Chern forms of singular metrics on vector bundles

In this paper, we present two kinds of total Chern forms [Formula: see text] and [Formula: see text] as well as a total Segre form [Formula: see text] of a holomorphic Finsler vector bundle [Formula: see text] expressed by the Finsler metric [Formula: see text], which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in Finsler Geometry, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133–144] to some extent. As some applications, we show that the signed Segre forms [Formula: see text] are positive [Formula: see text]-forms on [Formula: see text] when [Formula: see text] is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler–Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou’s one [Finsler geometry on complex vector bundles, in A Sampler of Riemann–Finsler Geometry, MSRI Publications, Vol. 50 (Cambridge University Press, 2004), pp. 83–105] and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.

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Banach Lie Algebroids

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.2478/v10157-011-0035-y ◽

2011 ◽

Vol 57 (2) ◽

pp. 409-416

Author(s):

Mihai Anastasiei

Keyword(s):

Differential Equations ◽

Smooth Manifold ◽

Vector Bundles ◽

Second Order ◽

Lie Algebroids ◽

Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

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A Stratification Theoretical Method of Construction of Holomorphic Vector Bundles

Complex Analytic Singularities ◽

10.2969/aspm/00810527 ◽

2018 ◽

Author(s):

Nobuo Sasakura

Keyword(s):

Vector Bundles ◽

Theoretical Method ◽

Holomorphic Vector Bundles

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Fano 3-folds from homogeneous vector bundles over Grassmannians

Revista Matemática Complutense ◽

10.1007/s13163-021-00401-2 ◽

2021 ◽

Author(s):

Lorenzo De Biase ◽

Enrico Fatighenti ◽

Fabio Tanturri

Keyword(s):

Vector Bundles ◽

Homogeneous Vector

AbstractWe rework the Mori–Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.

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EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s147474802100027x ◽

2021 ◽

pp. 1-66

Author(s):

Tom Bachmann ◽

Kirsten Wickelgren

Keyword(s):

Commutative Algebra ◽

Vector Bundles ◽

Euler Class ◽

Complete Intersections ◽

Bilinear Forms ◽

Euler Numbers ◽

Coherent Sheaves ◽

Linear Subspaces ◽

Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

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Higher rank Clifford indices of curves on a K3 surface

Selecta Mathematica ◽

10.1007/s00029-021-00664-z ◽

2021 ◽

Vol 27 (3) ◽

Author(s):

Soheyla Feyzbakhsh ◽

Chunyi Li

Keyword(s):

Vector Bundles ◽

Plane Curve ◽

Smooth Curve ◽

Line Bundles ◽

K3 Surface ◽

Clifford Index ◽

Smooth Plane ◽

Higher Rank ◽

Rank Vector ◽

Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

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THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

Nagoya Mathematical Journal ◽

10.1017/nmj.2020.37 ◽

2020 ◽

pp. 1-23

Author(s):

MICHELE BOLOGNESI ◽

NÉSTOR FERNÁNDEZ VARGAS

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Vector Bundles ◽

Hyperelliptic Curve ◽

Geometric Description ◽

Birational Model ◽

Rank 2 ◽

Semistable Vector Bundles ◽

Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

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