(E)$ be the projectivization of a holomorphic vector bundle $E$ over a compact complex curve $C$. We characterize the existence of an extremal Kähler metric on the ruled manifold $P(E)$ in terms of relative K-polystability and the fact that $E$ decomposes as a direct sum of stable bundles.

Higher codimensional Ueda theory for a compact submanifold with unitary flat normal bundle

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher-codimensional generalization of Ueda's result, we give a sufficient condition for the existence of a non-singular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semi-positive curvature on a nef line bundle.

Stability of symmetric powers of vector bundles of rank two with even degree on a curve

This paper treats the strict semi-stability of the symmetric powers $S^k E$ of a stable vector bundle $E$ of rank $2$ with even degree on a smooth projective curve $C$ of genus $g \geq 2$. The strict semi-stability of $S^2 E$ is equivalent to the orthogonality of $E$ or the existence of a bisection on the ruled surface $\mathbb{P}_C(E)$ whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of $E$ with strictly semi-stable $S^3 E$. Moreover, it is shown that when…

The Dolbeault dga of the formal neighborhood of the diagonal

A well-known theorem of Kapranov states that the Atiyah class of the tangent bundle $TX$ of a complex manifold $X$ makes the shifted tangent bundle $TX[-1]$ into a Lie algebra object in the derived category $D(X)$. Moreover, he showed that there is an $L_\infty$-algebra structure on the Dolbeault resolution of $TX[-1]$ and wrote down the structure maps explicitly in the case when $X$ is Kähler. The corresponding Chevalley-Eilenberg complex is isomorphic to the Dolbeault resolution of the jet bundle $\mathcal{J}^\infty_X$ via the construction of the holomorphic exponential map of the Kähler man…

Donaldson invariants of product ruled surfaces and two-dimensional gauge theories

Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of $Σ_g \times S^2$, where $Σ_g$ is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabo for g=1 to any genus g. We give two applications of our results: (1) We derive Thaddeus' formulae for the intersection pairings on the moduli space of rank two stable bundles over a Riemann surface. (2) We derive the eigenvalue spectrum of the Fukaya-Floer cohomology of $Σ_g \times S^1$.

关于可分解直纹曲面上曲线覆盖的 Tschirnhausen 模及其应用

On the Tschirnhausen module of coverings of curves on decomposable ruled surfaces and applications

本文证明了可分解直纹曲面上两类 $m$-割曲线覆盖的 Tschirnhausen 模可完全分解为线丛直和，并据此计算了法丛全局截面空间维数，进而构造了曲线 Hilbert 概型的新分支族。

Instantons on cylindrical manifolds and stable bundles

Let Sigma be a smooth complex curve, and let S be the product ruled surface Sigma \times CP^1. We prove a correspondence conjectured by Donaldson between finite energy U(2)-instantons over the cylinder Sigma \times S^1 \times R, and rank 2 holomorphic bundles over S whose restrictions to the divisors at infinity are stable.

Conic bundles in projective fourspace

P. Ellia and G.Sacchiero have shown that if $S$ is a smooth surface in $\Pn 4$ which is ruled in conics, then $S$ has degree 4 or 5. In this paper we give a proof of this result combining the ideas of Ellia and Sacchiero as they are used in the paper of the second author on plane curve fibrations and the recent work of G. Fløystad and the first author bounding the degree of smooth surfaces in $\Pn 4$ not of general type.