Smooth Quasi-projective Variety Research Articles

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.

A decomposition theorem for 0-cycles and applications to class field theory

We describe the abelian {e}tale fundamental group with modulus in terms of 0-cycles on a class of smooth quasi-projective varieties over finite fields which may not admit smooth compactifications. Using a limit process, this yields a cycle-theoretic analogue of the $K$-theoretic reciprocity theorem of Kato-Saito for such varieties. The proof is based on a decomposition theorem for the cohomological Chow group of 0-cycles on the double of a quasi-projective $R_1$-scheme over a field along a closed subscheme in terms of the Chow groups with and without modulus of the scheme. This also extends the decomposition theorem of Binda-Krishna to smooth schemes over imperfect fields.

Integrality of the Betti moduli space

If in a given rank r r , there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number ℓ \ell , there is an absolutely irreducible ℓ \ell -adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank r r if once there is an irreducible rank r r complex linear representation with determinant isomorphic to this torsion character, then for any ℓ \ell , there is an absolutely irreducible one of rank r r and determinant this given character, which is defined over Z ¯ ℓ \bar { \mathbb {Z}}_\ell . We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth quasi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne’s companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong’s conjecture (Gaitsgory for ℓ ≥ 3 \ell \ge 3 ). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of the Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, which shows the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple.

Finite presentation of the tame fundamental group

Let p be a prime number, and let k be an algebraically closed field of characteristic p. We show that the tame fundamental group of a smooth affine curve over k is a projective profinite group. We prove that the fundamental group of a smooth projective variety over k is finitely presented; more generally, the tame fundamental group of a smooth quasi-projective variety over k, which admits a good compactification, is finitely presented.

Free quotients of fundamental groups of smooth quasi-projective varieties

AbstractWe study the fundamental groups of the complements to curves on simply connected surfaces, admitting non-abelian free groups as their quotients. We show that given a subset of the Néron–Severi group of such a surface, there are only finitely many classes of equisingular isotopy of curves with irreducible components belonging to this subset for which the fundamental groups of the complement admit surjections onto a free group of a given sufficiently large rank. Examples of subsets of the Néron–Severi group are given with infinitely many isotopy classes of curves with irreducible components from such a subset and fundamental groups of the complements admitting surjections on a free group only of a small rank.

Perverse sheaves on semi-abelian varieties

We give a complete (global) characterization of $${\mathbb {C}}$$ -perverse sheaves on semi-abelian varieties in terms of their cohomology jump loci. Our results generalize Schnell’s work on perverse sheaves on complex abelian varieties, as well as Gabber–Loeser’s results on perverse sheaves on complex affine tori. We apply our results to the study of cohomology jump loci of smooth quasi-projective varieties, to the topology of the Albanese map, and in the context of homological duality properties of complex algebraic varieties.

Smooth covers of moduli stacks of Riemann surfaces with symmetry

We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety.

Semi-algebraic chains on projective varieties and the Abel-Jacobi map for higher Chow cycles

We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of delta-admissible chains. Roughly speaking, a delta-admissible chain is a simplicial semi-algebraic chain meeting the faces properly. As an application, we show that the Abel-Jacobi map for higher Chow cycles can be described via delta-admissible chains. As an example, we will describe the Hodge realization of the polylog cycles constructed by Bloch-Kriz in terms of the Abel-Jacobi map.

Log -modules and index theorems

Abstract We study log $\mathscr {D}$ -modules on smooth log pairs and construct a comparison theorem of log de Rham complexes. The proof uses Sabbah’s generalized b-functions. As applications, we deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. The log index theorem naturally generalizes the Dubson-Kashiwara index theorem on smooth projective varieties.

The equivalence of several conjectures on independence of $\ell$

We consider several conjectures on the independence of $\ell$ of the \'etale cohomology of (singular, open) varieties over $\bar{\mathbf F}_p$. The main result is that independence of $\ell$ of the Betti numbers $h^i_{\text{c}}(X,\mathbf Q_\ell)$ for arbitrary varieties is equivalent to independence of $\ell$ of homological equivalence $\sim_{\text{hom},\ell}$ for cycles on smooth projective varieties. We give several other equivalent statements. As a surprising consequence, we prove that independence of $\ell$ of Betti numbers for smooth quasi-projective varieties implies the same result for arbitrary separated finite type $k$-schemes.

The Hochschild cohomology ring of a global quotient orbifold

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields TXpoly on X, and is generated as a TXpoly-algebra by the sum of the determinants det⁡(NXg) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne–Mumford stacks in general. We discuss, in the case of a symplectic group action on a symplectic variety X, relationships with orbifold cohomology and Ruan's cohomological conjectures. In describing the Hochschild cohomology in the symplectic situation, we employ compatible trivializations of the determinants det⁡(NXg), which requires (for the cup product) a nontrivial normalization missing in previous literature.

Chow Filtration on Representation Rings of Algebraic Groups

Abstract We introduce and study a filtration on the representation ring $R(G)$ of an affine algebraic group $G$ over a field. This filtration, which we call Chow filtration, is an analogue of the coniveau filtration on the Grothendieck ring of a smooth variety. We compare it with the other known filtrations on $R(G)$ and show that all three define on $R(G)$ the same topology. For any $n\geq 1$, we compute the Chow filtration on $R(G)$ for the special orthogonal group $G:=O^+(2n+1)$. In particular, we show that the graded group associated with the filtration is torsion-free. On the other hand, the Chow ring of the classifying space of $G$ over any field of characteristic $\ne 2$ is known to contain non-zero torsion elements. As a consequence, any sufficiently good approximation of the classifying space yields an example of a smooth quasi-projective variety $X$ such that its Chow ring is generated by Chern classes and at the same time contains non-zero elements vanishing under the canonical homomorphism onto the graded ring associated with the coniveau filtration on the Grothendieck ring of $X$.

Mellin transformation, propagation, and abelian duality spaces

For arbitrary field coefficients K, we show that K-perverse sheaves on a complex affine torus satisfy the so-called propagation package, i.e., the generic vanishing property and the signed Euler characteristic property hold, and the corresponding cohomology jump loci satisfy the propagation property and codimension lower bound. The main ingredient used in the proof is Gabber–Loeser's Mellin transformation functor for K-constructible complexes on a complex affine torus, and the fact that it behaves well with respect to perverse sheaves.As a concrete topological application of our sheaf-theoretic results, we study homological duality properties of complex algebraic varieties, via abelian duality spaces. We provide new obstructions on abelian duality spaces by showing that their cohomology jump loci satisfy a propagation package. This is then used to prove that complex abelian varieties are the only complex projective manifolds which are abelian duality spaces. We also construct new examples of abelian duality spaces. For example, we show that if a smooth quasi-projective variety X satisfies a certain Hodge-theoretic condition and it admits a proper semi-small map (e.g., a closed embedding or a finite map) to a complex affine torus, then X is an abelian duality space.

The Künneth Formula for the Twisted de Rham and Higgs Cohomologies

We prove the K\"unneth formula for the irregular Hodge filtrations on the exponentially twisted de Rham and the Higgs cohomologies of smooth quasi-projective complex varieties. The method involves a careful comparison of the underlying chain complexes under a certain elimination of indeterminacy.

Cohomologically rigid local systems and integrality

We prove that the monodromy of an irreducible cohomologically complex rigid local system with finite determinant and quasi-unipotent local monodromies at infinity on a smooth quasiprojective complex variety X is integral. This answers positively a special case of a conjecture by Carlos Simpson. On a smooth projective variety, the argument relies on Drinfeld’s theorem on the existence of $$\ell $$ -adic companions over a finite field. When the variety is quasiprojective, one has in addition to control the weights and the monodromy at infinity.

The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties

Given any connected topological space X, assume that there exists an epimorphism \(\phi {:}\; \pi _1(X) \rightarrow {\mathbb {Z}}\). The deck transformation group \({\mathbb {Z}}\) acts on the associated infinite cyclic cover \(X^\phi \) of X, hence on the homology group \(H_i(X^\phi , {\mathbb {C}})\). This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring \({\mathbb {C}}[t,t^{-1}]\), which is a finite dimensional \({\mathbb {C}}\)-vector space. We study the sizes of the Jordan blocks of this linear automorphism. When X is a compact Kahler manifold, we show that all the Jordan blocks are of size one. When X is a smooth complex quasi-projective variety, we give an upper bound on the sizes of the Jordan blocks, which is an analogue of the Monodromy Theorem for the local Milnor fibration.

Equivalences of derived factorization categories of gauged Landau–Ginzburg models

For a given Fourier–Mukai equivalence of bounded derived categories of coherent sheaves on smooth quasi-projective varieties, we construct Fourier–Mukai equivalences of derived factorization categories of gauged Landau–Ginzburg (LG) models.As an application, we obtain some equivalences of derived factorization categories of K-equivalent gauged LG models. This result is an equivariant version of the result of Baranovsky and Pecharich, and it also gives a partial answer to Segal's conjecture. As another application, we prove that if the kernel of the Fourier–Mukai equivalence is linearizable with respect to a reductive affine algebraic group action, then the derived categories of equivariant coherent sheaves on the varieties are equivalent. This result is shown by Ploog for finite groups case.

On Grothendieck’s Riemann–Roch theorem

We prove that, for smooth quasi-projective varieties over a field, the K-theory K(X) of vector bundles is the universal cohomology theory where c1(L⊗L̄)=c1(L)+c1(L̄)−c1(L)c1(L̄). Then, we show that Grothendieck’s Riemann–Roch theorem is a direct consequence of this universal property, as well as the universal property of the graded K-theory GK•(X)⊗Q.

Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces

We develop a new approach to the study of supersymmetric gauge theories on ALE spaces using the theory of framed sheaves on root toric stacks, which illuminates relations with gauge theories on R4 and with two-dimensional conformal field theory. We construct a stacky compactification of the minimal resolution Xk of the Ak−1 toric singularity C2/Zk, which is a projective toric orbifold Xk such that Xk∖Xk is a Zk-gerbe. We construct moduli spaces of torsion free sheaves on Xk which are framed along the compactification gerbe. We prove that this moduli space is a smooth quasi-projective variety, compute its dimension, and classify its fixed points under the natural induced toric action. We use this construction to compute the partition functions and correlators of chiral BPS operators for N=2 quiver gauge theories on Xk with nontrivial holonomies at infinity. The partition functions are computed with and without couplings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas, which relate them to the corresponding Nekrasov partition functions on the affine toric open subsets of Xk. We compare our new partition functions with previous computations, explore their connections to the representation theory of affine Lie algebras, and find new constraints on fractional instanton charges in the coupling to fundamental matter. We show that the partition functions in the low energy limit are characterized by the Seiberg–Witten curves, and in some cases also by suitable blowup equations involving Riemann theta-functions on the Seiberg–Witten curve with characteristics related to the nontrivial holonomies.

The Euler characteristics of generalized Kummer schemes

We compute the Euler characteristics of the generalized Kummer schemes associated to $$A\times Y$$ , where A is an abelian variety and Y is a smooth quasi-projective variety. When Y is a point, our results prove a formula conjectured by Gulbrandsen.