Smooth Complex Projective Variety Research Articles

Quotient singularities in the Grothendieck ring of varieties

Let G G be a finite group, X X be a smooth complex projective variety with a faithful G G -action, and Y Y be a resolution of singularities of X / G X/G . Larsen and Lunts asked whether [ X / G ] − [ Y ] [X/G]-[Y] is divisible by [ A 1 ] [\mathbb {A}^1] in the Grothendieck ring of varieties. We show that the answer is negative if B G BG is not stably rational and affirmative if G G is abelian. The case when X = Z n X=Z^n for some smooth projective variety Z Z and G = S n G=S_n acts by permutation of the factors is of particular interest. We make progress on it by showing that [ Z n / S n ] − [ Z ⟨ n ⟩ / S n ] [Z^n/S_n]-[Z\langle n\rangle / S_n] is divisible by [ A 1 ] [\mathbb {A}^1] , where Z ⟨ n ⟩ Z\langle n\rangle is Ulyanov’s polydiagonal compactification of the n n th configuration space of Z Z .

Logarithmic bounds on Fujita's conjecture

AbstractLet be a smooth complex projective variety of dimension . We prove bounds on Fujita's basepoint freeness conjecture that grow as , where is the logarithm with natural base.

Infinite torsion in Griffiths groups

We show that there are smooth complex projective varieties with infinite 2 -torsion in their third Griffiths groups. It follows that the torsion subgroup of Griffiths groups is in general not finitely generated, thereby solving a problem of Schoen from 1992.

Twistor D-Modules and the Decomposition Theorem

The purpose of this Arbeitsgemeinschaft is to introduce the notion of twistor \mathcal{D} -modules and their main properties. The guiding principle leading this discussion is Simpson’s “meta-theorem”, which gives a heuristic for generalizing (mixed) Hodge-theoretic results into (mixed) twistor-theoretic results. The strength of the twistor approach is that it enables to enlarge the scope of Hodge theory not only to arbitrary semi-simple perverse sheaves, equivalently semi-simple regular holonomic \mathcal{D} -modules via the Riemann-Hilbert correspondence, but also to possibly semi-simple irregular holonomic \mathcal{D} -modules. An overarching goal for this session is Mochizuki’s proof of the decomposition theorem for semi-simple holonomic \mathcal{D} -modules on a smooth complex projective variety, first conjectured by Kashiwara in 1996.

On algebraic Chern classes of flat vector bundles

We show that under some assumptions on the monodromy group some combinations of higher Chern classes of a flat vector bundle are torsion in the Chow group. Similar results hold for flat vector bundles that deform to such flat vector bundles (also in case of quasi-projective varieties). The results are motivated by Bloch’s conjecture on Chern classes of flat vector bundles on smooth complex projective varieties but in some cases they give a more precise information. We also study Higgs version of Bloch’s conjecture and analogous problems in the positive characteristic case.

Refined unramified cohomology of schemes

We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This generalizes to cycles of arbitrary codimension previous results of Bloch–Ogus, Colliot-Thélène–Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch–Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel–Jacobi invariant has coniveau $1$. This establishes a torsion version of a conjecture of Jannsen originally formulated $\otimes \mathbb {Q}$. We further show that the group of homologically trivial torsion cycles modulo algebraic equivalence has a finite filtration (by coniveau) such that the graded quotients are determined by higher Abel–Jacobi invariants that we construct. This may be seen as a variant for torsion cycles modulo algebraic equivalence of a conjecture of Green. We also prove $\ell$-adic analogues of these results over any field $k$ which contains all $\ell$-power roots of unity.

Slope semistability and positive cones of Grassmann bundles

Let [Formula: see text] be a vector bundle of rank [Formula: see text] on a smooth complex projective variety [Formula: see text]. In this paper, we compute the nef and pseudoeffective cones of divisors on the Grassmann bundle Gr[Formula: see text] parametrizing [Formula: see text]-dimensional subspaces of the fibers of [Formula: see text], where [Formula: see text] rk([Formula: see text]), under assumptions on [Formula: see text] as well as on the vector bundle [Formula: see text]. In particular, we show that nef cone and the pseudoeffective cone of Gr[Formula: see text] coincide if and only if nef cone and pseudoeffective cone of [Formula: see text] coincide under the assumption that [Formula: see text] is a slope semistable bundle on [Formula: see text] with [Formula: see text](End([Formula: see text]))=0. We also discuss about the nefness and ampleness of the universal quotient bundle [Formula: see text] on Gr[Formula: see text].

Derived invariance of the Albanese relative canonical ring

We show the derived invariance of various geometric invariants of smooth complex projective varieties governed by the Albanese map, including the relative canonical ring and the class of the relative canonical model in a suitable variant of the Grothendieck ring of varieties. Then we derive some applications to the derived invariance of Hodge numbers.

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.

Bridgeland stability conditions on surfaces with curves of negative self-intersection

Abstract Let X be a smooth complex projective variety. In 2002, Bridgeland [6] defined a notion of stability for the objects in 𝔇 b (X), the bounded derived category of coherent sheaves on X, which generalised the notion of slope stability for vector bundles on curves. There are many nice connections between stability conditions on X and the geometry of the variety. We construct new stability conditions for surfaces containing a curve C whose self-intersection is negative. We show that these stability conditions lie on a wall of the geometric chamber of Stab(X), the stability manifold of X.We then construct the moduli space Mσ (ℴ X ) of σ-semistable objects of class [ℴ X ] in K 0(X) after wall-crossing.

Generalized polarized manifolds with low second class

On a smooth complex projective variety $X$ of dimension $n$, consider an ample vector bundle $\mathscr E$ of rank $r \leq n - 2$ and an ample line bundle $H$. A numerical character $m_2 = m_2(X,\mathscr E, H)$ of the triplet $(X,\mathscr E, H)$ is defined, extending the well-known second class of a polarized manifold $(X, H)$, when either $n = 2$ or $H$ is very ample. Under some additional assumptions on $\mathscr F:= \mathscr E \oplus H^{\oplus(n-r-2)}$, triplets $(X,\mathscr E, H)$ as above whose $m_2$ is small with respect to the invariants $d := c_{n-2}(\mathscr F)H^2$ and $g := 1 + \frac{1}{2}(K_X + c_1(\mathscr F)+H)\cdot c_{n-2}(\mathscr F)\cdot H$ are studied and classified.

Branes and moduli spaces of Higgs bundles on smooth projective varieties

Given a smooth complex projective variety M and a smooth closed curve \(X\, \subset \, M\) such that the homomorphism of fundamental groups \(\pi _1(X)\, \longrightarrow \, \pi _1(M)\) is surjective, we study the restriction map of Higgs bundles, namely from the Higgs bundles on M to those on X. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on M and X. We also consider the setup where a finite group is acting on M via holomorphic automorphisms or anti-holomorphic involutions, and the curve X is preserved by this action. Branes are studied in this context.

Smooth rational projective varieties with non-finitely generated discrete automorphism group and infinitely many real forms

We show, among other things, that for each integer \(n \ge 3\), there is a smooth complex projective rational variety of dimension n, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso.

Positivity of the second exterior power of the tangent bundles

Let X be a smooth complex projective variety with nef ⋀2TX and dim⁡X≥3. We prove that, up to a finite étale cover X˜→X, the Albanese map X˜→Alb(X˜) is a locally trivial fibration whose fibers are isomorphic to a smooth Fano variety F with nef ⋀2TF. As a bi-product, we see that TX is nef or X is a Fano variety. Moreover we study a contraction of a KX-negative extremal ray φ:X→Y. In particular, we prove that X is isomorphic to the blow-up of a projective space at a point if φ is of birational type. We also prove that φ is a smooth morphism if φ is of fiber type. As a consequence, we give a structure theorem of varieties with nef ⋀2TX.

Pseudo-effective cones of projective bundles and weak Zariski decomposition

In this article, we consider the projective bundle $\mathbb{P}_X(E)$ over a smooth complex projective variety $X$, where $E$ is a semistable bundle on $X$ with $c_2(End(E)) =0$. We give a necessary and sufficient condition to get the equality $ Nef^1\bigl(\mathbb{P}_X(E)\bigr) = \overline{Eff}^1\bigl(\mathbb{P}_X(E)\bigr)$ of nef cone and pseudoeffective cone of divisors in $\mathbb{P}_X(E)$. As an application of our result, we show the equality of nef and pseudoeffective cones of divisors of projective bundles over some special varieties. In particular, we show that weak Zariski decomposition exists on these projective bundles. We also show that a semistable bundle $E$ of rank $r \geq 2$ with $c_2\bigl(End(E)\bigr) = 0$ on a smooth complex projective variety of Picard number 1 is $k$-homogeneous i.e. $\overline{Eff}^k\bigl(\mathbb{P}_X(E)\bigr) = Nef^k\bigl(\mathbb{P}_{X}(E)\bigr)$ for all $1 \leq k < r$. Finally, we show that weak Zariski decomposition exists for a fibre product $\mathbb{P}_C(E)\times_C\mathbb{P}(E')$ over a smooth projective curve $C$.

A recursive formula for osculating curves

Let $X$ be a smooth complex projective variety. Using a construction devised to Gathmann, we present a recursive formula for some of the Gromov-Witten invariants of $X$. We prove that, when $X$ is homogeneous, this formula gives the number of osculating rational curves at a general point of a general hypersurface of $X$. This generalizes the classical well known pairs of inflexion (asymptotic) lines for surfaces in $\mathbb{P}^{3}$ of Salmon, as well as Darboux's $27$ osculating conics.

Defect of Euclidean distance degree

Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for “nearest” point problems of the algebraic variety. It is well known that the latter is an upper bound for the former. While this bound may be tight, many varieties appearing in optimization, engineering, statistics, and data science, have a significant gap between these two numbers. We call this difference the defect of the ED degree of an algebraic variety. In this paper we compute this defect by classical techniques in Singularity Theory, thereby deriving a new method for computing ED degrees of smooth complex projective varieties.

Criterion for existence of a logarithmic connection on a principal bundle over a smooth complex projective variety

Let $X$ be a connected smooth complex projective variety of dimension $n \geq 1$. Let $D$ be a simple normal crossing divisor on $X$. Let $G$ be a connected complex Lie group, and $E_G$ a holomorphic principal $G$-bundle on $X$. In this article, we give criterion for existence of a logarithmic connection on $E_G$ singular along $D$.

On the number and boundedness of log minimal models of general type

We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective klt pair (X,D) with $K_X+D$ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of D and the volume of $K_X+D$. We further show that all n-dimensional projective klt pairs (X,D), such that $K_X+D$ is big and nef of fixed volume and such that the coefficients of D are contained in a given DCC set, form a bounded family. It follows that in any dimension, minimal models of general type and bounded volume form a bounded family.

The construction problem for Hodge numbers modulo an integer

For any positive integer m and any dimension n, we show that any n-dimensional Hodge diamond with values in Z/mZ is attained by the Hodge numbers of an n-dimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of n-dimensional smooth complex projective varieties besides the ones induced by the Hodge symmetries, which answers a question raised by Koll\'ar in 2012.