Smooth Variety Research Articles

On the moment measure conjecture

AbstractThe moment measure conjecture of Białynicki-Birula and Sommese gives a combinatorial characterization of all open substacks of a global quotient stack for a torus action on a normal projective variety that admit a proper good moduli space, in other words it characterizes the invariant open subvarieties that admit a proper quotient. In this article we prove the conjecture for actions on smooth varieties. This gives an instance of stability conditions defined by cohomological invariants that are not given by Chern classes of line bundles.

The mirror Clemens–Schmid sequence

We introduce a four-term long exact sequence that relates the cohomology of a smooth variety admitting a projective morphism onto a projective base to the cohomology of the open set obtained by removing the preimage of a general linear section. We show that this sequence respects the perverse Leray filtration and induces exact sequences of mixed Hodge structures on its graded pieces. We conjecture that this exact sequence should be thought of as mirror to the Clemens–Schmid sequence, which describes the cohomology of degenerations. We exhibit this mirror relationship explicitly for all Type II and many Type III degenerations of K3 surfaces. In three dimensions, we show that for Tyurin degenerations of Calabi–Yau threefolds our conjecture is a consequence of existing mirror conjectures, and we explicitly verify our conjecture for a more complicated degeneration of the quintic threefold.

Extendability of differential forms via Cartier operators

Let X be a normal variety over a perfect field of positive characteristic and B a reduced divisor on X . We prove that if the Cartier isomorphism on the log smooth locus of (X,B) extends to the entire X , then (X,B) satisfies the logarithmic extension theorem for differential forms. As an application, we show that the logarithmic extension theorem holds for good quotients of smooth varieties by actions of reduced linearly reductive group schemes. In addition, the logarithmic extension theorem for one-forms holds for singularities of higher codimension under assumptions about Serre’s condition. We also prove that tame quotients satisfy the regular extension theorem.

Approximation of semistable bundles on smooth algebraic varieties

We prove some strong results on approximation of strongly semistable bundles with vanishing numerical Chern classes by filtrations, whose quotients are line bundles of similar slope. This generalizes some earlier results of Parameswaran–Subramanian in the curve case and Koley–Parameswaran in the surface case and it confirms the conjecture posed by Koley and Parameswaran.

Multiplicative Thom-Sebastiani for Bernstein-Sato polynomials

We show that if f∈OX(X) and g∈OY(Y) are nonzero regular functions on smooth complex algebraic varieties X and Y, then the Bernstein-Sato polynomial of the product function fg∈OX×Y(X×Y) is given by bfg(s)=bf(s)bg(s), answering a question of Budur and Popa.

Differential geometry and general relativity with algebraifolds

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of algebraifold, by which we mean a commutative algebra A for which the module of derivations of A is finitely generated projective. Over R as the base ring, this class of algebras includes the algebra C∞(M) of smooth functions on a manifold M, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on M, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion M→N makes C∞(M) into an algebraifold with C∞(N) as the base ring. Over any field k of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.

Investigating Secant Varieties

A variety is a geometric object that locally looks like the set of points that are the solutions of some polynomial equations. Any given variety gives rise to a sequence of secant varieties indexed by the nonnegative integers. The kth secant variety is the union of all planes spanned by k+1 points on the original variety. Like vector spaces in linear algebra, varieties have an intrinsic dimension. A naive parameter count produces an upper bound on the dimension of the kth secant variety called its expected dimension. A secant variety is called defective if its actual dimension is less than its expected dimension. Despite being well understood in a few interesting cases, the classification of varieties with defective secant varieties in general remains far from complete. For a specific class of varieties called smooth toric varieties, our goal is to determine which ones admit defective secant varieties. Using the computer algebra system Macaulay2, we conducted experiments on this class of varieties and collected data on defectiveness, focusing on varieties of low dimension and Picard rank. Toric varieties have strong connections to combinatorics and convex polytopes; for any toric variety, there is a corresponding polytope that contains data about the variety. From the polytope, we are able to construct a particular kind of matrix that encodes information about the variety and its secants, which allowed us to prove defectivity for various Picard rank 2 toric varieties. Conversely, applying results from a paper by Laface–Massarenti–Rischter (2022) to the polytopes, we were able to prove non-defectivity of all secant varieties of certain toric surfaces with Picard rank 2. These statements together yield a classification of defective secant varieties for a subset of 2-dimensional smooth toric varieties with Picard rank 2.

Twisting the Infinitesimal Site

Abstract We classify twistings of Grothendieck’s differential operators on a smooth variety $X$ in prime characteristic $p$. We prove that isomorphism classes of twistings are in bijection with $H^{2}(X,\mathbb{Z}_{p}(1))$, the degree 2, weight 1 syntomic cohomology of $X$. We also discuss the relationship between twistings of crystalline and Grothendieck differential operators. Twistings in mixed characteristic are also analyzed.

Weil cohomology theories and their motivic Hopf algebroids

In this paper we discuss a general notion of Weil cohomology theories, both in algebraic geometry and in rigid analytic geometry. We allow our Weil cohomology theories to have coefficients in arbitrary commutative ring spectra. Using the theory of motives, we give three equivalent viewpoints on Weil cohomology theories: as a cohomology theory on smooth varieties, as a motivic spectrum and as a realization functor. We also associate to every Weil cohomology theory a motivic Hopf algebroid generalizing the construction we gave in Ayoub (2014) for the Betti cohomology. Exploiting results and constructions from Ayoub (2020), we are able to prove that the motivic Hopf algebroids of all the classical Weil cohomology theories are connective. In particular, they give rise to motivic Galois groupoids which are spectral affine groupoid schemes.

Ecological niche modeling to identify cultivation areas for pineapple in the Republic of Benin

Pineapple is one of the most important tropical fruit species, widely cultivated and economically important in Benin. This study aimed to identify potentially favorable areas for the cultivation of pineapple under current and future environmental conditions in Benin. The two cultivars of pineapple grown in Benin were separately considered: Sugarloaf and Smooth Cayenne. Five (05) modeling algorithms such as Maxent, Random Forest (RF), Support-Vector Machines (SVM), Boosted Regression Trees (BRT) and Generalized Linear Model (GLM) were compared using the criteria: area under the curve (AUC), sensitivity, specificity, Cohen’s Kappa, deviance and True Skill Statistic (TSS). The future climate models available for Africa at horizon 2055 were used under the “Representative Concentration Pathways” scenario 4.5 and 8.5. Results suggested that pineapple suitable areas were governed by a combination of effects of climate (temperature and precipitation) and soils characteristics. Indeed, soil pH, temperature seasonality and precipitation of driest quarter were the main variables driving pineapple production in Benin. Results also indicated that RF was the most suitable technique to model the distribution of pineapples regardless of the variety. The current potential range of favorable areas for the two varieties was mainly found in the central and southern parts of the country. In the future, following the RCP4.5 scenario, there will be an increase in the area favorable for the cultivation of Smooth Cayenne variety by 5.28% compared to the current situation whereas, the area favorable for the cultivation of the Sugarloaf variety will be increased by 7.7%. However, suitable areas for cultivation of Smooth Cayenne and Sugarloaf following the RCP8.5 scenario will be increased, respectively by 21.82% and 31.64%. The low and medium suitability areas for the cultivation of smooth cayenne will decrease by 15.57% and 2.93%, respectively at the horizon 2055 with future conditions under RCP4.5, and 15.48% and 4.97%, respectively at the horizon 2055 with future conditions under RCP8.5. For sugarloaf, the low and medium suitable cultivation areas will decrease by 1.59% and 14.24, respectively at the horizon 2055 with future conditions under RCP4.5. According to RCP8.5, the low suitable areas will decrease by 5.08%. This study constitutes an initial step towards a sustainable scheme for planning exploration of the possibility of extending pineapple cultivation in Benin. Key words: Climate change, modeling, algorithms, pineapple, potential area distribution

Vanishing of local cohomology with applications to Hodge theory

Let H=((H,F•),L) be a polarized variation of Hodge structure on a smooth quasi-projective variety U. By M. Saito's theory of mixed Hodge modules, the variation of Hodge structure H can be viewed as a polarized Hodge module M∈HM(U). Let X be a compactification of U, and j:U↪X is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study j+M∈DbMHM(X). In particular, we study the graded pieces of the de Rham complex GrpFDR(j+M)∈Dcohb(X), and the Hodge structure of Hi(U,L) for i in sufficiently low degrees.

Extensions representing Nori-Srinivas obstruction

Let (X,F) be a pair of a smooth variety X over an algebraically closed field k of characteristic p>0 and its Frobenius morphism F. Given a Frobenius Wn(k)-lifting (X¯,F¯) of the pair (X,F) for n≥1, Nori and Srinivas [9] determined the obstruction obsX¯,F¯∈Ext(ΩX/k1,BF⁎ΩX/k1) to Frobenius Wn+1(k)-lifting of (X¯,F¯) in terms of Čech cohomology. The extension representing obsX¯,F¯ has been only known for n=1, which uses the Cartier operator. In this paper, we interpret obsX¯,F¯ in terms of Kato's version of de Rham-Witt Cartier operator [8] and determine the extension representing obsX¯,F¯ for n≥2.

Some remarks on the Kp,1$\mathcal {K}_{p,1}$ theorem

AbstractLet be a non‐degenerate projective irreducible variety of dimension , degree , and codimension over an algebraically closed field of characteristic 0. Let be the th graded Betti number of . Green proved the celebrating ‐theorem about the vanishing of for high values for and potential examples of nonvanishing graded Betti numbers. Later, Nagel–Pitteloud and Brodmann–Schenzel classified varieties with nonvanishing . It is clear that when there is an ‐dimensional variety of minimal degree containing , however, this is not always the case as seen in the example of the triple Veronese surface in .In this paper, we completely classify varieties with nonvanishing such that does not lie on an ‐dimensional variety of minimal degree. They are exactly cones over smooth del Pezzo varieties, whose Picard number is .

Reductive covers of klt varieties

In this article, we study G -covers of klt varieties, where G is a reductive group. First, we exhibit an example of a klt singularity admitting a \mathbb{P}{\mathrm{GL}}_{n}(\mathbb{K}) -cover that is not of klt type. Then, we restrict ourselves to G -quasi-torsors, a special class of G -covers that behave like G -torsors outside closed subsets of codimension two. Given a G -quasi-torsor X\rightarrow Y , where G is a finite extension of a torus \mathbb{T} , we show that X is of klt type if and only if Y is of klt type. We prove a structural theorem for \mathbb{T} -quasi-torsors over normal varieties in terms of Cox rings. As an application, we show that every sequence of \mathbb{T} -quasi-torsors over a variety with klt type singularities is eventually a sequence of \mathbb{T} -torsors. This is the torus version of a result due to Greb–Kebekus–Peternell regarding finite quasi-torsors of varieties with klt type singularities. On the contrary, we show that in any dimension there exists a sequence of finite quasi-torsors and \mathbb{T} -quasi-torsors over a klt type variety, such that infinitely many of them are not torsors. We show that every variety with klt type singularities is a quotient of a variety with canonical factorial singularities. We prove that a variety with Zariski locally toric singularities is indeed the quotient of a smooth variety by a solvable group. Finally, motivated by the work of Stibitz, we study the optimal class of singularities for which the previous results hold.

On the Smith–Thom deficiency of Hilbert squares

AbstractWe give an expression for the Smith–Thom deficiency of the Hilbert square of a smooth real algebraic variety in terms of the rank of a suitable Mayer– Vietoris mapping in several situations. As a consequence, we establish a necessary and sufficient condition for the maximality of in the case of projective complete intersections, and show that with a few exceptions, no real nonsingular projective complete intersection of even dimension has maximal Hilbert square. We also provide new examples of smooth real algebraic varieties with maximal Hilbert square.

Milnor–Wood inequality for klt varieties of general type and uniformization

AbstractWe generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor–Wood‐type inequality and we characterize the corresponding maximal representations.

Singular contact varieties

In this note we propose the generalization of the notion of a holomorphic contact structure on a manifold (smooth variety) to varieties with rational singularities and prove basic properties of such objects. Natural examples of singular contact varieties come from the theory of nilpotent orbits: every projectivization of the closure of a nilpotent orbit in a semisimple Lie algebra satisfies our definition after normalization. We show the correspondence between symplectic varieties with the structure of a C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^*$$\\end{document}-bundle and the contact ones along with the existence of stratification à la Kaledin. In the projective case we demonstrate the equivalence between crepant and contact resolutions of singularities, show the uniruledness and give a full classification of projective contact varieties in dimension 3.

𝑉-filtrations and minimal exponents for local complete intersections

Abstract We define and study a notion of minimal exponent for a local complete intersection subscheme 𝑍 of a smooth complex algebraic variety 𝑋, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara–Malgrange 𝑉-filtration associated to 𝑍. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology H Z r ⁢ ( O X ) \mathcal{H}^{r}_{Z}(\mathcal{O}_{X}) , where 𝑟 is the codimension of 𝑍 in 𝑋. We also study its relation to the Bernstein–Sato polynomial of 𝑍. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension 1 case. A key ingredient for our main result is a description of the Kashiwara–Malgrange 𝑉-filtration associated to any ideal ( f 1 , … , f r ) (f_{1},\ldots,f_{r}) in terms of the microlocal 𝑉-filtration associated to the hypersurface defined by ∑ i = 1 r f i ⁢ y i \sum_{i=1}^{r}f_{i}y_{i} .

On the Brauer groups of fibrations

Let X→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {X}}\\rightarrow C$$\\end{document} be a flat k-morphism between smooth integral varieties over a finitely generated field k such that the generic fiber X is smooth, projective and geometrically connected. Assuming that C is a curve with function field K, we build a relation between the Tate-Shafarevich group of PicX/K0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Pic}^0_{X/K}$$\\end{document} and the geometric Brauer groups of X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {X}}$$\\end{document} and X, generalizing a theorem of Artin and Grothendieck for fibered surfaces to higher relative dimensions.

Hochschild cohomology of differential operators in positive characteristic

For k a field of positive characteristic and X a smooth variety over k, we compute the Hochschild cohomology of Grothendieck's differential operators on X. The answer involves the derived inverse limit of the Frobenius acting on the cohomology of the structure sheaf of X.