Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

In this paper, we define the virtual moduli cycle of moduli spaces with perfect tangent-obstruction theory. The two interesting moduli spaces of this type are moduli spaces of vector bundles over surfaces and moduli spaces of stable morphisms from curves to projective varieties. As an application, we define the Gromov-Witten invariants of smooth projective varieties and prove all its basic properties.

     

Grothendieck Groups of Bundles on Varieties over Finite Fields

Let X be an irreducible, projective variety over a finite field, and let A be a sheaf of rings on X. In this paper, we study Grothendieck groups of categories of vector bundles over certain types of ringed spaces (X,A).     

The Dolbeault complex in infinite dimensions I

In this paper we introduce certain basic notions concerning infinite dimensional complex manifolds, and prove that the Dolbeault cohomology groups of infinite dimensional projective spaces, with values in finite rank vector bundles, vanish. Some applications of such vanishing theorems are discussed; e.g., we classify vector bundles of finite rank over infinite dimensional projective spaces. Finally, we prove a sharp theorem on solving the inhomogeneous Cauchy-Riemann equations on affine spaces.

     

Bubble tree compactification of moduli spaces of vector bundles on surfaces

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c ₁ = 0, c ₁ = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.     

Syzygies using vector bundles

This paper studies syzygies of curves that have been embedded in projective space by line bundles of large degree. The proofs take advantage of the relationship between syzygies and spaces of section of vector bundles associated to the given line bundles.

     

On stable and uniform rank-2 vector bundles on P2 in characteristic p

The uniform rank-2 vector bundles on Pⁿ are determined and the behaviour of the stable rank-2 vector bundles on P² under restriction to a general line is studied, where Pⁿ denotes the n-dimensional projective space over an algebraically closed field of positive characteristic.     

A Universal Construction for Moduli Spaces of Decorated Vector Bundles over Curves

Let $X$ be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation $\rho\colon \mathop{\rm GL}(r)\rightarrow \mathop{\rm GL}(V)$. Then one can associate to every vector bundle $E$ of rank $r$ over $X$ a vector bundle $E_\rho$ with fibre $V$. We would like to study triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ over $X$, $L$ is a line bundle over $X$, and $\phi\colon E_\rho\rightarrow L$ is a nontrivial homomorphism. This setup comprises well known objects such as framed vector bundles, Higgs bundles, and conic bundles. In this paper, we will formulate a general (parameter dependent) semistability concept for such triples, which generalizes the classical Hilbert--Mumford criterion, and we establish the existence of moduli spaces for the semistable objects. In the examples which have been studied so far, our semistability concept reproduces the known ones. Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.     

Moduli of Vector Bundles on Curves in Positive Characteristics

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.     

非负曲率流形上的丛

本文主要证明了射影空间乃至齐性空间上各种向量丛的全空间可以赋予截面曲率非负的完备度量。     

The Hodge conjecture for certain moduli varieties

For smooth projective varietiesX over ℂ, the Hodge Conjecture states that every rational Cohomology class of type (p, p) comes from an algebraic cycle. In this paper, we prove the Hodge conjecture for some moduli spaces of vector bundles on compact Riemann surfaces of genus 2 and 3.     

Projektive Einbettung nicht lokal projektiver Räume

It is well known that every locally projective linear space (M,M) with dimM 3, fulfilling the Bundle Theorem (B) can be embedded in a projective space. We give here a new construction for the projective embedding of linear spaces which need not be locally projective. Essentially for this new construction are the assumptions (A) and (C) that for any two bundles there are two points on every line which are incident with a line of each of these bundles. With the Embedding Theorem (7.4) of this note for example a [0,m]-space can be embedded in a projective space.

     

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

Equivariant Vector Bundles Over Quantum Projective Spaces

Theoretical and Mathematical Physics - We construct equivariant vector bundles over quantum projective spaces using parabolic Verma modules over the quantum general linear group. Using an...     

A Sufficient Condition for Splitting of Arithmetically Cohen–Macaulay Bundles on General Hypersurfaces

A vector bundle on a hypersurface is arithmetically Cohen–Macaulay if its intermediate cohomologies vanish. On projective spaces, such bundles coincide with those which split into a direct sum of line bundles, but this fails on hypersurfaces of higher degree in general. In this article, we give an inequality which gives a sufficient condition for splitting of arithmetically Cohen–Macaulay bundles on general hypersurfaces.     

Embedding polynomial matrices of one variable

A non square matrix with coefficients in K[z] can (if a condition on its minors is satisfied) be embedded into a square matrix with determinant 1. Finding theoretically and in an algorithmic way an embedding of small degree is solved by a construction with vector bundles on the projective line over K.     

Maximal subbundles of vector bundles on a curve

For a rank 2 vector bundle E on a non-singular projective curve of genus g, the theorem of Nagata tells us that deg E-2 max deg Fg where the maximum is taken over all sub line bundles F of E. We generalize this result to vector bundles of arbitrary rank.     

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class c₁ = O_Q(-1,-1){c_1= \mathcal{O}_Q(-1,-1)} with respect to the ample line bundle O_Q(1,1){\mathcal {O}_Q(1,1)} . We show that the set of jumping conics of E is a hypersurface of degree c ₂(E) − 1 in \mathbb P₃^*{\mathbb {P}_3^{*}} . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c ₂(E).     

ON SOME ROTATIONALLY SYMMETRIC GRADIENT PSEUDO-KHLER-RICCI SOLITONS

In this paper, we explicitly construct some rotationally symmetric gradient pseudoK¨ahler-Ricci solitons which depend on some parameters, on some line bundles and other bundles over projective spaces. We also discuss the "phase change" phenomenon caused by the variation of parameters.     

Reduction theory for a rational function field

LetG be a split reductive group over a finite field F_q. LetF = F_q(t) and let A denote the adèles ofF. We show that every double coset inG(F)/G(A)/K has a representative in a maximal split torus ofG. HereK is the set of integral adèlic points ofG. WhenG ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.