Projective Complete Intersection Research Articles

Period integrals of smooth projective complete intersections as exponential periods

Let X be a smooth projective complete intersection over Q of dimension n−k in the projective space PQn defined by the zero locus of f_(x_)=(f1(x_),⋯,fk(x_)), for given positive integers n and k. For a given primitive homology cycle [γ]∈Hn−k(X(C),Z)0, the period integral is defined to be a linear map from the primitive de Rham cohomology group HdR,primn−k(X(C);Q) to C given by [ω]↦∫γω. The goal of this article is to interpret this period integral as Feynman's path integral of 0-dimensional quantum field theory with the action functional S=∑ℓ=1kyℓfℓ(x_) (in other words, the exponential period with the action functional S) and use this interpretation to develop a formal deformation theory of period integrals of X, which can be viewed as a modern deformation theoretic treatment of the period integrals based on the Maurer-Cartan equation of a dgla (differential graded Lie algebra).

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.

Conjecture 𝒪 for Projective Complete Intersections

Abstract Conjecture ${{\mathcal{O}}}$ and Gamma conjectures for quantum cohomology of Fano manifolds were proposed by Galkin, Golyshev, and Iritani [ 16]. We prove Conjecture ${{\mathcal{O}}}$ for Fano complete intersections in projective spaces. The main tools to compute relevant genus-zero Gromov–Witten invariants with primitive insertions are the genus-one topological recursion relation and Zinger’s standard-versus-reduced formula. We also prove a related conjecture of Galkin [ 15] for projective Fano complete intersections.

Framed duality and mirror symmetry for toric complete intersections

This paper is devoted to systematically extend f-mirror symmetry between families of hypersurfaces in complete toric varieties, as introduced in [17], to families of complete intersections subvarieties. Namely, f-mirror symmetry is induced by framed duality of framed toric varieties extending Batyrev-Borisov polar duality between Fano toric varieties. Framed duality has been defined and essentially well described for families of hypersurfaces in toric varieties in the previous [17]. Here it is developed for families of complete intersections, allowing us to strengthen some previous results on hypersurfaces. In particular, the class of projective complete intersections and their mirror partners are studied in detail. Moreover, a (generalized) Landau-Ginzburg/Complete-Intersection correspondence is discussed, extending to the complete intersection setup the LG/CY correspondence firstly studied Chiodo-Ruan and Krawitz.

Systematics of perturbatively flat flux vacua for CICYs

In this paper, we extend the analysis of scanning the perturbatively flat flux vacua (PFFV) for the type IIB orientifold compactifications on the mirror of the projective complete intersection Calabi-Yau (pCICY) 3-folds, which are realized as hypersurfaces in the product of complex projective spaces. The main objective of this scan is to investigate the behaviour of PFFV depending on the nature of CY 3-folds in the light of the observations made in [1] where it has been found that K3-fibered CY 3-folds have significantly large number of physical vacua as compared to other geometries. For this purpose, we present the PFFV statistics for all the 36 pCICYs with h1,1 = 2 and classify them into two categories of being K3-fibered model and non K3-fibered model. We subsequently confirm that all the K3-fibered models have a significantly large number of PFFV leading to physical vacua by fixing the axio-dilaton by non-perturbative effects, while only a couple of non K3-fibered models have such physical vacua. For h1,1 = 2 case, we have found that there are five pCICY 3-folds with the suitable exchange symmetry leading to the so-called exponentially flat flux vacua (EFFV) which are protected against non-perturbative prepotential effects as well. By exploring the underlying exchange symmetries in the favorable CY 3-folds with h1,1 ≥ 3 in the dataset of 7820 pCICYs, we have found that there are only 13 spaces which can result in EFFV configurations, and therefore most of the CY 3-folds are a priory suitable for fixing the dilaton valley of the flat vacua using the non-perturbative prepotential contributions.

Divisor topologies of CICY 3-folds and their applications to phenomenology

In this article, we present a classification for the divisor topologies of the projective complete intersection Calabi-Yau (pCICY) 3-folds realized as hypersurfaces in the product of complex projective spaces. There are 7890 such pCICYs of which 7820 are favorable, and can be subsequently useful for phenomenological purposes. To our surprise we find that the whole pCICY database results in only 11 (so-called coordinate) divisors (D) of distinct topology and we classify those surfaces with their possible deformations inside the pCICY 3-fold, which turn out to be satisfying 1 ≤ h2,0(D) ≤ 7. We also present a classification of the so-called ample divisors for all the favorable pCICYs which can be useful for fixing all the (saxionic) Kähler moduli through a single non-perturbative term in the superpotential. We argue that this relatively unexplored pCICY dataset equipped with the necessary model building ingredients, can be used for a systematic search of physical vacua. To illustrate this for model building in the context of type IIB CY orientifold compactifications, we present moduli stabilization with some preliminary analysis of searching possible vacua in simple models, as a template to be adopted for analyzing models with a larger number of Kähler moduli.

LAURENT POLYNOMIAL LANDAU–GINZBURG MODELS FOR COMINUSCULE HOMOGENEOUS SPACES

In this article we construct Laurent polynomial Landau–Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau–Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.

On holomorphic distributions on Fano threefolds

This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds X which is threedimensional and with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on X. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable.

The degree of irrationality of hypersurfaces in various Fano varieties

The purpose of this paper is to compute the degree of irrationality of hypersurfaces of sufficiently high degree in various Fano varieties: quadrics, Grassmannians, products of projective space, cubic threefolds, cubic fourfolds, and complete intersection threefolds of type (2,2). This extends the techniques of Bastianelli, De Poi, Ein, Lazarsfeld, and the second author who computed the degree of irrationality of hypersurfaces of sufficiently high degree in projective space. A theme in the paper is that the fibers of low degree rational maps from the hypersurfaces to projective space tend to lie on curves of low degree contained in the Fano varieties. This allows us to study these maps by studying the geometry of curves in these Fano varieties.

$${\mathbb A}^1$$ A 1 -curves on affine complete intersections

We generalize the results of Clemens, Ein, and Voisin regarding rational curves and zero cycles on generic projective complete intersections to the logarithmic setup.

Riemann–Roch for homotopy invariant K-theory and Gysin morphisms

We prove the Riemann–Roch theorem for homotopy invariant K-theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions. We also prove a new Riemann–Roch theorem for the relative cohomology of a morphism.In order to do so, we construct and characterize new Gysin morphisms for regular immersions between cohomologies represented by spectra (examples include homotopy invariant K-theory, motivic cohomology, their arithmetic counterparts, real absolute Hodge and Deligne–Beilinson cohomology, rigid syntomic cohomology, mixed Weil cohomologies) and we use this construction to prove a motivic version of the Riemann–Roch.

Koppelman formulas on affine cones over smooth projective complete intersections

In the present paper, we study regularity of the Andersson-Samuelsson Koppelman integral operator on affine cones over smooth projective complete intersections. Particularly, we prove $L^p$- and $C^\alpha$-estimates, and compactness of the operator, when the degree is sufficiently small. As applications, we obtain homotopy formulas for different $\overline{\partial}$-operators acting on $L^p$-spaces of forms, including the case $p=2$ if the varieties have canonical singularities. We also prove that the $\mathcal{A}$-forms introduced by Andersson-Samuelsson are $C^\alpha$ for $\alpha < 1$.

A direct algorithm to compute the topological Euler characteristic and Chern–Schwartz–MacPherson class of projective complete intersection varieties

Let V be a possibly singular scheme-theoretic complete intersection subscheme of Pn over an algebraically closed field of characteristic zero. Using a recent result of Fullwood (“On Milnor classes via invariants of singular subschemes”, Journal of Singularities) we develop an algorithm to compute the Chern–Schwartz–MacPherson class and Euler characteristic of V. This algorithm complements existing algorithms by providing performance improvements in the computation of the Chern–Schwartz–MacPherson class and Euler characteristic for certain types of complete intersection subschemes of Pn.

The genus 0 Gromov–Witten invariants of projective complete intersections

We describe the structure of mirror formulas for genus 0 Gromov-Witten invariants of projective complete intersections with any number of marked points and provide an explicit algorithm for obtaining the relevant structure coefficients. The structural description alone suffices for some qualitative applications, such as vanishing results and the bounds on the growth of these invariants predicted by R. Pandharipande.

NON-FACTORIAL NODAL COMPLETE INTERSECTION THREEFOLDS

We give a bound on the minimal number of singularities of a nodal projective complete intersection threefold which contains a smooth complete intersection surface that is not a Cartier divisor.

The genus one Gromov-Witten invariants of Calabi-Yau complete intersections

We obtain mirror formulas for the genus 1 Gromov-Witten invariants of projective Calabi-Yau complete intersections. We follow the approach previously used for projective hypersurfaces by extending the scope of its algebraic results; there is little change in the geometric aspects. As an application, we check the genus 1 BPS integrality predictions in low degrees for all projective complete intersections of dimensions 3, 4, and 5.

NOETHER–LEFSCHETZ THEORY AND NÉRON–SEVERI GROUP

Let Z be a closed subscheme of a smooth complex projective complete intersection variety Y ⊆ ℙN, with dim Y = 2r + 1 ≥ 3. We describe the Néron–Severi group NSr(X) of a general smooth hypersurface X ⊂ Y of sufficiently large degree containing Z.

Non-genericity of infinitesimal variations of Hodge structures arising in some geometric contexts

AbstractWe prove that the infinitesimal variations of Hodge structure arising in a number of geometric situations are non-generic. In particular, we consider the case of generic hypersurfaces in complete smooth projective toric varieties, generic hypersurfaces in weighted projective spaces and generic complete intersections in projective space and show that, for sufficiently high degrees, the corresponding infinitesimal variations are non-generic.

Topological invariants of isolated complete intersection curve singularities

In this paper we present some formulae for topological invariants of projective complete intersection curves with isolated singularities in terms of the Milnor number, the Euler characteristic and the topological genus. We also present some conditions, involving the Milnor number and the degree of the curve, for the irreducibility of complete intersection curves.

Homeomorphism classification of complex projective complete intersections of dimensions 5, 6 and 7

In this paper we prove that two complete intersections \({X_n(\underline{d})}\) and \({X_n(\underline{d}^\prime)}\) are homeomorphic if and only if they have the same total degree, Pontrjagin classes and Euler characteristics, provided n = 5, 6, 7. This extends earlier result of Fang and Klaus (Manuscr Math 90:139–147, 1996).