Cohomology of toric line bundles via simplicial Alexander duality

@article{Jow2011CohomologyOT,
  title={Cohomology of toric line bundles via simplicial Alexander duality},
  author={Shin-Yao Jow},
  journal={Journal of Mathematical Physics},
  year={2011},
  volume={52},
  pages={033506-033506}
}
  • Shin-Yao Jow
  • Published 4 June 2010
  • Mathematics
  • Journal of Mathematical Physics
We give a rigorous mathematical proof for the validity of the toric sheaf cohomology algorithm conjectured in the recent paper by Blumenhagen, Jurke, Rahn, and Roschy (arXiv:1003.5217). We actually prove not only the original algorithm but also a speed-up version of it. Our proof is independent from (in fact appeared earlier on the arXiv than) the proof by Roschy and Rahn (arXiv:1006.2392), and has several advantages such as being shorter and cleaner and can also settle the additional… 
EULER CHARACTERISTIC OF COHERENT SHEAVES ON SIMPLICIAL TORICS VIA THE
We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustaţǎ-Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for
EULER CHARACTERISTIC OF LINE BUNDLES ON SIMPLICIAL TORICS VIA THE
We combine work of Cox on the homogeneous coordinate ring of a toric variety and results of Eisenbud-Mustaţǎ-Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for
Euler characteristic of coherent sheaves on simplicial torics via the Stanley–Reisner ring
We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud–Mustaţǎ–Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for
Cohomology of Line Bundles: Proof of the Algorithm
We present a proof of the algorithm for computing line bundle valued cohomology classes over toric varieties conjectured by R.~Blumenhagen, B.~Jurke and the authors (arXiv:1003.5217) and suggest a
Index Formulae for Line Bundle Cohomology on Complex Surfaces
We conjecture and prove closed‐form index expressions for the cohomology dimensions of line bundles on del Pezzo and Hirzebruch surfaces. Further, for all compact toric surfaces we provide a simple
Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds
We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The
Computing Cohomology on Toric Varieties
In these notes a recently developed technique for the computation of line bundle-valued sheaf cohomology group dimensions on toric varieties is reviewed. The key result is a vanishing theorem for the
Cohomology of line bundles: Applications
Massless modes of both heterotic and Type II string compactifications on compact manifolds are determined by vector bundle valued cohomology classes. Various applications of our recent algorithm for
Computational Tools for Cohomology of Toric Varieties
TLDR
Applications to the computation of chiral massless matter spectra in string compactifications are discussed, and using the software package cohomCalg, its utility is highlighted on a new target space dual pair of (0,2) heterotic string models.
Formulae for Line Bundle Cohomology on Calabi‐Yau Threefolds
We present closed form expressions for the ranks of all cohomology groups of holomorphic line bundles on several Calabi‐Yau threefolds realised as complete intersections in products of projective
...
...

References

SHOWING 1-10 OF 15 REFERENCES
Cohomology of Line Bundles: Proof of the Algorithm
We present a proof of the algorithm for computing line bundle valued cohomology classes over toric varieties conjectured by R.~Blumenhagen, B.~Jurke and the authors (arXiv:1003.5217) and suggest a
Cohomology on Toric Varieties and Local Cohomology with Monomial Supports
TLDR
A grading on R which is coarser than the Z^n-grading such that each component of H^i_B(R) is finite dimensional is given and an effective way to compute these components is given.
Multigraded Castelnuovo-Mumford Regularity
We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the
Note: Combinatorial Alexander Duality—A Short and Elementary Proof
TLDR
A self-contained proof from first principles accessible to a nonexpert is given that the combinatorial Alexander duality of X is isomorphic to the (|V|−i−3)th reduced cohomology group of X* (over a given commutative ring R).
Introduction to Toric Varieties.
Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic
Cohomology of line bundles: A computational algorithm
We present an algorithm for computing line bundle valued cohomology classes over toric varieties. This is the basic starting point for computing massless modes in both heterotic and type IIB/F-theory
The homogeneous coordinate ring of a toric variety
This paper will introduce the homogeneous coordinate ring S of a toric variety X . The ring S is a polynomial ring with one variable for each one-dimensional cone in the fan ∆ determining X , and S
The Alexander Duality Functors and Local Duality with Monomial Support
Abstract Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated N n-graded module. The functors associated with Alexander duality provide a
Torus Actions and their Applications in Topology and Combinatorics
Here, the study of torus actions on topological spaces is presented as a bridge connecting combinatorial and convex geometry with commutative and homological algebra, algebraic geometry, and
...
...