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1.
Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective
torus actions on normal, algebraic varieties. Based on the notion of a “proper polyhedral divisor” introduced in earlier work,
we develop the concept of a “divisorial fan” and show that these objects encode the equivariant gluing of affine varieties
with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and
we study examples like *-surfaces and projectivizations of (nonsplit) vector bundles over toric varieties. 相似文献
2.
M. Franz 《Proceedings of the Steklov Institute of Mathematics》2006,252(1):53-62
We prove that the integral cohomology of a smooth, not necessarily compact, toric variety X
Σ is determined by the Stanley-Reisner ring of Σ. This follows from a formality result for singular cochains on the Borel construction
of X
Σ. As a consequence, we show that the cycle map from Chow groups to Borel-Moore homology is split injective. 相似文献
4.
We study the problem of when the collection of the recession cones of a polyhedral complex also forms a complex. We exhibit
an example showing that this is no always the case. We also show that if the support of the given polyhedral complex satisfies
a Minkowski–Weyl-type condition, then the answer is positive. As a consequence, we obtain a classification theorem for proper
toric schemes over a discrete valuation ring in terms of complete strongly convex rational polyhedral complexes. 相似文献
5.
E. Shustin 《Proceedings of the Steklov Institute of Mathematics》2007,258(1):218-246
The Welschinger invariants of real rational algebraic surfaces are natural analogs of the Gromov-Witten invariants, and they
estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a
tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces equipped with a nonstandard real structure.
Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric
structure) was established by Mikhalkin and the author. As a consequence we prove that for any real ample divisor D on a surface Σ under consideration, through any generic configuration of c
1(Σ)D − 1 generic real points, there passes a real rational curve belonging to the linear system |D|.
To Vladimir Igorevich Arnold on the occasion of his 70th birthday 相似文献
6.
We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties.
For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result
to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category
of quasi–projective varieties.
Received October 29, 1998; in final form December 28, 1998 相似文献
7.
Thomas Hüttemann 《Geometriae Dedicata》2010,148(1):175-203
Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U
σ
= Spec A
σ
. A quasi-coherent sheaf on X gives rise, by taking sections over the U
σ
, to a diagram of modules over the coordinate rings A
σ
, indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent
sheaves on X can be obtained from a category of Σop-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits.
Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the
machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves
in the sense that chain complexes over different open sets U
σ
agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves
is equivalent to the derived category of X. 相似文献
8.
Victor P. Palamodov 《Journal d'Analyse Mathématique》1999,78(1):177-204
The system of differential equations for polymonogenic functions of several quaternionic variables is an analogue of the
-equation in complex analysis. We give a representation of polymonogenic functions by means of integration of a family of
σ-holomorphic functions as σ runs over the variety Σ of all complex structures ℍ ≅ ℂ2 which are consistent with the metric and an orientation in ℍ. The variety Σ is isomorphic to the manifold of all proper right
ideals in the complexified quaternionic algebra and has a natural complex analytic structure. We construct a
-complex on Σ that provides a resolvennt for the sheaf of polymonogenic functions. 相似文献
9.
Hye Sook Park 《Journal of Applied Mathematics and Computing》1996,3(1):65-78
The properties of a toric variety have strong connection with the combinatorial structure of the corresponding fan and the relations among the generators. Using this fact, we have described explicitly the Chow ring for aQ-factorial toric variety as the Stanley-Reisner ring for the corresponding fan modulo the linear equivalence relation. In this paper, we calculate the Chow ring for 3-dimensionalQ-factorial toric varieties having one bad isolated singularity. 相似文献
10.
In this work, we describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented
punctured surface Σ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of
a triple of flags in the complex hyperbolic plane
H2\mathbb C{\bf H^2_{\mathbb {C}}} . We establish a bijection between a set of decorations of an ideal triangulation of Σ and a subset of the PU(2,1)-representation
variety of π
1(Σ). 相似文献
11.
Eva Maria Feichtner 《Proceedings of the American Mathematical Society》2003,131(6):1695-1704
We show that the real cohomology algebra of a compact toric variety of complex dimension is determined, up to isomorphism, by the combinatorial data of its defining fan. Surprisingly enough, this is no longer the case when taking rational coefficients. Moreover, we show that neither the rational nor the real or complex cohomology algebras of compact quasi-smooth toric varieties are combinatorial invariants in general.
12.
In this paper we study the integral Chow ring of toric Deligne–Mumford stacks. We prove that the integral Chow ring of a semi-projective
toric Deligne–Mumford stack is isomorphic to the Stanley–Reisner ring of the associated stacky fan. The integral orbifold
Chow ring is also computed. Our results are illustrated with several examples. 相似文献
13.
T. J. Ford 《Results in Mathematics》1999,35(1-2):44-69
A toroidal embedding is defined which does not assume the fan consists of rational cones. For a rational fan, the toroidal embedding is the usual toric variety. If the fan is not rational, the toroidal embedding is in general a quasi-compact noetherian locally ringed space which is not a scheme. A divisor theory exists and a class group is defined. A second construction is also carried out which mimics the gluing construction of the usual toric variety, but which makes no reference to a lattice. The resulting scheme is separated but infinite dimensional. The Picard group is described in terms of the group of real valued locally linear support functions on the fan and the Brauer group is shown to be trivial. Many examples are given throughout the paper; in particular, it is shown that there is associated to a real hyperplane arrangement of full rank a toroidal embedding. 相似文献
14.
This paper presents algorithms for computing the Gröbner fan of an arbitrary polynomial ideal. The computation involves enumeration of all reduced Gröbner bases of the ideal. Our algorithms are based on a uniform definition of the Gröbner fan that applies to both homogeneous and non-homogeneous ideals and a proof that this object is a polyhedral complex. We show that the cells of a Gröbner fan can easily be oriented acyclically and with a unique sink, allowing their enumeration by the memory-less reverse search procedure. The significance of this follows from the fact that Gröbner fans are not always normal fans of polyhedra, in which case reverse search applies automatically. Computational results using our implementation of these algorithms in the software package Gfan are included.
15.
We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y\{0})/C* such that the toric residue becomes the local residue at 0 in Y. 相似文献
16.
Nathan Owen Ilten 《manuscripta mathematica》2011,134(1-2):123-137
For a complete, smooth toric variety Y, we describe the graded vector space ${T_Y^1}$ . Furthermore, we show that smooth toric surfaces are unobstructed and that a smooth toric surface is rigid if and only if it is Fano. For a given toric surface we then construct homogeneous deformations by means of Minkowski decompositions of polyhedral subdivisions, compute their images under the Kodaira-Spencer map, and show that they span ${T_Y^1}$ . 相似文献
17.
We consider a finite dimensional k-algebraA and associate to each tilting module a cone in the Grothendieck groupK
0 of finitely generated A-modules. We prove that the set of cones associated to tilting modules of projective dimension at
most one defines a, not necessarily finite, fan Σ(A). IfA is of finite global dimension, the fan Σ(A) is smooth. Moreover, using the cone of a tilting module, we can associate a volume
to each tilting module. Using the fan and the volume, we obtain new proofs for several classical results; we obtain certain
convergent sums naturally associated to the algebraA and obtain criteria for the completeness of a list of tilting modules. Finally, we consider several examples.
Dedicated to O. Riemenschneider on the occasion of his 65th birthday 相似文献
18.
Following Sam Payne?s work, we study the existence problem of nontrivial vector bundles on toric varieties. The first result we prove is that every complete fan admits a nontrivial conewise linear multivalued function. Such functions could potentially be the Chern classes of toric vector bundles. Then we use the results of Cortiñas, Haesemeyer, Walker and Weibel to show that the (non-equivariant) Grothendieck group of the toric 3-fold studied by Payne is large, so the variety has a nontrivial vector bundle. Using the same computation, we show that every toric 3-fold X either has a nontrivial line bundle, or there is a finite surjective toric morphism from Y to X, such that Y has a large Grothendieck group. 相似文献
19.
It is shown that the integral cohomology groups of a smooth complex 3-dimensional toric variety are free (and we compute their dimensions), under the assumption that the spherical section of its fan is homeomorphic to a closed disk. As a consequence, this gives a partial positive answer to the torsion problem in a conjecture of Reid [3] about McKay correspondence in dimension 3. 相似文献
20.
T. E. Panov 《Proceedings of the Steklov Institute of Mathematics》2008,263(1):150-162
In the theory of algebraic group actions on affine varieties, the concept of a Kempf-Ness set is used to replace the categorical
quotient by the quotient with respect to a maximal compact subgroup. Using recent achievements of “toric topology,” we show
that an appropriate notion of a Kempf-Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate
subspace arrangement complements) arising in the Batyrev-Cox “geometric invariant theory” approach to toric varieties. We
proceed by studying the cohomology of these “toric” Kempf-Ness sets. In the case of projective nonsingular toric varieties
the Kempf-Ness sets can be described as complete intersections of real quadrics in a complex space.
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 159–172. 相似文献