Cohomology Of Line Bundles Research Articles

Generating Functions for Line Bundle Cohomology Dimensions on Complex Projective Varieties

This paper explores the possibility of constructing multivariate generating functions for all cohomology dimensions of all holomorphic line bundles on certain complex projective varieties of Fano, Calabi-Yau and general type in various dimensions and Picard numbers. Most of the results are conjectural and rely on explicit cohomology computations. We first propose a generating function for the Euler characteristic of all holomorphic line bundles on complete intersections in products of projective spaces and toric varieties. This generating function is constructed by expanding the Hilbert-Poincaré series associated with the coordinate ring of the variety around all possible combinations of zero and infinity and then summing up the resulting contributions with alternating signs. Similar generating functions are proposed for the individual cohomology dimensions of all holomorphic line bundles on certain complete intersections, including examples of Mori and non-Mori dream spaces. Surprisingly, the examples studied indicate that a single generating function encodes both the zeroth and all higher cohomologies upon expansion around different combinations of zero and infinity, raising the question whether such generating functions determine the variety uniquely.

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 obtained formulae express cohomologies in terms of divisor class intersections, and are adapted to the decomposition of the effective cone into Zariski chambers. In particular, we show this occurs on generalised del Pezzo surfaces, toric surfaces, and K3 surfaces. In the second part we use these surface results to derive formulae for all line bundle cohomology on a simple class of elliptically fibered Calabi–Yau three-folds. Computing such quantities is a crucial step in deriving the massless spectrum in string compactifications.

Cohomology of line bundles on the incidence correspondence

For a finite dimensional vector space V V of dimension n n , we consider the incidence correspondence (or partial flag variety) X ⊂ P V × P V ∨ X\subset \mathbb {P}V \times \mathbb {P}V^{\vee } , parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X X in characteristic p > 0 p>0 . If n = 3 n=3 then X X is the full flag variety of V V , and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0 0 , the cohomology groups are described for all V V by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When n = 3 n=3 , we recover the recursive description of characters from recent work of Linyuan Liu, while for general n n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.

Cohomology of holomorphic line bundles and Hodge symmetry on Oeljeklaus–Toma manifolds

Cohomology of holomorphic line bundles and Hodge symmetry on Oeljeklaus–Toma manifolds

On Picard groups of perfectoid covers of toric varieties

Let X be a proper smooth toric variety over a perfectoid field of prime residue characteristic p. We study the perfectoid space which covers X constructed by Scholze, showing that is canonically isomorphic to $$\textrm{Pic}(X)[p^{-1}]$$ . We also compute the cohomology of line bundles on and establish analogs of Demazure and Batyrev–Borisov vanishing. This generalizes the first author’s analogous results for projectivoid space.

Brill-Noether-general limit root bundles: absence of vector-like exotics in F-theory Standard Models

Root bundles appear prominently in studies of vector-like spectra of 4d F-theory compactifications. Of particular importance to phenomenology are the Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vector-like spectra for the matter representations (3,2)1/6, ( overline{textbf{3}} ,1)−2/3 and (1,1)1. For the family B3( {Delta }_4^{{}^{circ}} ) consisting of mathcal{O} (1011) F-theory QSM geometries, we argue that more than 99.995% of the roots in this superset have no vector-like exotics. This indicates that absence of vector-like exotics in those representations is a very likely scenario in the mathcal{O} (1011) QSM geometries B3( {Delta }_4^{{}^{circ}} ).The QSM geometries come in families of toric 3-folds B3(∆°) obtained from triangulations of certain 3-dimensional polytopes ∆°. The matter curves in XΣ ∈ B3(∆°) can be deformed to nodal curves which are the same for all spaces in B3(∆°). Therefore, one can probe the vector-like spectra on the entire family B3(∆°) from studies of a few nodal curves. We compute the cohomologies of all limit roots on these nodal curves.In our applications, for the majority of limit roots the cohomologies are determined by line bundle cohomology on rational tree-like curves. For this, we present a computer algorithm. The remaining limit roots, corresponding to circuit-like graphs, are handled by hand. The cohomologies are independent of the relative position of the nodes, except for a few circuits. On these jumping circuits, line bundle cohomologies can jump if nodes are specially aligned. This mirrors classical Brill-Noether jumps. B3( {Delta }_4^{{}^{circ}} ) admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens.

REPRESENTATION THEORY VIA COHOMOLOGY OF LINE BUNDLES

Let G be a reductive algebraic group over a field k and let B be a Borel subgroup in G. We demonstrate how a number of results on the cohomology of line bundles on the ag manifold G/B have had interesting consequences in the representation theory for G, and vice versa. Our focus is on the case where the characteristic of k is positive. In this case, both the vanishing behavior of the cohomology modules for a line bundle on G/B and the G-structures of the nonzero cohomology modules are still very much open problems. We give an account of the developments over the years, trying to illustrate what is now known and what is still not known today.

Flops, Gromov-Witten invariants and symmetries of line bundle cohomology on Calabi-Yau three-folds

The zeroth line bundle cohomology on Calabi-Yau three-folds encodes information about the existence of flop transitions and the genus zero Gromov-Witten invariants. We illustrate this claim by studying several Picard number 2 Calabi-Yau three-folds realised as complete intersections in products of projective spaces. Many of these manifolds exhibit certain symmetries on the Picard lattice which preserve the zeroth cohomology.

On the cohomology of line bundles over certain flag schemes

Let G be the group scheme SLd+1 over Z and let Q be the parabolic subgroup scheme corresponding to the simple roots α2,⋯,αd−1. Then G/Q is the Z-scheme of partial flags {D1⊂Hd⊂V}. We will calculate the cohomology modules of line bundles over this flag scheme. We will prove that the only non-trivial ones are isomorphic to the kernel or the cokernel of certain matrices with multinomial coefficients.

Topological Formulae for the Zeroth Cohomology of Line Bundles on del Pezzo and Hirzebruch Surfaces

Abstract We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.

Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations

We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group $G$. These manifolds are obtained as quotients of the domains of discontinuity in generalized flag varieties $G/P$ constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For $G$-Fuchsian representations and their Anosov deformations, where $G$ is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For $G$-Fuchsian and $G$-quasi-Fuchsian representations in simple $G$ of rank at least two, we show that the quotient manifolds are not K\"{a}hler. We also describe the Picard groups of these quotient manifolds, compute the cohomology of line bundles on them, and show that for $G$ of sufficiently large rank these manifolds admit nonconstant meromorphic functions. In a final section, we apply our topological results to several explicit families of domains and derive closed formulas for topological invariants in some cases. We also show that the quotient manifold for a $G$-Fuchsian representation in $\mathrm{PSL}_3(\mathbb{C})$ is a fiber bundle over a surface, and we conjecture that this holds for all simple $G$.

On the cohomology of line bundles over certain flag schemes II

Over a field K of characteristic p, let Z be the incidence variety in Pd×(Pd)⁎ and let L be the restriction to Z of the line bundle O(−n−d)⊠O(n), where n=p+f with 0≤f≤p−2. We prove that Hd(Z,L) is the simple GLd+1-module corresponding to the partition λf=(p−1+f,p−1,f+1). When f=0, using the first author's description of Hd(Z,L) and Jantzen's sum formula, we obtain as a by-product that the sum of the monomial symmetric functions mλ, for all partitions λ of 2p−1 less than (p−1,p−1,1) in the dominance order, is the alternating sum of the Schur functions Sp−1,p−1−i,1i+1 for i=0,…,p−2.

Displaying the cohomology of toric line bundles

There is a standard approach to calculate the cohomology of torus-invariant sheaves on a toric variety via the simplicial cohomology of the associated subsets of the space of 1-parameter subgroups of the torus. For a line bundle represented by a formal difference of polyhedra in the character space , [1] contains a simpler formula for the cohomology of , replacing by the set-theoretic difference . Here, we provide a short and direct proof of this formula.

The Abel map for surface singularities II. Generic analytic structure

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants includes: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincaré series associated with the divisorial filtration, the analytic semigroup, the maximal ideal cycle.The first part contains the definition of ‘generic structure’ based on the work of Laufer [14]. The second technical ingredient is the Abel map developed in [21].The results can be compared with certain parallel statements from the Brill–Noether theory and from the theory of Abel map associated with projective smooth curves (see e.g. [1] and [6]), though the tools and machineries are very different.

The growth of dimension of cohomology of semipositive line bundles on Hermitian manifolds

In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic (0, q)-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we estimate the dimension of cohomology of semipositive line bundles on q-convex manifolds, pseudo-convex domains, weakly 1-complete manifolds and complete manifolds. We also obtain the estimate of cohomology on compact manifolds with semipositive line bundles endowed with a Hermitian metric with analytic singularities and related results.

Index Formulae for Line Bundle Cohomology on Complex Surfaces

AbstractWe 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 algorithm which allows expression of any line bundle cohomology in terms of an index. These formulae follow from general theorems we prove for a wider class of surfaces. In particular, we construct a map that takes any effective line bundle to a nef line bundle while preserving the zeroth cohomology dimension. For complex surfaces, these results explain the appearance of piecewise polynomial equations for cohomology and they are a first step towards understanding similar formulae recently obtained for Calabi‐Yau three‐folds.

Displaying the cohomology of toric line bundles

Имеется стандартный подход к вычислению когомологий торически инвариантных пучков $\mathcal{L}$ на торическом многообразии с помощью симплициальных когомологий ассоциированных с ними подмножеств $V(\mathcal{L})$ пространства $N_\mathbb{R}$ всех однопараметрических подгрупп тора. В то же время для когомологий линейного расслоения $\mathcal{L}$, заданного формальной разностью $\Delta^+-\Delta^-$ многогранников из пространства характеров $M_\mathbb{R}$ данного тора, в работе [1] приведена более простая формула, в которой $V(\mathcal{L})$ заменяется на теоретико-множественную разность $\Delta^- \setminus \Delta^+$. Мы даем короткое и прямое доказательство этой формулы. Библиография: 11 наименований.

Machine Learning Line Bundle Cohomology

AbstractWe investigate different approaches to machine learning of line bundle cohomology on complex surfaces as well as on Calabi‐Yau three‐folds. Standard function learning based on simple fully connected networks with logistic sigmoids is reviewed and its main features and shortcomings are discussed. It has been observed recently that line bundle cohomology can be described by dividing the Picard lattice into certain regions in each of which the cohomology dimension is described by a polynomial formula. Based on this structure, we set up a network capable of identifying the regions and their associated polynomials, thereby effectively generating a conjecture for the correct cohomology formula. For complex surfaces, we also set up a network which learns certain rigid divisors which appear in a recently discovered master formula for cohomology dimensions.

Cohomology of line bundles on horospherical varieties

A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties. The main tool in this article is the machinery of Grothendieck–Cousin complexes, and we also prove a Kunneth-like formula for local cohomology.

Formulae for Line Bundle Cohomology on Calabi‐Yau Threefolds

AbstractWe 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 spaces. The formulae have been obtained by systematising and extrapolating concrete calculations and they have been checked computationally. Although the intermediate calculations often involve laborious computations of ranks of Leray maps in the Koszul spectral sequence, the final results for cohomology follow a simple pattern. The space of line bundles can be divided into several different regions, and in each such region the ranks of all cohomology groups can be expressed as polynomials in the line bundle integers of degree at most three. The number of regions increases and case distinctions become more complicated for manifolds with a larger Picard number. We also find explicit cohomology formulae for several non‐simply connected Calabi‐Yau threefolds realised as quotients by freely acting discrete symmetries. More cases may be systematically handled by machine learning algorithms.