Smooth Variety Research Articles

Quantisation of derived Lagrangians

We investigate quantisations of line bundles $\mathcal{L}$ on derived Lagrangians $X$ over $0$-shifted symplectic derived Artin $N$-stacks $Y$. In our derived setting, a deformation quantisation consists of a curved $A_{\infty}$ deformation of the structure sheaf $\mathcal{O}_{Y}$, equipped with a curved $A_{\infty}$ morphism to the ring of differential operators on $\mathcal{L}$; for line bundles on smooth Lagrangian subvarieties of smooth symplectic algebraic varieties, this simplifies to deforming $(\mathcal{L}, \mathcal{O}_{Y})$ to a DQ module over a DQ algebroid. For each choice of formality isomorphism between the $E_2$ and $P_2$ operads, we construct a map from the space of non-degenerate quantisations to power series with coefficients in relative cohomology groups of the respective de Rham complexes. When $\mathcal{L}$ is a square root of the dualising line bundle, this leads to an equivalence between even power series and certain anti-involutive quantisations, ensuring that the deformation quantisations always exist for such line bundles. This gives rise to a dg category of algebraic Lagrangians, an algebraic Fukaya category of the form envisaged by Behrend and Fantechi. We also sketch a generalisation of these quantisation results to Lagrangians on higher $n$-shifted symplectic derived stacks.

A -adic monodromy theorem for de Rham local systems

We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a$p$-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms.

Curve classes on Calabi–Yau complete intersections in toric varieties

AbstractWe prove the integral Hodge conjecture for curve classes on smooth varieties of dimension at least three constructed as a complete intersection of ample hypersurfaces in a smooth projective toric variety, such that the anticanonical divisor is the restriction of a nef divisor. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric minimal model program, we prove that in each case, is generated by classes of rational curves.

Algebraic foliations and derived geometry: the Riemann–Hilbert correspondence

This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations defined in terms of differential ideals in the algebra of forms. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety X is formally integrable at any point, and, if we suppose that its singular locus has codimension ge 2, its analytification is a locally integrable singular foliation on the associated complex manifold X^h. We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on X, and prove a Riemann-Hilbert correspondence for them when X is proper. We discuss several examples and applications.

Equivariant Cobordism of Smooth Projective Spherical Varieties

AbstractWe study the equivariant cobordism rings for the action of a torusTon smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rationalT-equivariant cobordism rings of smooth projectiveG-spherical varieties with the action of a maximal torusTofG. As an application, we obtain explicit presentations for the rational equivariant cobordism rings of smooth projective horospherical varieties of Picard number one.

On the characterization of abelian varieties for log pairs in zero and positive characteristic

Let (X,Δ) be a pair. We study how the values of the log Kodaira dimension and log plurigenera relates to surjectivity and birationality of the Albanese map and the Albanese morphism of X in both characteristic 0 and characteristic p>0. In particular, we generalize some well known results for smooth varieties in both zero and positive characteristic to varieties with various types of singularities. Moreover, we show that if X is a normal projective threefold in characteristic p>0, the coefficients of the components of Δ are ≤1 and -(K X +Δ) is semiample, then the Albanese morphism of X is surjective under certain assumptions on p and the singularities of the general fibers of the Albanese morphism. This is a positive characteristic analogue in dimension 3 of a result of Zhang on a conjecture of Demailly–Peternell–Schneider.

K-orbit closures and Barbasch–Evens–Magyar varieties

We define the Barbasch-Evens-Magyar varieties. We show they are isomorphic to the smooth varieties defined in [D.~Barbasch-S.~Evens '94] that map generically finitely to symmetric orbit closures, thereby giving resolutions of singularities in certain cases. Our definition parallels [P.~Magyar '98]'s construction of the Bott-Samelson varieties [H.~C.~Hansen '73, M.~Demazure '74]. From this alternative viewpoint, one deduces a graphical description in type $A$, stratification into closed subvarieties of the same kind, and determination of the torus-fixed points. Moreover, we explain how these manifolds inherit a natural symplectic structure with Hamiltonian torus action. We then express the moment polytope in terms of the moment polytope of a Bott-Samelson variety.

Special MMP for log canonical generalised pairs (with an appendix joint with Xiaowei Jiang)

We show that minimal models of mathbb {Q}-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a mathbb {Q}-factorial NQC log canonical generalised pair (X,B+M) we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that K_X+B+M is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces.

Deformations of Semi-Smooth Varieties

Abstract For a singular variety $X$, an essential step to determine its smoothability and study its deformations is the understanding of the tangent sheaf and of the sheaf $\mathcal {T}^{1}_{X}:={{\mathcal {E}}xt}^{1}(\Omega _{X},\mathcal {O}_{X})$. A variety is semi-smooth if its singularities are étale locally the product of a double crossing point ($uv=0$) or a pinch point ($u^2-v^2w=0$) with affine space; equivalently, if it can be obtained by gluing a smooth variety along a smooth divisor via an involution with smooth quotient. Our main result is the explicit computation of the tangent sheaf and the sheaf $\mathcal {T}^{1}_{X}$ for a semi-smooth variety $X$ in terms of the gluing data.

Existence of embeddings of smooth varieties into linear algebraic groups

We prove that every smooth affine variety of dimension d d embeds into every simple algebraic group of dimension at least 2 d + 2 2d+2 . We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and build upon parametric transversality results for flexible affine varieties due to Kaliman. By adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show that our result is optimal up to a possible improvement of the bound to 2 d + 1 2d+1 . In order to study the limits of our embedding method, we use rational homology group calculations of homogeneous spaces and we establish a domination result for rational homology of complex smooth varieties.

Saturation bounds for smooth varieties

We prove bounds on the saturation degrees of homogeneous ideals (and their powers) defining smooth complex projective varieties. For example, we show that a classical statement due to Macualay for zero-dimensional complete intersection ideals holds for any smooth variety. For curves, we bound the saturation degree of powers in terms of the regularity.

Compatibility of weak approximation for zero-cycles on products of varieties

The Brauer-Manin obstruction is conjectured to be the only obstruction to weak approximation for zero-cycles on proper smooth varieties defined over number fields. We prove that the conjecture is compatible for products of rationally connected varieties, K3 surfaces, Kummer varieties and one curve.

On the isomorphism problem for the rings of differential operators on smooth affine varieties

We show that given two smooth affine varieties over C such that their rings of differential operators are Morita equivalent, then corresponding cotangent bundles are isomorphic as symplectic varieties.

Universal birational invariants and -homology

We prove that the functor of zeroth -homology is universal as a functorial birational invariant of smooth proper varieties taking values in a category enriched by abelian groups, when the ground field admits resolution of singularities. Communicated by Tony Varilly-Alvarado

Moduli of Stokes torsors and singularities of differential equations

Let \mathcal{M} be a meromorphic connection with poles along a smooth divisor D in a smooth algebraic variety. Let \operatorname{Sol} \mathcal{M} be the solution complex of \mathcal{M} . We prove that the good formal structure locus of \mathcal{M} coincides with the locus where the restrictions to D of \operatorname{Sol} \mathcal{M} and \operatorname{Sol} \operatorname{End}\mathcal{M} are local systems. Despite the very different natures of these loci (the first one is defined via algebra, and the second via analysis), the proof of their coincidence is geometric. It relies on moduli of Stokes torsors.

Evaluation of various varieties of peas as raw material for deep processing

The article provides an overview of the criteria for choosing pea varieties to evaluate economically valuable traits. For the effective use of peas as raw materials for deep processing, two directions can be considered: a joint increase in the content of protein and starch or the choice of varieties with the required indicators. The results of laboratory studies of processing for starch and protein isolate of 37 variety samples of peas from the collection of the Federal State Budgetary Scientific Institution «Federal Scientific Center of Legumes and Groat Crops» are presented. As evaluation criteria determining high technological properties, providing the most complete extraction of starch and protein isolate there have been selected coefficients of extraction of these components. When selecting a variety of peas, it is also necessary to consider the physicochemical properties of starch, which depend on the mass fraction of amylose. According to the results obtained, the most promising varieties for the processing of grain into starch and protein isolate are Orlovchanin, Alla, Rodnik, Sibirsky 1, Pharaon, Jaguar, Sophia. The most efficient extraction of the main processed products ‒ protein isolate and starch is established for the variety Alla, protein extraction coefficient 81.2 %, starch 90.6 %. Among the wrinkled pea varieties for industrial processing, the Amius 1241 breeding line is the most interesting, with protein isolate and starch extraction coefficients of 70.3 and 69.1 % and a mass fraction of amylose in starch of 81.5 %. It is found that the mass fraction of starch and protein in pea grains does not correlate with the extraction coefficients of protein and starch isolate during processing by two-stage extraction of protein from pea flour using the alkaline method. When processing wrinkled pea varieties, there are difficulties in separating the starch and residual protein undissolved in alkaline medium, since these pea varieties are found to have complex starch granules, which are easily crushed into small irregular and polygonal granules. In smooth pea varieties, the starch granules are oval in shape with a high percentage of large size and therefore there are no problems with separation of the starch-protein suspension. The dependence of viscosity of 3.0 % pea starch paste on the mass fraction of amylose in pea starch has been determined. The choice of pea variety for processing makes it possible to produce starch that has lifetime modification, i.e., that has the required physical and chemical properties and resistance. In addition, depending on the choice of pea variety, it is possible to predict the yield of isolate and protein concentrate.

Smooth weight structures and birationality filtrations on motivic categories

In various triangulated motivic categories, a vast family of aisles (these are certain classes of objects) is introduced. These aisles are defined in terms of the corresponding “motives” (or motivic spectra) of smooth varieties; it is proved that they are expressed in terms of the corresponding homotopytt-structures. The aisles in question are described in terms of stalks at function fields, and it is shown that they widely generalize the ones corresponding to slice filtrations. Further, the filtrations on the “homotopy hearts”Ht_homeff{\underline {Ht}}_{\mathrm {hom}}^{\mathrm {eff}}of the corresponding effective subcategories that are induced by these aisles can be described in terms of (Nisnevich) sheaf cohomology as well as in terms of the Voevodsky contractions−−1-_{-1}. Respectively, the condition for an object ofHt_homeff{\underline {Ht}}_{\mathrm {hom}}^{\mathrm {eff}}to be weakly birational (i.e., for its(n+1)(n+1)st contraction to be trivial, or equivalently, its Nisnevich cohomology to vanish in degrees strictly greater thannnfor somen≥0n\ge 0) are expressed in terms of these aisles; this statement generalizes well-known results of Kahn and Sujatha. Next, these classes give rise to weight structureswSmoothsw_{\mathrm {Smooth}}^{s}(where thes=(sj)s=(s_{j})are nondecreasing sequences parametrizing our aisles) that vastly generalize the Chow weight structureswChoww_{\operatorname {Chow}}defined earlier. By using general abstract nonsense, the correspondingadjacenttt-structurestSmoothst_{\mathrm {Smooth}}^{s}are constructed and it is proved that they give the birationality filtrations onHt_homeff{\underline {Ht}}^{\mathrm {eff}}_{\mathrm {hom}}.Moreover, some of these weight structures induce weight structures on the correspondingnn-birational motivic categories (these are the localizations by the levels of the slice filtrations). The results also yield some new unramified cohomology calculations.

The numerical algebraic geometry of bottlenecks

This is a computational study of bottlenecks on algebraic varieties. The bottlenecks of a smooth variety X⊆Cn are the lines in Cn which are normal to X at two distinct points. The main result is a numerical homotopy that can be used to approximate all isolated bottlenecks. This homotopy has the optimal number of paths under certain genericity assumptions. In the process we prove bounds on the number of bottlenecks in terms of the Euclidean distance degree. Applications include the optimization problem of computing the distance between two real varieties. Also, computing bottlenecks may be seen as part of the problem of computing the reach of a smooth real variety and efficient methods to compute the reach are still to be developed. Relations to triangulation of real varieties and meshing algorithms used in computer graphics are discussed in the paper. The resulting algorithms have been implemented with Bertini [4] and Macaulay2 [17].

Minimal slope conjecture of F-isocrystals

The minimal slope conjecture, which was proposed by K. S. Kedlaya, asserts that two irreducible overconvergent F-isocrystals on a smooth variety are isomorphic to each other if both minimal slope constitutions of slope filtrations are isomorphic to each other. We affirmatively solve the minimal slope conjecture for overconvergent F-isocrystals on curves and for overconvergent $$\overline{{\mathbb {Q}}}_p$$ -F-isocrystals on smooth varieties over finite fields.

Bit complexity for computing one point in each connected component of a smooth real algebraic set

We analyze the bit complexity of an algorithm for the computation of at least one point in each connected component of a smooth real algebraic set. This work is a continuation of our analysis of the hypersurface case (On the bit complexity of finding points in connected components of a smooth real hypersurface, ISSAC'20). In this paper, we extend the analysis to more general cases.Let F=(f1,…,fp) in Z[X1,…,Xn]p be a sequence of polynomials with V=V(F)⊂Cn a smooth and equidimensional variety and 〈F〉⊂C[X1,…,Xn] a radical ideal. To compute at least one point in each connected component of V∩Rn, our starting point is an algorithm by Safey El Din and Schost (Polar varieties and computation of one point in each connected component of a smooth real algebraic set, ISSAC'03). This algorithm uses random changes of variables that are proven to generically ensure certain desirable geometric properties. The cost of the algorithm was given in an algebraic complexity model; here, we analyze the bit complexity and the error probability, and we provide a quantitative analysis of the genericity statements. In particular, we are led to use Lagrange systems to describe polar varieties, as they make it simpler to rely on techniques such as weak transversality and an effective Nullstellensatz.