The order and the characteristic classes of a stable vector bundle

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

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Tangent cones to generalised theta divisors and generic injectivity of the theta map

Compositio Mathematica ◽

10.1112/s0010437x17007539 ◽

2017 ◽

Vol 153 (12) ◽

pp. 2643-2657 ◽

Cited By ~ 4

Author(s):

George H. Hitching ◽

Michael Hoff

Keyword(s):

Vector Bundle ◽

Tangent Cone ◽

Constructive Proof ◽

Tangent Cones ◽

Stable Vector Bundle ◽

Theta Divisor ◽

General Curve

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^{0}(C,E)=r+1$. For $g\geqslant (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the generalised theta divisor $\unicode[STIX]{x1D6E9}_{E}$ at ${\mathcal{O}}_{C}$. We use this to give a constructive proof and a sharpening of Brivio and Verra’s theorem that the theta map $\mathit{SU}_{C}(r){\dashrightarrow}|r\unicode[STIX]{x1D6E9}|$ is generically injective for large values of $g$.

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Analytic Evaluation of Certain Characteristic Classes(1)

Canadian Mathematical Bulletin ◽

10.4153/cmb-1973-038-5 ◽

1973 ◽

Vol 16 (2) ◽

pp. 219-223

Author(s):

Clark D. Jeffries

Keyword(s):

Vector Bundle ◽

Orthonormal Frame ◽

Alternative Proof ◽

Characteristic Classes ◽

Real Vector ◽

Analytic Evaluation ◽

Riemann Structure

We have an alternative proof of the following result of Kervaire [2]:Let V→M be a real vector bundle with fibre dimension n≥4k+l over a compact 4k-manifold. Suppose V restricted to M — {x} is trivial. Choose a Riemann structure for V and an orthonormal frame for V restricted to M —{x}. Thus the obstruction to extending the frame smoothly over M is an element λ in π4k+1SO(n))≅Z. Then up to sign the evaluation of the kth Pontrayagin class Pk on M is ak(2k— 1)!. λ, where ak is 1 or 2 depending upon whether k is even or odd.

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Stable Higgs bundles over positive principal elliptic fibrations

Complex Manifolds ◽

10.1515/coma-2018-0012 ◽

2018 ◽

Vol 5 (1) ◽

pp. 195-201

Author(s):

Indranil Biswas ◽

Mahan Mj ◽

Misha Verbitsky

Keyword(s):

Vector Bundle ◽

Line Bundle ◽

Complex Manifold ◽

Higgs Bundle ◽

Compact Complex Manifold ◽

Holomorphic Line Bundle ◽

Higgs Bundles ◽

Stable Vector Bundle ◽

Hermitian Metric ◽

The Third

AbstractLet M be a compact complex manifold of dimension at least three and Π : M → X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In [14], the third author proved that every stable vector bundle on M is of the form L⊕ Π ⃰ B0, where B0 is a stable vector bundle on X, and L is a holomorphic line bundle on M. Here we prove that every stable Higgs bundle on M is of the form (L ⊕ Π ⃰B0, Π ⃰ ɸX), where (B0, ɸX) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

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Ample vector bundles on a rational surface (higher rank)

Nagoya Mathematical Journal ◽

10.1017/s0027763000017724 ◽

1977 ◽

Vol 66 ◽

pp. 77-88

Author(s):

Toshio Hosoh

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Rational Surface ◽

Sufficient Condition ◽

Chern Classes ◽

Stable Vector Bundle ◽

Rank 2 ◽

Higher Rank

In the previous paper [1], we showed that the set of simple vector bundles of rank 2 on a rational surface with fixed Chern classes is bounded and we gave a sufficient condition for an H-stable vector bundle of rank 2 on a rational surface to be ample. In this paper, we shall extend the results of [1] to the case of higher rank.

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Ample vector bundles on a rational surface

Nagoya Mathematical Journal ◽

10.1017/s0027763000016846 ◽

1975 ◽

Vol 59 ◽

pp. 135-148 ◽

Cited By ~ 4

Author(s):

Toshio Hosoh

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Rational Surface ◽

Ruled Surface ◽

Necessary Condition ◽

Abelian Surface ◽

Closed Field ◽

Stable Vector Bundle ◽

Singular Curve ◽

Rank 2

On a complete non-singular curve defined over the complex number field C, a stable vector bundle is ample if and only if its degree is positive [3]. On a surface, the notion of the H-stability was introduced by F. Takemoto [8] (see § 1). We have a simple numerical sufficient condition for an H-stable vector bundle on a surface S defined over C to be ample; let E be an H-stable vector bundle of rank 2 on S with Δ(E) = c1(E)2 - 4c2(E) ≧ 0, then E is ample if and only if c1(E) > 0 and c2(E) > 0, provided S is an abelian surface, a ruled surface or a hyper-elliptic surface [9]. But the assumption above concerning Δ(E) evidently seems too strong. In this paper, we restrict ourselves to the projective plane P2 and a rational ruled surface Σn defined over an algebraically closed field k of arbitrary characteristic. We shall prove a finer assertion than that of [9] for an H-stable vector bundle of rank 2 to be ample (Theorem 1 and Theorem 3). Examples show that our result is best possible though it is not a necessary condition (see Remark (1) §2).

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A REMARK ON THE STABLE REAL FORMS OF COMPLEX VECTOR BUNDLES OVER MANIFOLDS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972717000132 ◽

2017 ◽

Vol 96 (1) ◽

pp. 69-76

Author(s):

HUIJUN YANG

Keyword(s):

Vector Bundle ◽

Smooth Manifold ◽

Vector Bundles ◽

Characteristic Classes ◽

Dimensional Manifold ◽

Real Form ◽

Complex Vector Bundle ◽

Complex Vector ◽

Simply Connected ◽

Real Forms

Let$M$be an$n$-dimensional closed oriented smooth manifold with$n\equiv 4\;\text{mod}\;8$, and$\unicode[STIX]{x1D702}$be a complex vector bundle over$M$. We determine the final obstruction for$\unicode[STIX]{x1D702}$to admit a stable real form in terms of the characteristic classes of$M$and$\unicode[STIX]{x1D702}$. As an application, we obtain the criteria to determine which complex vector bundles over a simply connected four-dimensional manifold admit a stable real form.

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Stability of tautological bundles on the Hilbert scheme of two points on a surface

Nagoya Mathematical Journal ◽

10.1215/00277630-2416410 ◽

2014 ◽

Vol 214 ◽

pp. 79-94 ◽

Cited By ~ 3

Author(s):

Malte Wandel

Keyword(s):

Vector Bundle ◽

Hilbert Scheme ◽

Projective Surface ◽

Stable Vector Bundle ◽

Free Sheaf ◽

Smooth Projective Surface ◽

Rank 2 ◽

Torsion Free

AbstractLet (X, H) be a polarized smooth projective surface satisfyingH1(Χ, OΧ) = 0, and letƑbe either a rank 1 torsion-free sheaf or a rank 2μH-stable vector bundle onΧ. Assume thatc1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheafƑ[2]associated withƑon the Hilbert squareΧ[2]isμ-stable with respect to a certain polarization.

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An Integral Formula for the Chern from of a Hermitian Bundle

Nagoya Mathematical Journal ◽

10.1017/s0027763000014306 ◽

1971 ◽

Vol 42 ◽

pp. 135-172 ◽

Cited By ~ 1

Author(s):

Hideo Omoto

Keyword(s):

Vector Bundle ◽

Chern Class ◽

Basic Characteristic ◽

Grassmann Manifold ◽

Integral Formula ◽

Characteristic Classes ◽

Characteristic Class ◽

Chern Classes ◽

Simplicial Decomposition ◽

Hermitian Bundle

We shall consider a Hermitian n-vector bundle E over a complex manifold X. When X is compact (without boundary), S.S. Chern defined in his paper [3] the Chern classes (the basic characteristic classes of E) Ĉi(E), i = 1, · · ·, n, in terms of the basic forms Φi on the Grassmann manifold H(n, N) and the classifying map f of X into H(n, N). Moreover he proved ([3], [4]) that if Ek denotes the k-general Stiefel bundle associated with E, the (n — k + 1)-th Chern class Ĉn-k+1(E) coincides with the characteristic class C(Ek) of Ek defined as follows: Let K be a simplicial decomposition of X and K2(n-k)+1 the 2(n — k) + 1 — shelton of K.

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Stable vector bundles on algebraic surfaces II

Nagoya Mathematical Journal ◽

10.1017/s0027763000015968 ◽

1973 ◽

Vol 52 ◽

pp. 173-195 ◽

Cited By ~ 16

Author(s):

Fumio Takemoto

Keyword(s):

Vector Bundle ◽

Line Bundle ◽

Vector Bundles ◽

Ample Line Bundle ◽

Algebraic Surfaces ◽

Complex Numbers ◽

Stable Vector Bundle ◽

Projective Varieties ◽

Stable Vector Bundles ◽

Direct Sum

This paper is a continuation of “Stable vector bundles on algebraic surfaces” [10]. For simplicity we deal with non-singular projective varieties over the field of complex numbers. Let W be a variety whose fundamental group is solvable, let H be an ample line bundle on W, and let f: V → W be an unramified covering. Then we show in section 1 that if E is an f*H-stable vector bundle on V then f*E is a direct sum of H-stable vector bundles. In particular f*L is a direct sum of simple vector bundles if L is a line bundle on V.

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