Complex Quasi-projective Varieties Research Articles

Generating Series of the Classes of Exotic Unordered Configuration Spaces

The concept of exotic (ordered) configuration spaces of points in a space was proposed by Baryshnikov. He obtained formulas for the (exponential) generating series of their Euler characteristics. We explore unordered analogs of the spaces. Considering a complex quasiprojective variety, we give a formula for the generating series of the classes of these configuration spaces in the Grothendieck ring of complex quasiprojective varieties. We state the result in terms of the natural power structure over this ring. This yields formulas of generating series of additive invariants of configuration spaces like the Hodge–Deligne polynomial and the Euler characteristic.

Equidistribution of Hodge loci II

Let $\mathbb {V}$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, the pull–push form. In particular, it is always analytically dense when the pull–push form does not vanish. When the weight is two, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q$th Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal {A}_g$, equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal {A}_g$. These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull–push form appears in this greater generality, we provide several tools to determine it, and we compute it in many examples.

Bivariant Class of Degree One

Let f:Xrightarrow Y be a projective birational morphism, between complex quasi-projective varieties. Fix a bivariant class theta in H^0(X{mathop {rightarrow }limits ^{f}}Y)cong Hom_{D^{b}_{c}(Y)}(Rf_*{mathbb {A}}_X, {mathbb {A}}_Y) (here {mathbb {A}} is a Noetherian commutative ring with identity, and {mathbb {A}}_X and {mathbb {A}}_Y denote the constant sheaves). Let theta _0:H^0(X)rightarrow H^0(Y) be the induced Gysin morphism. We say that theta has degree one if theta _0(1_X)= 1_Yin H^0(Y). This is equivalent to say that theta is a section of the pull-back f^*: {mathbb {A}}_Yrightarrow Rf_*{mathbb {A}}_X, i.e. theta circ f^*={text {id}}_{{mathbb {A}}_Y}, and it is also equivalent to say that {mathbb {A}}_Y is a direct summand of Rf_*{mathbb {A}}_X. We investigate the consequences of the existence of a bivariant class of degree one. We prove explicit formulas relating the (co)homology of X and Y, which extend the classic formulas of the blowing-up. These formulas are compatible with the duality morphism. Using which, we prove that the existence of a bivariant class theta of degree one for a resolution of singularities, is equivalent to require that Y is an {mathbb {A}}-homology manifold. In this case theta is unique, and the Betti numbers of the singular locus {text {Sing}}(Y) of Y are related with the ones of f^{-1}({text {Sing}}(Y)).

Regulator maps for higher Chow groups via current transforms

We show how to use equidimensional algebraic correspondences between complex algebraic varieties to construct pull-backs and transforms of certain classes of geometric currents. Using this construction we produce explicit formulas at the level of complexes for a regulator map from the higher Chow groups of smooth complex quasi-projective algebraic varieties to Deligne–Beilinson cohomology with integral coefficients. A distinct aspect of our approach is the use of Suslin's complex n ↦ Z Δ , eq p ( X , n ) $n \mapsto \mathcal {Z}^p_{\Delta , \text{eq}}(X,n)$ of equidimensional cycles over Δ n $ \Delta ^n$ to compute Bloch's higher Chow groups. We calculate explicit examples involving the Mähler measure of Laurent polynomials.

Note on absolute sets of rigid local systems

In this note we give a description up to a quasi-finite morphism of the absolute sets of simple cohomologically rigid local systems on a smooth complex quasi-projective algebraic variety. In dimension one or rank two, this proves a conjecture of Budur–Wang on the structure of these sets.

Actions of Nilpotent Groups on Complex Algebraic Varieties

AbstractWe study nilpotent groups that act faithfully on complex algebraic varieties. In the finite case, we show that when $\textbf {k}$ is a number field, a finite $p$-subgroup of the group of polynomial automorphisms of $\textbf {k}^d$ is isomorphic to a subgroup of $\textrm {GL}_d(\textbf {k})$. In the case of infinite nilpotent group actions, we show that a finitely generated nilpotent group $H$ acting on a complex quasiprojective variety $X$ of dimension $d$ can be embedded in a $p$-adic Lie group that acts faithfully and analytically on $\textbf {Z}_p^d$. As a consequence, we show that the virtual derived length of $H$ is at most the dimension of $X$.

GEOMETRY AND TOPOLOGY OF EXTERNAL AND SYMMETRIC PRODUCTS OF VARIETIES

We give a brief overview of recent developments on the calculation of generating series for invariants of external products of suitable coefficients (e.g., constructible or coherent sheaves, or mixed Hodge modules) on complex quasi-projective varieties.

On the geometric André–Oort conjecture for variations of Hodge structures

Abstract Let 𝕍 {{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for ( S , 𝕍 ) {(S,{\mathbb{V}})} , which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.

On the closure of the Hodge locus of positive period dimension

Given {{mathbb {V}}} a polarizable variation of {{mathbb {Z}}}-Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for {{mathbb {V}}}^otimes is the set of closed points s of S where the fiber {{mathbb {V}}}_s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for {{mathbb {V}}}^otimes is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for {{mathbb {V}}}. Under the assumption that the adjoint group of the generic Mumford–Tate group of {{mathbb {V}}} is simple we prove that the union of the special subvarieties for {{mathbb {V}}} whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space {{mathcal {A}}}_g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of {{mathcal {A}}}_g is either a closed algebraic subvariety of S or is Zariski-dense in S.

Serre polynomials of [formula omitted]- and [formula omitted]-character varieties of free groups

Let G be a complex reductive group and XrG denote the G-character variety of the free group of rank r. Using geometric methods, we prove that E(XrSLn)=E(XrPGLn), for any n,r∈N, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety X, settling a conjecture of Lawton–Muñoz in Lawton and Muñoz (2016). The proof involves the stratification by polystable type introduced in Florentino et al. (2019), and shows moreover that the equality of E-polynomials holds for every stratum and, in particular, for the irreducible stratum of XrSLn and XrPGLn. We also present explicit computations of these polynomials, and of the corresponding Euler characteristics, based on our previous results and on formulas of Mozgovoy–Reineke for GLn-character varieties over finite fields.

Plethysm and cohomology representations of external and symmetric products

We prove refined generating series formulae for characters of (virtual) cohomology representations of external products of suitable coefficients, e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules on spaces such as (possibly singular) complex quasi-projective varieties. These formulae generalize our previous results for symmetric and alternating powers of such coefficients, and apply also to other Schur functors. The proofs of these results are reduced via an equivariant Künneth formula to a more general generating series identity for abstract characters of tensor powers V⊗n of an element V in a suitable symmetric monoidal category A. This abstract approach applies directly also in the equivariant context for spaces with additional symmetries (e.g., finite group actions, finite order automorphisms, resp., endomorphisms), as well as for introducing an abstract plethysm calculus for symmetric sequences of objects in A.

Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups

The notion of the orbifold Euler characteristic came from physics at the end of the 1980s. Coincidence (up to sign) of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric. There were defined higher order versions of the orbifold Euler characteristic and generalized (“motivic”) versions of them. In a previous paper, the authors defined a notion of the Grothendieck ring K 0 fGr ( Var C ) of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here, we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from K 0 fGr ( Var C ) to the Grothendieck ring K 0 ( Var C ) of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space.

Power structure over the Grothendieck ring of maps

A power structure over a ring is a method to give sense to expressions of the form $$(1+a_1t+a_2t^2+\cdots )^m$$ , where $$a_i$$ , $$i=1, 2,\ldots $$ , and m are elements of the ring. The (natural) power structure over the Grothendieck ring of complex quasi-projective varieties appeared to be useful for a number of applications. We discuss new examples of $$\lambda $$ -and power structures over some Grothendieck rings. The main example is for the Grothendieck ring of maps of complex quasi-projective varieties. We describe two natural $$\lambda $$ -structures on it which lead to the same power structure. We show that this power structure is effective. In the terms of this power structure we write some equations containing classes of Hilbert–Chow morphisms. We describe some generalizations of this construction for maps of varieties with some additional structures.

LOCAL SYSTEMS ON COMPLEMENTS OF ARRANGEMENTS OF SMOOTH, COMPLEX ALGEBRAIC HYPERSURFACES

We consider smooth, complex quasiprojective varieties$U$that admit a compactification with a boundary, which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on$U$vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.

Equivariant characteristic classes of external and symmetric products of varieties

We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasi-projective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel-Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products, and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.

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.

Equivariant analogues of the Euler characteristic and Macdonald type equations

One of the simplest and, at the same time, most important invariants of a topological space is the Euler characteristic. A generalization of the notion of the Euler characteristic to the equivariant setting, that is, to spaces with an action of a group (say, finite) is far from unique. An equivariant analogue of the Euler characteristic can be defined as an element of the ring of representations of the group or as an element of the Burnside ring of the group. From physics came the notion of the orbifold Euler characteristic, and this was generalized to orbifold Euler characteristics of higher orders. The main property of the Euler characteristic (defined in terms of the cohomology with compact support) is its additivity. On some classes of spaces there are additive invariants other than the Euler characteristic, and they can be regarded as generalized Euler characteristics. For example, the class of a variety in the Grothendieck ring of complex quasi-projective varieties is a universal additive invariant on the class of complex quasi-projective varieties. Generalized analogues of the Euler characteristic can also be defined in the equivariant setting. There is a simple formula — the Macdonald equation — for the generating series of the Euler characteristics of the symmetric powers of a space: it is equal to the series independent of the space, raised to a power equal to the Euler characteristic of the space itself. Equations of a similar kind for other invariants (`equivariant and generalized Euler characteristics') are called Macdonald type equations. This survey discusses different versions of the Euler characteristic in the equivariant setting and describes some of their properties and Macdonald type equations. Bibliography: 59 titles.

Compatibility of t-structures for quantum symplectic resolutions

Let W be a smooth complex quasiprojective variety with the action of a connected reductive group G. Adapting the stratification approach of Teleman to a microlocal context, we prove a vanishing theorem for the functor of G-invariant sections---i.e., of quantum Hamiltonian reduction---for G-equivariant twisted D-modules on W. As a consequence, when W is affine we establish an effective combinatorial criterion for exactness of the global sections functors of microlocalization theory. When combined with our earlier derived equivalence results, this gives precise criteria for "microlocalization of representation categories."

Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting

We define and construct mixed Hodge structures on real schematic homotopy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x-i), giving new results even for simply connected varieties. The mixed Hodge structures can thus be recovered from the Gysin spectral sequence of cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these structures all become real variations of mixed Hodge structure.

Higher order generalized Euler characteristics and generating series

For a complex quasi-projective manifold with a finite group action, we define higher order generalized Euler characteristics with values in the Grothendieck ring of complex quasi-projective varieties extended by the rational powers of the class of the affine line. We compute the generating series of generalized Euler characteristics of a fixed order of the Cartesian products of the manifold with the wreath product actions on them.