THE BOGOMOLOV–MIYAOKA–YAU INEQUALITY FOR LOG CANONICAL SURFACES

2001 ◽  
Vol 64 (2) ◽  
pp. 327-343 ◽  
Author(s):  
ADRIAN LANGER
Keyword(s):  
Vector Bundles ◽  
Universal Cover ◽  
Kodaira Dimension ◽  
Projective Surface ◽  
Chern Classes ◽  
Singular Surfaces ◽  
Smooth Projective Surface ◽  
Log Canonical ◽  
Surface Singularities ◽  
Negative Kodaira Dimension

Let X be a smooth projective surface of non-negative Kodaira dimension. Bogomolov [1, Theorem 5] proved that c21 [les ] 4c2. This was improved to c21 [les ] 3c2 by Miyaoka [12, Theorem 4] and Yau [19, Theorem 4]. Equality c21 [les ] 3c2 is attained, for example, if the universal cover of X is a ball (if κ(X) = 2 then this is the only possibility). Further generalizations of inequalities for Chern classes for some singular surfaces with (fractional) boundary were obtained by Sakai [16, Theorem 7.6], Miyaoka [13, Theorem 1.1], Kobayashi [6, Theorem 2; 7, Theorem 12], Wahl [18, Main Theorem] and Megyesi [10, Theorem 10.14; 11, Theorem 0.1].In [8] we introduced Chern classes of reflexive sheaves, using Wahl's local Chern classes of vector bundles on resolutions of surface singularities. Here we apply them to obtain the following generalization of the Bogomolov–Miyaoka–Yau inequality.

scholarly journals On the Connectedness of the Real Part of Moduli Spaces of Vector Bundles on Real Algebraic Surfaces

2000 ◽  
Vol 68 (1) ◽  
pp. 41-54
Author(s):  
Edoardo Ballico
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Algebraic Surfaces ◽  
Stable Rank ◽  
Projective Surface ◽  
Chern Classes ◽  
The Real ◽  
Smooth Projective Surface ◽  
Real Algebraic Surfaces

AbstractLet X be a smooth projective surface with q(X) = 0 defined over R and M(X;r;c1;c2;H) the moduli space of H-stable rank r vector bundles on X with Chern classes c1 and c2. Assume either r = 3 and X(R) connected or r = 3 and X(R) =ø or r=2 and X(R) = ø. We prove that quite often M is connected.

A CRITERION FOR GETTING A BIG COMPONENT OF THE MODULI OF VECTOR BUNDLES BY CHANGING A POLARIZATION

2001 ◽  
Vol 12 (08) ◽  
pp. 927-942
Author(s):  
KIMIKO YAMADA
Keyword(s):  
Vector Bundles ◽  
Line Bundles ◽  
Closed Field ◽  
Projective Surface ◽  
Chern Classes ◽  
Algebraically Closed Field ◽  
Stable Vector Bundles ◽  
Normal Surfaces ◽  
Smooth Projective Surface ◽  
Moduli Of Vector Bundles

Let MH(c1, c2) be a coarse moduli scheme parameterizing all rank-two H-μ-stable vector bundles with Chern classes (c1, c2) on a smooth projective surface X over an algebraically closed field. For fixed two ample line bundles H and H′, it is known that if c2 is greater than some constant p(X, H, H′) depending on H and H′ then MH(c1, c2) and MH′(c1, c2) are birationally equivalent. In this paper we show that this constant p(X, H, H′) generally does depends on the choice of H and H′. More precisely, we give some example of surface (and c1) on which, for any number K, there exists an integer c2 with c2≥K such that sup H: ample dim MH(c1, c2) = + ∞. This result is available also for normal surfaces.

scholarly journals A Lower Bound For KxL Of Quasi-Polarized Surfaces (X, L) With Non-Negative Kodaira Dimension

1998 ◽  
Vol 50 (6) ◽  
pp. 1209-1235 ◽  
Author(s):  
Yoshiaki Fukuma
Keyword(s):  
Lower Bound ◽  
Number Field ◽  
Complex Number ◽  
Kodaira Dimension ◽  
Projective Surface ◽  
Smooth Projective Surface ◽  
Complex Number Field ◽  
Negative Kodaira Dimension

AbstractLet X be a smooth projective surface over the complex number field and let L be a nef-big divisor on X. Here we consider the following conjecture; If the Kodaira dimension ≥ 0, then KXL ≥ 2q(X) - 4, where q(X) is the irregularity of X. In this paper, we prove that this conjecture is true if (1) the case in which = 0 or 1, (2) the case in which = 2 and h0(L) ≥ 2, or (3) the case in which = 2, X is minimal, h0(L) = 1, and L satisfies some conditions.

scholarly journals On vector bundles on algebraic surfaces and 0-cycles

1993 ◽  
Vol 130 ◽  
pp. 19-23 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Algebraic Structure ◽  
Vector Bundles ◽  
Algebraic Surfaces ◽  
Projective Surface ◽  
Classical Theorem ◽  
Chern Classes ◽  
Complex Projective Surface ◽  
Holomorphic Structure ◽  
Rank 2

Let X be an algebraic complex projective surface equipped with the euclidean topology and E a rank 2 topological vector bundle on X. It is a classical theorem of Wu ([Wu]) that E is uniquely determined by its topological Chern classes . Viceversa, again a classical theorem of Wu ([Wu]) states that every pair (a, b) ∈ (H (X, Z), Z) arises as topological Chern classes of a rank 2 topological vector bundle. For these results the existence of an algebraic structure on X was not important; for instance it would have been sufficient to have on X a holomorphic structure. In [Sch] it was proved that for algebraic X any such topological vector bundle on X has a holomorphic structure (or, equivalently by GAGA an algebraic structure) if its determinant line bundle has a holomorphic structure. It came as a surprise when Elencwajg and Forster ([EF]) showed that sometimes this was not true if we do not assume that X has an algebraic structure but only a holomorphic one (even for some two dimensional tori (see also [BL], [BF], or [T])).

scholarly journals FUNDAMENTAL GROUP OF SOME GENUS-2 FIBRATIONS AND APPLICATIONS

2012 ◽  
Vol 23 (08) ◽  
pp. 1250080 ◽  
Author(s):  
R. V. GURJAR ◽  
SAGAR KOLTE
Keyword(s):  
Fundamental Group ◽  
Elliptic Curve ◽  
Universal Cover ◽  
Projective Surface ◽  
Smooth Projective Surface ◽  
Shafarevich Conjecture ◽  
Genus 2 ◽  
Holomorphic Convexity

We will prove that given a genus-2 fibration f : X → C on a smooth projective surface X such that b1(X) = b1(C) + 2, the fundamental group of X is almost isomorphic to π1(C) × π1(E), where E is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces X with genus-2 fibration X → C such that b1(X) > b1(C).

scholarly journals Universal polynomials for singular curves on surfaces

2014 ◽  
Vol 150 (7) ◽  
pp. 1169-1182 ◽  
Author(s):  
Jun Li ◽  
Yu-jong Tzeng
Keyword(s):  
Vector Bundle ◽  
Linear System ◽  
Line Bundle ◽  
Vector Bundles ◽  
Projective Surface ◽  
Analytic Singularity ◽  
Smooth Projective Surface ◽  
Chern Numbers ◽  
Cutting Out ◽  
Curves On Surfaces

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a complex smooth projective surface and $L$ be a line bundle on $S$. For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system $|L|$ with prescribed singularities is a universal polynomial of Chern numbers of $L$ and $S$, assuming $L$ is sufficiently ample. More generally, we show for vector bundles of any rank and smooth varieties of any dimension, similar universal polynomials also exist and equal the number of singular subvarieties cutting out by sections of the vector bundle. This work is a generalization of Göttsche’s conjecture.

scholarly journals Degeneracy loci, virtual cycles and nested Hilbert schemes II

2020 ◽  
Vol 156 (8) ◽  
pp. 1623-1663
Author(s):  
Amin Gholampour ◽  
Richard P. Thomas
Keyword(s):  
Chern Class ◽  
Vector Bundles ◽  
Projective Surface ◽  
Hilbert Schemes ◽  
Counting Problems ◽  
Degeneracy Loci ◽  
Smooth Projective Surface ◽  
Witten Theory

We express nested Hilbert schemes of points and curves on a smooth projective surface as ‘virtual resolutions’ of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa–Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom–Porteous-like Chern class formulae.

scholarly journals A Construction of Chern Classes of Parabolic Vector Bundles

2013 ◽  
Vol 42 (3) ◽  
pp. 1111-1122 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon
Keyword(s):  
Vector Bundles ◽  
Chern Classes

scholarly journals Chern classes of tensor products

2016 ◽  
Vol 27 (10) ◽  
pp. 1650079 ◽  
Author(s):  
Laurent Manivel
Keyword(s):  
Vector Bundles ◽  
Tensor Products ◽  
Chern Classes ◽  
Explicit Formulas

We prove explicit formulas for Chern classes of tensor products of virtual vector bundles, whose coefficients are given by certain universal polynomials in the ranks of the two bundles.

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