Natural Cohomology Research Articles

Ulrich bundles of arbitrary rank on Segre-Veronese varieties

We generalize the results by Eisenbud and Schreyer about Ulrich bundles over Veronese varieties to Segre-Veronese varieties. We discuss the range where we have natural cohomology and we construct multigraded resolutions and monads for Ulrich bundles of any rank. Moreover we give cohomological characterizations for significant families of bundles.

From β to η: a new cohomology for deformed Sasaki-Einstein manifolds

We discuss in detail the different analogues of Dolbeault cohomology groups on Sasaki-Einstein manifolds and prove a new vanishing result for the transverse Dolbeault cohomology groups {H}_{overline{partial}}^{left(p,0right)}(k) graded by their charge under the Reeb vector. We then introduce a new cohomology, η-cohomology, which is defined by a CR structure and a holomorphic function f with non-vanishing η ≡ df. It is the natural cohomology associated to a class of supersymmetric type IIB flux backgrounds that generalise the notion of a Sasaki-Einstein manifold. These geometries are dual to finite deformations of the 4d mathcal{N} = 1 SCFTs described by conventional Sasaki-Einstein manifolds. As such, they are associated to Calabi-Yau algebras with a deformed superpotential. We show how to compute the η-cohomology in terms of the transverse Dolbeault cohomology of the undeformed Sasaki-Einstein space. The gauge-gravity correspondence implies a direct relation between the cyclic homologies of the Calabi-Yau algebra, or equivalently the counting of short multiplets in the deformed SCFT, and the η-cohomology groups. We verify that this relation is satisfied in the case of S5, and use it to predict the reduced cyclic homology groups in the case of deformations of regular Sasaki-Einstein spaces. The corresponding Calabi-Yau algebras describe non-commutative deformations of ℙ2, ℙ1× ℙ1 and the del Pezzo surfaces.

On q-de Rham cohomology via Lambda -rings

We show that Aomoto’s q-deformation of de Rham cohomology arises as a natural cohomology theory for Lambda -rings. Moreover, Scholze’s (q-1)-adic completion of q-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in mixed characteristic equipped with a suitable lift of Frobenius. If we attach p-power roots of q, the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.

Embedding of a compactified Jacobian and theta divisors

Using a general stable vector bundle, we give an embedding $$\alpha _Y$$ of the compactified Jacobian $$\bar{J}(Y)$$ of an integral nodal curve Y into the moduli space $$U_Y(r,d)$$ of semistable torsion free sheaves of rank r and degree d on Y. We also give an embedding of the normalisation $$\tilde{J}(Y)$$ of $$\bar{J}(Y)$$ in the normalisation P(r, d) of $$U_Y(r,d)$$ . We determine a relation between the restriction of the theta line bundle on P(r, d) to $$\tilde{J}(Y)$$ and the theta line bundle on $$\tilde{J}(Y)$$ . We show that the restriction of the Picard bundle $$E_{r,d}$$ on $$U_Y(r,d)$$ to $$\bar{J}(Y)$$ is stable with respect to any theta divisor $$\theta _{\bar{J}(Y)}$$ on $$\bar{J}(Y)$$ if $$d > r(2g-1)$$ and it is semistable if $$d= r(2g-1)$$ . Finally we prove some results on natural cohomology.

A canonical torsion theory for pro-p Iwahori–Hecke modules

Let F be a locally compact nonarchimedean field with residue characteristic p and G the group of F-rational points of a connected split reductive group over F. We define a torsion pair in the category Mod(H) of modules over the pro-p-Iwahori Hecke k-algebra H of G, where k is an arbitrary field. We prove that, under a certain hypothesis, the torsionfree class embeds fully faithfully into the category ModI(G) of smooth k-representations of G generated by their pro-p-Iwahori fixed vectors.If the characteristic of k is different from p then this hypothesis is always satisfied and the torsionfree class is the whole category Mod(H).If k contains the residue field of F then we study the case G=SL2(F). We show that our hypothesis is satisfied, and we describe explicitly the torsionfree and the torsion classes. If F≠Qp and p≠2, then an H-module is in the torsion class if and only if it is a union of supersingular finite length submodules; it lies in the torsionfree class if and only if it does not contain any nonzero supersingular finite length module. If F=Qp, the torsionfree class is the whole category Mod(H), and we give a new proof of the fact that Mod(H) is equivalent to ModI(G). These results are based on the computation of the H-module structure of certain natural cohomology spaces for the pro-p-Iwahori subgroup I of G.

Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds

Given an irregular compact Kahler manifold X, one can form the derivative complex of X, which governs the deformation theory of the groups H i(X,α) as α varies over Pic(X). Together with its variants, it plays a central role in a body of work involving generic vanishing theorems. The purpose of this paper is to present two new applications of this complex. First, we show that it fits neatly into the setting of the so-called Bernstein-Gel’fand-Gel’fand (BGG) correspondence between modules over an exterior algebra and linear complexes over a symmetric algebra (cf. [BGG], [EFS], [Eis], [Co]). What comes out here is that some natural cohomology modules associated to X have a surprisingly simple algebraic structure, and that conversely one can read off from these modules some basic geometric invariants associated to the Albanese mapping of X. Secondly, under an additional hypothesis, the derivative complex determines a vector bundle on the projectivized tangent space to Pic(X) at the origin. We show that this bundle, which encodes the infinitesimal behavior of the Hilbert scheme of paracanonical divisors along the canonical linear series, can be used to generate inequalities among numerical invariants of X.

Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology

Abstract Here we study vector bundles E on the Hirzebruch surface F e such that their twists by a spanned, but not ample, line bundle M = $$ \mathcal{O} $$ Fe(h + ef) have natural cohomology, i.e. h 0(F e, E(tM)) > 0 implies h 1(F e, E(tM)) = 0.

A desingularization of the main component of the moduli space of genus-one stable maps into ℙn

We construct a desingularization of the ``main component'' $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$ of the moduli space $\bar{\mathfrak M}_{1,k}(\Bbb{P}^n,d)$ of genus-one stable maps into the complex projective space $\Bbb{P}^n$. As a bonus, we obtain desingularizations of certain natural sheaves over $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$. Such desingularizations are useful for integrating natural cohomology classes on $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$ using localization. In turn, these classes can be used to compute the genus-one Gromov-Witten invariants of complete intersections and classical enumerative invariants of projective spaces involving genus-one curves. The desingularization of $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$ is obtained by sequentially blowing up $\bar{\mathfrak M}_{1,k}(\Bbb{P}^n,d)$ along ``bad'' subvarieties. At the end of the process, we are left with a modification of the main component, which turns out to be nonsingular.

Diagrams of Lie algebras

Diagrams of Lie algebras have natural cohomology and deformation theories. The relationship between Lie and Hochschild cohomology allows one to reduce these to the associative case, where the cohomology comparison theorem asserts that for every diagram of associative algebras there is a single associative algebra whose cohomology and deformation theories are the same as those of the entire diagram. We show that the cohomology of a diagram of Lie algebras with coefficients in a diagram of Lie modules is canonically isomorphic to that of a single associative algebra with coefficients in a single bimodule, but the existence of operations analogous to those in the associative case remains unknown. In the last section, we construct a single cochain complex from a diagram of cochain complexes and derive the usual mapping cone and cylinder.

The moduli space of torsion-free G2G2 structures

Let M be the moduli space of torsion-free G2 structures on a compact oriented G2 manifold M. The natural cohomology map π3:M→H3(M,ℝ) is known to be a local diffeomorphism [Compact Manifolds with Special Holonomy, Oxford University Press, 2000]. Let M1⊂M be the subset of G2 structures with volume (M)=1. We show every nonzero element of H4(M,ℝ)=H3(M,ℝ)* is a Morse function on M1 when composed with π3, and we compute its Hessian. The result implies a special case of Torelli’s theorem: if H1(M,ℝ)=0 and dimH3(M,ℝ)=2, the cohomology map π3:M→H3(M,ℝ) is one to one on each connected component of M. We formulate a compactness conjecture on the set of G2 structures of volume (M)=1 with bounded L2 norm of curvature. If this conjecture were true, it would imply that every connected component of M is contractible, and that every compact G2 manifold supports a G2 structure whose fundamental 4-form represents the negative of the (nonzero) first Pontryagin class of M. We also observe that when H1(M,ℝ)=0, and the volume of the torus H3(M,ℝ)/H3(M,ℤ) is constant along M1, the locus π3(M1)⊂H3(M,ℝ) is a hyperbolic affine sphere.

The moduli space of torsion-free [formula omitted] structures

Let M be the moduli space of torsion-free G 2 structures on a compact oriented G 2 manifold M . The natural cohomology map π 3 : M → H 3 ( M , ℝ ) is known to be a local diffeomorphism [Compact Manifolds with Special Holonomy, Oxford University Press, 2000]. Let M 1 ⊂ M be the subset of G 2 structures with volume ( M ) = 1 . We show every nonzero element of H 4 ( M , ℝ ) = H 3 ( M , ℝ ) * is a Morse function on M 1 when composed with π 3 , and we compute its Hessian. The result implies a special case of Torelli’s theorem: if H 1 ( M , ℝ ) = 0 and dim H 3 ( M , ℝ ) = 2 , the cohomology map π 3 : M → H 3 ( M , ℝ ) is one to one on each connected component of M . We formulate a compactness conjecture on the set of G 2 structures of volume ( M ) = 1 with bounded L 2 norm of curvature. If this conjecture were true, it would imply that every connected component of M is contractible, and that every compact G 2 manifold supports a G 2 structure whose fundamental 4-form represents the negative of the (nonzero) first Pontryagin class of M . We also observe that when H 1 ( M , ℝ ) = 0 , and the volume of the torus H 3 ( M , ℝ ) / H 3 ( M , ℤ ) is constant along M 1 , the locus π 3 ( M 1 ) ⊂ H 3 ( M , ℝ ) is a hyperbolic affine sphere.

On the geometric quantization of Jacobi manifolds

The geometric quantization of Jacobi manifolds is discussed. A natural cohomology (termed Lichnerowicz–Jacobi) on a Jacobi manifold is introduced, and using it the existence of prequantization bundles is characterized. To do this, a notion of contravariant derivatives is used, in such a way that the procedure developed by Vaisman for Poisson manifolds is naturally extended. A notion of polarization is discussed and the quantization problem is studied. The existence of prequantization representations is also considered.

On curves with natural cohomology and their deficiency modules

The minimal free resolution of the Hartshorne-Rao module of a curve with natural cohomology is studied, and conditions are given on the degrees and the ranks of the terms of this resolution.

Chern classes and cohomology for rank 2 reflexive sheaves on P3

The paper shows that, if F is a nonsplit rank 2 reflexive sheaf on P 3, then the knowledge of the numbers dn = h2(F(n)) - h\F(n)) gives an explicit algorithm to compute the Chern classes C\, Cι, c?> and the dimensions h°(F(n)), for all n (in particular the first integer a such that the sheaf F(a) has some nonzero section). If the sheaf is a vector bundle it is also proved that the knowledge of the numerical sequence {hι(F(n))} together with the first Chern class gives all the information as above. In some special cases, i.e. when hx(F(n)) Φ 0 for at most three values of n, an algorithm is also produced to compute the first Chern class from the sequence {hι(F(n))} . Vector bundles with natural cohomology are also discussed. It must be remarked that, if one knows not only the dimensions hι (F(n)), for all n, but also the whole structure of the Rao-module Q)Hι(F(n)), then the first Chern class C is uniquely determined (as it is shown in a paper by P. Rao).