Smooth Complex Algebraic Variety Research Articles

𝑉-filtrations and minimal exponents for local complete intersections

Abstract We define and study a notion of minimal exponent for a local complete intersection subscheme 𝑍 of a smooth complex algebraic variety 𝑋, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara–Malgrange 𝑉-filtration associated to 𝑍. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology H Z r ⁢ ( O X ) \mathcal{H}^{r}_{Z}(\mathcal{O}_{X}) , where 𝑟 is the codimension of 𝑍 in 𝑋. We also study its relation to the Bernstein–Sato polynomial of 𝑍. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension 1 case. A key ingredient for our main result is a description of the Kashiwara–Malgrange 𝑉-filtration associated to any ideal ( f 1 , … , f r ) (f_{1},\ldots,f_{r}) in terms of the microlocal 𝑉-filtration associated to the hypersurface defined by ∑ i = 1 r f i ⁢ y i \sum_{i=1}^{r}f_{i}y_{i} .

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.

Regular-singular connections on relative complex schemes

Deligne's celebrated "Riemann--Hilbert correspondence" relates representations of the fundamental group of a smooth complex algebraic variety and regular-singular integrable connections. In this work, we show how to arrive at a similar statement in the case of a smooth scheme $X$ over the spectrum of a ring $R=\mathbb C[[t_1,\ldots, t_r]]/I$. On one side of the correspondence we have representations on $R$-modules of the fundamental group of the special fibre, and on the other we have certain integrable $R$-connections admitting logarithmic models. The correspondence is then applied to give explicit examples of differential Galois groups of $\mathbb C[[t]]$--connections.

The Du Bois complex of a hypersurface and the minimal exponent

We study the Du Bois complex Ω_Z∙ of a hypersurface Z in a smooth complex algebraic variety in terms of its minimal exponent α˜(Z). The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein–Sato polynomial of Z, and refining the log-canonical threshold. We show that if α˜(Z)≥p+1, then the canonical morphism ΩZp→Ω_Zp is an isomorphism, where Ω_Zp is the pth associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, if Z is singular and α˜(Z)>p≥2, we obtain non-vanishing results for some higher cohomologies of Ω_Zn−p.

On a Conjecture of Bitoun and Schedler

Abstract Suppose that $X$ is a smooth complex algebraic variety of dimension $\geq 3$ and $f$ defines a hypersurface $Z$ in $X$, with a unique singular point $P$. Bitoun and Schedler conjectured that the ${\mathcal {D}}$-module generated by $\tfrac {1}{f}$ has length equal to $g_{P}(Z)+2$, where $g_{P}(Z)$ is the reduced genus of $Z$ at $P$. We prove that this length is always $\geq g_{P}(Z)+2$ and equality holds if and only if $\tfrac {1}{f}$ lies in the ${\mathcal {D}}$-module generated by $I_{0}(f)\tfrac {1}{f}$, where $I_{0}(f)$ is the multiplier ideal ${\mathcal {J}}(f^{1-\epsilon })$, with $0<\epsilon \ll 1$. In particular, we see that the conjecture holds if the pair $(X,Z)$ is log canonical. We can also recover, with an easy proof, the result of Bitoun and Schedler saying that the conjecture holds for weighted homogeneous isolated singularities. On the other hand, we give an example (a polynomial in $3$ variables with an ordinary singular point of multiplicity $4$) for which the conjecture does not hold.

On the structure of metabelian Galois coverings of complex algebraic varieties

We describe the general structure of metabelian branched covers of smooth complex algebraic varieties. The main question that we consider is the following: given a smooth algebraic variety $Y$, which “building data” on $Y$ determine a metabelian cover $f:

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.

Logarithmic Vector Fields and Freeness of Divisors and Arrangements: New perspectives and applications

The central topic of the workshop was the notion of logarithmic vector fields along a divisor in a smooth complex analytic or algebraic variety, i.e., the vector fields on the ambient variety tangent to the divisor. Following their introduction by K. Saito for the purpose of studying the universal unfolding of an isolated singularity, this fundamental object has been the focus of studies in a wide range of mathematical fields such as algebra, algebraic geometry, singularity theory, root systems, (geometric) representation theory, combinatorics, (toric) topology, or symplectic geometry. In the last few years the logarithmic vector field approach has seen some unexpected and striking advances and deep applications. The aim of the workshop was to provide reports and to share these various new developments in the field.

Hodge theory on Alexander invariants – A survey

We survey recent developments in the study of Hodge theoretic aspects of Alexander-type invariants associated with smooth complex algebraic varieties.

Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension

Abstract We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated to Z.

Generic simplicity of quantum Hamiltonian reductions

Let a reductive group G act on a smooth affine complex algebraic variety X. Let 𝔤 be the Lie algebra of G and μ:T * (X)→𝔤 * be the moment map. If the moment map is flat, and for a generic character χ:𝔤→ℂ, the action of G on μ -1 (χ) is free, then we show that for very generic characters χ the corresponding quantum Hamiltonian reduction of the ring of differential operators D(X) is simple.

Convergence of polarizations, toric degenerations, and Newton–Okounkov bodies

Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$ satisfying some natural technical hypotheses, we construct a deformation $\{J_s\}$ of the complex structure on $X$ and bases $\mathcal{B}_s$ of $H^0(X,L, J_s)$ so that $J_0$ is the standard complex structure and, in the limit as $s \to \infty$, the basis elements approach dirac-delta distributions centered at Bohr-Sommerfeld fibers of a moment map associated to $X$ and its toric degeneration. The theory of Newton-Okounkov bodies and its associated toric degenerations shows that the technical hypotheses mentioned above hold in some generality. Our results significantly generalize previous results in geometric quantization which prove independence of polarization between Kahler quantizations and real polarizations. As an example, in the case of general flag varieties $X=G/B$ and for certain choices of $\lambda$, our result geometrically constructs a continuous degeneration of the (dual) canonical basis of $V_{\lambda}^*$ to a collection of dirac delta functions supported at the Bohr-Sommerfeld fibres corresponding exactly to the lattice points of a Littelmann-Berenstein-Zelevinsky string polytope $\Delta_{\underline{w}_0}(\lambda) \cap \mathbb{Z}^{\dim(G/B)}$.

Absolute sets and the decomposition theorem

We give a framework to produce constructible functions from natural functors between categories, without need of a morphism of moduli spaces to model the functor. We show using the Riemann-Hilbert correspondence that any natural (derived) functor on constructible sheaves on smooth complex algebraic varieties can be used to construct a kind of constructible sets, called absolute sets, generalizing a notion introduced by Simpson in presence of moduli. We conjecture that the absolute sets of local systems satisfy a special varieties package, among which is an analog of the Manin-Mumford, Mordell-Lang, and Andre-Oort conjectures. The conjecture gives a simple proof of the Decomposition Theorem for all semi-simple perverse sheaves, assuming the Decomposition Theorem for the geometric ones. We prove the conjecture in the rank one case by showing that the closed absolute sets in this case are finite unions of torsion-translated affine tori. This extends a structure result of the authors for cohomology jump loci to any other natural jump loci. For example, to jump loci of intersection cohomology and Leray filtrations. We also show that the Leray spectral sequence for the open embedding in a good compactification degenerates for all rank one local systems at the usual page, not just for unitary local systems.

OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

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.

A criterion for quadraticity of a representation of the fundamental group of an algebraic variety

Let $\Gamma$ be a finitely presented group and $G$ a linear algebraic group over $\mathbb{R}$. A representation $\rho:\Gamma\rightarrow G(\mathbb{R})$ can be seen as an $\mathbb{R}$-point of the representation variety $\mathfrak{R}(\Gamma, G)$. It is known from the work of Goldman and Millson that if $\Gamma$ is the fundamental group of a compact Kahler manifold and $\rho$ has image contained in a compact subgroup then $\rho$ is analytically defined by homogeneous quadratic equations in $\mathfrak{R}(\Gamma, G)$. When $X$ is a smooth complex algebraic variety, we study a certain criterion under which this same conclusion holds.

Gromov–Witten gauge theory

We introduce a modular completion of the stack of maps from stable marked curves to the quotient stack [pt/C×], and use this stack to construct some gauge-theoretic analogues of the Gromov–Witten invariants. We also indicate the generalization of these invariants to the quotient stacks [X/C×], where X is a smooth proper complex algebraic variety.

Purity, Formality, and Arrangement Complements

We prove a purity implies statement in the context of the rational homotopy theory of smooth complex algebraic varieties, and apply it to complements of hypersurface arrangements. In particular, we prove that the complement of a toric arrangement is formal. This is analogous to the classical formality theorem for complements of hyperplane arrangements, due to Brieskorn, and generalizes a theorem of De Concini and Procesi.

Rank One Local Systems and Forms of Degree One

Cohomology support loci of rank one local systems of a smooth quasiprojective complex algebraic variety are finite unions of torsion-translated complex subtori of the character variety of the fundamental group. Tangent spaces of the character variety are (partially) represented by logarithmic 1-forms. In this paper, we give a relation between cohomology support loci and the natural strata of 1-forms given by the dimension of the vanishing locus. This relation generalizes the one for the projective case due to Green and Lazarsfeld and also generalizes the partial relation due to Dimca in the quasi-projective case.

ON REDUCTIVE AUTOMORPHISM GROUPS OF REGULAR EMBEDDINGS

Let G be a connected reductive complex algebraic group acting on a smooth complete complex algebraic variety X. We assume that X is a regular embedding, a condition satisfied in particular by smooth toric varieties and flag varieties. For any set D of G-stable prime divisors, we study the action on X of the group Aut°(X, D), the connected automorphism group of X stabilizing all elements of D. We determine a Levi subgroup A(X, D) of Aut°(X, D), and also relevant invariants of X as a spherical A(X, D)-variety. As a byproduct, we obtain a complete description of the inclusion relation between closures of A(X, D)-orbits on X.