Projective Algebraic Variety Research Articles

System of Hodge bundles and generalized opers on smooth projective varieties

Let k be an algebraically closed field of any characteristic. Let X be a polarized irreducible smooth projective algebraic variety over k. We give criterion for semistability and stability of system of Hodge bundles on X. We define notion of generalized opers on X, and prove semistability of the Higgs bundle associated to generalized opers. We also show that existence of partial oper structure on a vector bundle E together with a connection ∇ over X implies semistability of the pair (E,∇).

Rational differential forms on the variety of flexes of plane cubics

We prove that for every positive integer $d$, there are no nonzero regular differential $d$-forms on every smooth irreducible projective algebraic variety birationally isomorphic to the variety of flexes of plane cubics.

Completeness of Finite-Rank Differential Varieties

Differential algebraic geometry offers tantalizing similarities to the algebraic version as well as puzzling anomalies. This thesis builds on results of Kolchin, Blum, Morrison, van den Dries, and Pong to study the problem of completeness for projective differential varieties. The classical fundamental theorem of elimination theory asserts that if V is a projective algebraic variety defined over an algebraically closed field K and W is any algebraic variety defined over K, then the projection VxW -> W takes Zariski-closed sets to Zariski-closed sets. Differential varieties defined by differential polynomial equations over a differentially closed field are more complicated. We give the first example of an incomplete finite-rank differential variety, as well as new instances of complete differential varieties. We also explain how model theory yields multiple versions of Pong's valuative criterion for completeness and reduces the differential completeness problem to one involving algebraic varieties over the complex numbers.

Fixed point theorems involving numerical invariants

We exhibit invariants of smooth projective algebraic varieties with integer values, whose nonvanishing modulo$p$prevents the existence of an action without fixed points of certain finite$p$-groups. The case of base fields of characteristic$p$is included. Counterexamples are systematically provided to test the sharpness of our results.

Examples of non-algebraic classes in the Brown–Peterson tower

For every $$n\ge 0$$ , we construct classes in the Brown–Peterson cohomology $$BP\langle n \rangle $$ of smooth projective complex algebraic varieties which are not in the image of the cycle map from the corresponding motivic Brown–Peterson cohomology. This generalizes the examples of Atiyah and Hirzebruch to all finite levels in the Brown–Peterson tower.

Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound

We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Grobner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre’s inequality for the number of points of projective hypersurfaces over finite fields is included.

Algebraic stratified general position and transversality

The method of Whitney interpolation is used to construct, for any real or complex projective algebraic variety, a stratified submersive family of self-maps that yields stratified general position and transversality theorems for semialgebraic subsets.

Towards the Green–Griffiths–Lang conjecture via equivariant localisation

The Green-Griffiths-Lang conjecture says that for every complex projective algebraic variety $X$ of general type there exists a proper algebraic subvariety of $X$ containing all nonconstant entire holomorphic curves $f:\mathbb{C} \to X$. Using equivariant localisation on the Demailly-Semple jet differentials bundle we give an affirmative answer to this conjecture for generic projective hypersurfaces $X \subset \mathbb{P}^{n+1}$ of degree $\mathrm{deg}(X) \ge n^{9n}$.

Reducibility in Sasakian geometry

The purpose of this paper is to study reducibility properties in Sasakian geometry. First we give the Sasaki version of the de Rham decomposition theorem; however, we need a mild technical assumption on the Sasaki automorphism group which includes the toric case. Next we introduce the concept of cone reducible and consider S 3 S^3 bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2-torus of Reeb type. In particular, we can classify all such Sasakian structures up to contact isotopy on S 3 S^3 bundles over a Riemann surface of genus greater than zero. Finally, we show that in the toric case an extremal Sasaki metric on a Sasaki join always splits.

The minimal resolution conjecture on a general quartic surface in [formula omitted

Mustaţă has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in P3 this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.

A non-integrated hypersurface defect relation for meromorphic maps over complete Kähler manifolds into projective algebraic varieties

In this paper, a non-integrated defect relation for meromorphic maps from complete K\"ahler manifolds $M$ into smooth projective algebraic varieties $V$ intersecting hypersurfaces located in $k$-subgeneral position is proved. The novelty of this result lies in that both the upper bound and the truncation level of our defect relation depend only on $k$, $\dim_{\,\mathbb{C}}(V)$ and the degrees of the hypersurfaces considered. In addition, this defect relation recovers Hirotaka Fujimoto [Theorem 1.1; MR0884636 (88m:32049); Japan. J. Math. (N.S.) 11 (1985), no. 2, 233-264.] when subjected to the same conditions.

Representations of E7, equivalent combinatorics and algebraic varieties

In our earlier work on a new functor from [Formula: see text]-Mod to [Formula: see text]-Mod, we found a one-parameter ([Formula: see text]) family of inhomogeneous first-order differential operator representations of the simple Lie algebra of type [Formula: see text] in [Formula: see text] variables. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector [Formula: see text], we prove that the space forms an irreducible [Formula: see text]-module for any [Formula: see text] if [Formula: see text] is not on an explicitly given projective algebraic variety. Certain equivalent combinatorial properties of the basic oscillator representation of [Formula: see text] over its 27-dimensional module play key roles in our proof. Our result can also be used to study free bosonic field irreducible representations of the corresponding affine Kac–Moody algebra.

Representations of Weil–Deligne groups and Frobenius conjugacy classes

Let X be a smooth projective algebraic variety over a number field F, with an embedding τ:F↪C. The action of Gal(F¯/F) on ℓ-adic cohomology groups Heti(X/F¯,Qℓ), induces Galois representations ρℓi:Gal(F¯/F)→GL(Heti(X/F¯,Qℓ)). Fix a non-archimedean valuation v on F, of residual characteristic p. Let Fv be the completion of F at v and ′Wv be the Weil–Deligne group of Fv. We establish new cases, for which the linear representations ρℓi_ of ′Wv, associated to ρℓi, form a compatible system of representations of ′Wv defined over Q. Under suitable hypotheses, we show that in some cases, these representations actually form a compatible system of representations of ′Wv, with values in the Mumford–Tate group of HBi(τX(C),Q). When X has good reduction at v, we establish a motivic relationship between the compatibility of the system {ρℓi}ℓ≠p and the conjugacy class of the crystalline Frobenius of the reduction of X at v.

On the algebraic stringy Euler number

We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler number is one of such invariants. In the present paper, we prove this conjecture for varieties having an action of a connected algebraic group G and admitting equivariant desingularizations with only finitely many G-orbits. In particular, we prove our conjecture for arbitrary projective spherical varieties.

A unified approach to the Galois closure problem

In this paper we give a unified approach in categorical setting to the problem of finding the Galois closure of a finite cover, which includes as special cases the familiar finite separable field extensions, finite unramified covers of a connected undirected graph, finite covering spaces of a locally connected topological space, finite étale covers of a smooth projective irreducible algebraic variety, and finite covers of normal varieties. We present two algorithms whose outputs are shown to be desired Galois closures. An upper bound of the degree of the Galois closure under each algorithm is also obtained.

Integral Chow motives of threefolds with $K$-motives of unit type

We prove that if a smooth projective algebraic variety of dimension less or equal to three has a unit type integral $K$-motive, then its integral Chow motive is of Lefschetz type. As a consequence, the integral Chow motive is of Lefschetz type for a smooth projective variety of dimension less or equal to three that admits a full exceptional collection.

Thom Polynomials and the Green–Griffiths–Lang Conjecture for Hypersurfaces with Polynomial Degree

Abstract Green and Griffiths [25] and Lang [29] conjectured that for every complex projective algebraic variety X of general type there exists a proper algebraic subvariety of X containing all nonconstant entire holomorphic curves $f:{\mathbb{C}} \to X$. We construct a compactification of the invariant jet differentials bundle over complex manifolds motivated by an algebraic model of Morin singularities and we develop an iterated residue formula using equivariant localisation for tautological integrals over it. Using this we show that the polynomial Green–Griffiths–Lang conjecture for a generic projective hypersurface of degree $\deg (X)>2n^{9}$ follows from a positivity conjecture for Thom polynomials of Morin singularities.

New examples (and counterexamples) of complete finite-rank differential varieties

ABSTRACTDifferential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both differential algebra and model theory have investigated the property of completeness of differential varieties. After reviewing their results, we extend that work by proving several versions of a “differential valuative criterion" and using them to give new examples of complete differential varieties. We conclude by analyzing the first examples of incomplete, finite-rank projective differential varieties, demonstrating a clear difference from projective algebraic varieties.

Algebraic embeddings of smooth almost complex structures

The goal of this work is to prove an embedding theorem for compact almost complex manifolds into complex algebraic varieties. It is shown that every almost complex structure can be realized by the transverse structure to an algebraic distribution on an affine algebraic variety, namely an algebraic subbundle of the tangent bundle. In fact, there even exist universal embedding spaces for this problem, and their dimensions grow quadratically with respect to the dimension of the almost complex manifold to embed. We give precise variation formulas for the induced almost complex structures and study the related versality conditions. At the end, we discuss the original question raised by F. Bogomolov: can one embed every compact complex manifold as a \mathcal C^\infty smooth subvariety that is transverse to an algebraic foliation on a complex projective algebraic variety?

On Linear Dependence Over Complete Differential Algebraic Varieties

We extend Kolchin's results from [12] on linear dependence over the constant points of projective algebraic varieties to linear dependence over arbitrary complete differential algebraic varieties. We show that in this more general setting, the notion of linear dependence still has necessary and sufficient conditions given by the vanishing of a certain system of differential-polynomials equations. We also discuss some conjectural questions around completeness and the catenary problem.