488 research outputs found
Cohomology of toric line bundles via simplicial Alexander duality
We give a rigorous mathematical proof for the validity of the toric sheaf
cohomology algorithm conjectured in the recent paper by R. Blumenhagen, B.
Jurke, T. Rahn, and H. 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 H. Roschy and T.
Rahn (arXiv:1006.2392), and has several advantages such as being shorter and
cleaner and can also settle the additional conjecture on "Serre duality for
Betti numbers" which was raised but unresolved in arXiv:1006.2392.Comment: 9 pages. Theorem 1.1 and Corollary 1.2 improved; Abstract and
Introduction modified; References updated. To appear in Journal of
Mathematical Physic
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-Mustata-Stillman and Mustata on cohomology of toric and
monomial ideals to obtain a formula for computing the Euler characteristic of a
Weil divisor D on a complete simplicial toric variety in terms of graded pieces
of the Cox ring and Stanley-Reisner ring. The main point is to use Alexander
duality to pass from the toric irrelevant ideal, which appears in the
computation of the Euler characteristic of D, to the Stanley-Reisner ideal of
the fan, which is used in defining the Chow ring. The formula also follows from
work of Maclagan-Smith.Comment: 9 pages 1 figur
A^1-homotopy groups, excision, and solvable quotients
We study some properties of A^1-homotopy groups: geometric interpretations of
connectivity, excision results, and a re-interpretation of quotients by free
actions of connected solvable groups in terms of covering spaces in the sense
of A^1-homotopy theory. These concepts and results are well-suited to the study
of certain quotients via geometric invariant theory.
As a case study in the geometry of solvable group quotients, we investigate
A^1-homotopy groups of smooth toric varieties. We give simple combinatorial
conditions (in terms of fans) guaranteeing vanishing of low degree A^1-homotopy
groups of smooth (proper) toric varieties. Finally, in certain cases, we can
actually compute the "next" non-vanishing A^1-homotopy group (beyond
\pi_1^{A^1}) of a smooth toric variety. From this point of view, A^1-homotopy
theory, even with its exquisite sensitivity to algebro-geometric structure, is
almost "as tractable" (in low degrees) as ordinary homotopy for large classes
of interesting varieties.Comment: 48 pages, To appear Adv. Math, typographical and grammatical update
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 the computation of line bundle
valued cohomology classes over toric varieties are presented. For the heterotic
string, the prime examples are so-called monad constructions on Calabi-Yau
manifolds. In the context of Type II orientifolds, one often needs to compute
equivariant cohomology for line bundles, necessitating us to generalize our
algorithm to this case. Moreover, we exemplify that the different terms in
Batyrev's formula and its generalizations can be given a one-to-one
cohomological interpretation.
This paper is considered the third in the row of arXiv:1003.5217 and
arXiv:1006.2392.Comment: 56 pages, 8 tables, cohomCalg incl. Koszul extension available at
http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg
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 standard graded case, our
definition of multigraded regularity involves the vanishing of graded
components of local cohomology. We establish the key properties of regularity:
its connection with the minimal generators of a module and its behavior in
exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove
that its multigraded regularity bounds the equations that cut out the
associated subvariety. We also provide a criterion for testing if an ample line
bundle on X gives a projectively normal embedding.Comment: 30 pages, 5 figure
Bass Numbers of Semigroup-Graded Local Cohomology
Given a module M over a ring R which has a grading by a semigroup Q, we
present a spectral sequence that computes the local cohomology of M at any
Q-graded ideal I in terms of Ext modules. This method is used to obtain
finiteness results for the local cohomology of graded modules over semigroup
rings; in particular we prove that for a semigroup Q whose saturation is
simplicial, the Bass numbers of such local cohomology modules are finite.
Conversely, if the saturation of Q is not simplicial, one can find a graded
ideal I and a graded R-module M whose local cohomology at I in some degree has
an infinite-dimensional socle. We introduce and exploit the combinatorially
defined essential set of a semigroup.Comment: 19 pages LaTeX, 1 figure (.eps) Definition 5.1 corrected;
transcription error in Theorem 7.1.3 fixe
On some local cohomology spectral sequences
We introduce a formalism to produce several families of spectral sequences
involving the derived functors of the limit and colimit functors over a finite
partially ordered set. The first type of spectral sequences involves the left
derived functors of the colimit of the direct system that we obtain applying a
family of functors to a single module. For the second type we follow a
completely different strategy as we start with the inverse system that we
obtain by applying a covariant functor to an inverse system. The spectral
sequences involve the right derived functors of the corresponding limit. We
also have a version for contravariant functors. In all the introduced spectral
sequences we provide sufficient conditions to ensure their degeneration at
their second page. As a consequence we obtain some decomposition theorems that
greatly generalize the well-known decomposition formula for local cohomology
modules given by Hochster.Comment: 63 pages, comments are welcome. To appear in International
Mathematics Research Notice
- …