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1.
Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.). The quotient depends on a choice of an ample linearized line bundle. Two choices are equivalent if they give rise to identical quotients. A priori, there are infinitely many choices since there are infinitely many isomorphism classes of linearized ample line bundles. Hence several natural questions arise. Is the set of equivalence classes, and hence the set of non-isomorphic quotients, finite? How does the quotient vary under change of the equivalence class? In this paper we give partial answers to these questions in the case of actions of reductive algebraic groups on nonsingular projective algebraic varieties. We shall show that among ample line bundles which give projective geometric quotients there are only finitely many equivalence classes. These classes span certain convex subsets (chambers) in a certain convex cone in Euclidean space, and when we cross a wall separating one chamber from another, the corresponding quotient undergoes a birational transformation which is similar to a Mori flip. 相似文献
2.
Lin Weng 《Mathematische Annalen》2001,320(2):239-283
In Part I, Deligne-Riemann-Roch isometry is generalized for punctured Riemann surfaces equipped with quasi-hyperbolic metrics.
This is achieved by proving the Mean Value Lemmas, which explicitly explain how metrized Deligne pairings for -admissible metrized line bundles depend on . In Part II, we first introduce several line bundles over Knudsen-Deligne-Mumford compactification of the moduli space (or
rather the algebraic stack) of stable N-pointed algebraic curves of genus g, which are rather natural and include Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles. Then we use
Deligne-Riemann-Roch isomorphism and its metrized version (proved in Part I) to establish some fundamental relations among
these line bundles. Finally, we compute first Chern forms of the metrized Weil-Petersson, Takhtajan-Zograf and logarithmic
Mumford line bundles by using results of Wolpert and Takhtajan-Zograf, and show that the so-called Takhtajan-Zograf metric
on the moduli space is algebraic.
Received February 14, 2000 / Accepted August 18, 2000 / Published online February 5, 2001 相似文献
3.
By way of intersection theory on \(\overline{\mathcal {M}}_{g,n}\), we show that geometric interpretations for conformal blocks, as sections of ample line bundles over projective varieties, do not have to hold at points on the boundary. We show such a translation would imply certain recursion relations for first Chern classes of these bundles. While recursions can fail, geometric interpretations are shown to hold under certain conditions. 相似文献
4.
Siegmund Kosarew Paul Lupascu 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2000,70(1):265-274
In this note we identify two complex structures (one is given by algebraic geometry, the other by gauge theory) on the set
of isomorphism classes of holomorphic bundles with section on a given compact complex manifold. In the case ofline bundles, these complex spaces are shown to be isomorphic to a space of effective divisors on the manifold.
The second author was partially supported by SNF, nr. 2000-055290.98/1. 相似文献
5.
We establish a generalization of Breen's theory of cubic structures on line bundles over group schemes. We study such “n-cubic structures” inductively using multiextensions. As a result we obtain information on the set of isomorphism classes of line bundles with n-cubic structures over finite multiplicative group schemes over Spec(Z) by relating this set to certain corresponding eigenspaces of ideal class groups of cyclotomic fields. 相似文献
6.
In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising
from L
2 metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum
line bundle turns out to be Quillen’s determinant line bundle with a modified Quillen metric. Next, as in the case of abelian
vortices, we construct line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrized
by Ψ0, a section of a certain bundle. The equivalence of these prequantum bundles are discussed. 相似文献
7.
In this paper, we prove a Gauss-Bonnet theorem for the higher algebraic K-theory of smooth complex algebraic varieties. To each exact n-cube of hermitian vector bundles, we associate a higher Bott-Chen form, generalizing the Bott-Chern forms associated to exact
sequences. These forms allow us to define characteristic classes from K-theory to absolute Hodge cohomology. Then we prove that these characteristic classes agree with Beilinson's regulator map.
Oblatum 21-III-1997 & 12-VI-1997 相似文献
8.
Ronen Peretz 《Applied Mathematics Letters》2011,24(6):850-853
In this paper we give a geometric proof for a version of the Pinchuk solution of the Strong Real Jacobian Conjecture. Moreover, we compute the location of the zero sets on the Pinchuk surface of the determinant of the Jacobians of the corresponding étale mappings. 相似文献
9.
For all spherical homogeneous spaces G/H, where G is a simply connected semisimple algebraic group and H a connected solvable subgroup of G, we compute the spectra of representations of G on spaces of regular sections of homogeneous line bundles over G/H. 相似文献
10.
《Journal of Pure and Applied Algebra》2022,226(6):106977
The purpose of this article is to show that the bivariant algebraic A-cobordism groups considered previously by the author are independent of the chosen base ring A. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define the corresponding homology theory called algebraic bordism. We also generalize projective bundle formula, the theory of Chern classes, the Conner–Floyd theorem and the Grothendieck–Riemann–Roch theorem to this setting. The general definitions of bivariant cobordism are based on the careful study of ample line bundles and quasi-projective morphisms of Noetherian derived schemes, also undertaken in this work. 相似文献
11.
We compute the Picard group of the moduli spaceU′ of semistable vector bundles of rankn and degreed on an irreducible nodal curveY and show thatU′ is locally factorial. We determine the canonical line bundles ofU′ andU
L
′, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata
in the compactification ofU′. 相似文献
12.
The object of this paper is to study continuous vector bundles, over real algebraic varieties, admitting an algebraic structure. For large classes of real varieties, we obtain explicit information concerning the Grothendieck group of algebraic vector bundles. We show that in many cases this group is small compared to the corresponding group of continuous vector bundles. These results are used elsewhere to study the geometry of real algebraic varieties.Dedicated to Professor Alexander Grothendieck on the occasion of his 60th birthdaySupported by the NSF Grant DMS-8602672. 相似文献
13.
14.
Nitin Nitsure 《Proceedings Mathematical Sciences》1986,95(1):61-77
The purpose of this paper is to compute the Betti numbers of the moduli space ofparabolic vector bundles on a curve (see Seshadri [7], [8] and Mehta & Seshadri [4]), in the case where every semi-stable parabolic bundle is necessarily
stable. We do this by generalizing the method of Atiyah and Bott [1] in the case of moduli of ordinary vector bundles. Recall
that (see Seshadri [7]) the underlying topological space of the moduli of parabolic vector bundles is the space of equivalence
classes of certain unitary representations of a discrete subgroup Γ which is a lattice in PSL (2,R). (The lattice Γ need not necessarily be co-compact).
While the structure of the proof is essentially the same as that of Atiyah and Bott, there are some difficulties of a technical
nature in the parabolic case. For instance the Harder-Narasimhan stratification has to be further refined in order to get
the connected strata. These connected strata turn out to have different codimensions even when they are part of the same Harder-Narasimhan
strata.
If in addition to ‘stable = semistable’ the rank and degree are coprime, then the moduli space turns out to be torsion-free
in its cohomology.
The arrangement of the paper is as follows. In § 1 we prove the necessary basic results about algebraic families of parabolic
bundles. These are generalizations of the corresponding results proved by Shatz [9]. Following this, in § 2 we generalize
the analytical part of the argument of Atiyah and Bott (§ 14 of [1]). Finally in § 3 we show how to obtain an inductive formula
for the Betti numbers of the moduli space. We illustrate our method by computing explicitly the Betti numbers in the special
case of rank = 2, and one parabolic point. 相似文献
15.
A toric origami manifold, introduced by Cannas da Silva, Guillemin and Pires,
is a generalization of a toric symplectic manifold. For a toric symplectic manifold, its
equivariant Chern classes can be described in terms of the corresponding Delzant polytope
and the stabilization of its tangent bundle splits as a direct sum of complex line bundles.
But in general a toric origami manifold is not simply connected, so the algebraic topology
of a toric origami manifold is more difficult than a toric symplectic manifold. In this paper
they give an explicit formula of the equivariant Chern classes of an oriented toric origami
manifold in terms of the corresponding origami template. Furthermore, they prove the
stabilization of the tangent bundle of an oriented toric origami manifold also splits as a
direct sum of complex line bundles. 相似文献
16.
《Indagationes Mathematicae》2023,34(3):488-580
The aim of the present paper is to lay the foundation for a theory of Ehresmann structures in positive characteristic, generalizing the Frobenius-projective and Frobenius-affine structures defined in the previous work. This theory deals with atlases of étale coordinate charts on varieties modeled on homogeneous spaces of algebraic groups, which we call Frobenius–Ehresmann structures. These structures are compared with Cartan geometries in positive characteristic, as well as with higher-dimensional generalizations of dormant indigenous bundles. In particular, we investigate the conditions under which these geometric structures are equivalent to each other. Also, we consider the classification problem of Frobenius–Ehresmann structures on algebraic curves. The latter half of the present paper discusses the deformation theory of indigenous bundles in the algebraic setting. The tangent and obstruction spaces of various deformation functors are computed in terms of the hypercohomology groups of certain complexes. As a consequence, we formulate and prove the Ehresmann–Weil–Thurston principle for Frobenius–Ehresmann structures. This fact asserts that deformations of a variety equipped with a Frobenius–Ehresmann structure are completely determined by their monodromy crystals. 相似文献
17.
In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological
invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams
in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant
is derived from a Morse-Conley homotopy index.?In the second half of the paper we apply this technology to second order Lagrangians
via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence
of infinitely many braid classes of closed orbits.
Oblatum 11-V-2001 & 13-XI-2002?Published online: 24 February 2003
RID="*"
ID="*"The first author was supported by NSF DMS-9971629 and NSF DMS-0134408. The second author was supported by an EPSRC Fellowship.
The third author was supported by NWO Vidi-grant 639.032.202. 相似文献
18.
Ulrich Bunke 《manuscripta mathematica》2002,109(3):263-287
The holonomy of an unitary line bundle with connection over some base space B is a U(1)-valued function on the loop space LB. In a parallel manner, the holonomy of a gerbe with connection on B is a line bundle with connection over LB.
Given a family of graded Dirac operators on B and some additional geometric data one can define the determinant line bundle with Quillen metric and Bismut-Freed connection.
According to Witten, Bismut-Freed the holonomy of this determinant bundle can be expressed in terms of an adiabatic limit
of eta invariants of an associated family of Dirac operators over LB.
Recently, for a family of ungraded Dirac operators on B. Lott constructed an index gerbe with connection. In the present paper we show, in analogy to the holonomy formula for the
determinant bundle, that the holonomy of the index gerbe coincides with an adiabatic limit of determinant bundles of the associated
family of Dirac operators over LB.
Received: 17 October 2001 / Revised version: 5 August 2002 相似文献
19.
We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of semistable vector bundles over an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli space of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit coordinates on the moduli space. The finiteness results give low-dimensional parametrizations of Jacobians (in for generic curves), described by 2Θ functions or second logarithmic derivatives of theta.We interpret the Klein and Wirtinger maps in terms of opers on X. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The Wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the Szegö kernel. This generalizes the Wirtinger projective connections associated to theta characteristics, and the associated Klein bidifferentials. 相似文献
20.
Max Karoubi 《Topology》2003,42(4):715-742
An algebraic variety defined over the real numbers has an associated topological space with involution, and algebraic vector bundles give rise to Real vector bundles. We show that the associated map from algebraic K-theory to Atiyah's Real K-theory is, after completion at two, an isomorphism on homotopy groups above the dimension of the variety. 相似文献