Smooth Algebraic Variety Research Articles

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.

Note on the filtrations of the K-theory

Let $X$ be a (colimit of) smooth algebraic variety over a subfield $k$ of $\mathbf C$. Let $K_{alg}^0(X)$ (resp. $K_{top}^0(X(\mathbf C)))$ be the algebraic (resp. topological) $K$-theory of $k$ (resp. complex) vector bundles over $X$ (resp. $X(\mathbf C)))$. When $K_{alg}^0(X) \cong K_{top}^0(X(\mathbf C))$, we study the differences of its three (gamma, geometrical and topological) filtrations. In particular, we consider in the cases $X = BG$ for algebraic group $G$ over algebraically closed fields $k$, and $X = \mathbf G_k/T_k$ the twisted form of flag varieties $G/T$ for non-algebraically closed field $k$.

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.

Monads for framed sheaves on Hirzebruch surfaces

Abstract We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.

Gromov-Witten theory of root Gerbes I: Structure of genus 0 moduli spaces

Let $X$ be a smooth complex projective algebraic variety. Given a line bundle $\mathcal{L}$ over $X$ and an integer $r \gt 1$, one defines the stack $\sqrt[r]{\mathcal{L} / X}$ of $r$-th roots of $\mathcal{L}$. Motivated by Gromov-Witten theoretic questions, in this paper we analyze the structure of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L} / X}$. Our main results are explicit constructions of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L} / X}$ starting from moduli stacks of genus $0$ stable maps to $X$. As a consequence, we prove an exact formula expressing genus $0$ Gromov-Witten invariants of $\sqrt[r]{\mathcal{L} / X}$ in terms of those of $X$.

Monodromy and the Lefschetz fixed point formula

We give a new proof - not using resolution of singularities - of a formula of Denef and the second author expressing the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. Our proof uses l-adic cohomology of non-archimedean spaces, motivic integration and the Lefschetz fixed point formula for finite order automorphisms. We also consider a generalization due to Nicaise and Sebag and at the end of the paper we discuss connections with the motivic Serre invariant and the motivic Milnor fiber.

DG-modules over de Rham DG-algebra

For a morphism of smooth schemes over a regular affine base we define functors of derived direct image and extraordinary inverse image on coderived categories of DG-modules over de Rham DG-algebras. Positselski proved that for a smooth algebraic variety \(X\) over a field \(k\) of characteristic zero the coderived category of DG-modules over \(\Omega ^\bullet _{X/k}\) is equivalent to the unbounded derived category of quasi-coherent right \({\fancyscript{D}}_X\)-modules. We prove that our functors correspond to the functors of the same name for \({\fancyscript{D}}_X\)-modules under Positselski equivalence.

Cleanliness and log-characteristic cycles for vector bundles with flat connections

Let $$X$$ be a proper smooth algebraic variety over a field $$k$$ of characteristic zero and let $$D$$ be a divisor with simple normal crossings. Let $$M$$ be a vector bundle over $$X-D$$ equipped with a flat connection with possible irregular singularities along $$D$$ . We define a cleanliness condition which roughly says that the singularities of the connection are controlled by the singularities at the generic points of $$D$$ . When this condition is satisfied, we compute explicitly the associated log-characteristic cycle, and relate it to the so-called refined irregularities. As a corollary of a log-variant of Kashiwara–Dubson formula, we obtain the Euler characteristic of the de Rham cohomology of the vector bundle, under a mild technical hypothesis on $$M$$ .

E1–formality of complex algebraic varieties

Let X be a smooth complex algebraic variety. Morgan [Mor78] showed that the rational homotopy type of X is a formal consequence of the differential graded algebra defined by the first term of its weight spectral sequence. In the present work we generalize this result to arbitrary nilpotent complex algebraic varieties (possibly singular and/or non-compact) and to algebraic morphisms between them. The result for algebraic morphisms generalizes the Formality Theorem of [DGMS75] for compact K\"ahler varieties, filling a gap in Morgan's theory concerning functoriality over the rational numbers. As an application, we study the Hopf invariant of certain algebraic morphisms using intersection theory.

Hodge filtered complex bordism

We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and $\A^1$-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.

Twisted deformation quantization of algebraic varieties

Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf OX. These are stack-like versions of usual deformations. We prove that there is a twisted quantization operation from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.

On the Lie algebroid of a derived self-intersection

Let i:X↪Y be a closed embedding of smooth algebraic varieties. Denote by N the normal bundle of X in Y. The present paper contains two constructions of certain Lie structure on the shifted normal bundle N[−1] encoding the information of the formal neighborhood of X in Y. We also present a few applications of these Lie theoretic constructions in understanding the algebraic geometry of embeddings.

ℓ-Independence for a system of motivic representations

Let $X$ be an smooth projective algebraic variety over a number field $F \subset \mathbb{C}$. Suppose that the Absolute Hodge(AH) motive $M:= h^i(X)$ is contained in the Tannakian category generated by the AH motives of abelian varieties. For every prime number $\ell$ the Galois group $\Gamma_F:= Gal(\bar{F}/F)$ acts on $H_\ell(M)$, the $\ell $-adic realization of $M$. Over a finite extension of $F$ this action factorizes as $\rho_{M,\ell}:\Gamma_F\rightarrow G_M(\ql)$, where $G_M$ is the Mumford-Tate group of $M$. Fix a valuation $v$ of $F$ and suppose $v(\ell)=0 $. The restriction $ \rho_{M,\ell} \vert _{\Gamma_{F_v}}$ defines a representation ${}'W_v \rightarrow G_{M/\ql}$ of the Weil-Deligne group of $F_v$. J-P Serre and J-M Fontaine (independently) have made conjectures that indicates that ${}'W_v \rightarrow G_{M/\ql}$ should be defined over $\mathbb{Q}$ for $\ell$ fixed and that these representations form a compatible system for variable $\ell $. Under certain additional hypothesis , we answer these questions in affirmative, when $X$ has good reduction or Semi-Stable reduction at $v$.

Generalized holomorphic analytic torsion

In this paper we extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for projective morphisms. The extension of the holomorphic analytic torsion classes of Bismut and K¨ohler is obtained as the theory of generalized analytic torsion classes associated to –R=2 , R being the R -genus. As an application of the axiomatic characterization, we give new simpler proofs of known properties of holomorpic analytic torsion classes, we give a characterization of the R -genus, and we construct a direct image of hermitian structures for projective morphisms.

Irreducible components of Hurwitz spaces parameterizing Galois coverings of curves of positive genus

Let Y be a smooth, projective, irreducible complex curve. A G-covering p : C → Y is a Galois covering, where C is a smooth, projective, irreducible curve and an isomorphism G ∼ −→ Aut(C/Y ) is fixed. Two G-coverings are equivalent if there is a G-equivariant isomorphism between them. We are concerned with the Hurwitz spaces H n (Y ) and H G n (Y, y0). The first one parameterizes Gequivalence classes of G-coverings of Y branched in n points. The second one, given a point y0 ∈ Y , parameterizes G-equivalence classes of pairs [p : C → Y, z0], where p : C → Y is a G-covering unramified at y0 and z0 ∈ p (y0). When G = Sd one can equivalently consider coverings f : X → Y of degree d with full monodromy group Sd. The Hurwitz spaces are smooth algebraic varieties and associating to a covering its branch divisor yields finite etale morphisms H n (Y ) → Y (n) − ∆ and H n (Y ) → (Y − y0) (n) − ∆, where ∆ is the codimension one subvariety of Y (n) whose points correspond to effective non simple divisors of Y . The main result of the present paper is the explicit calculation of the monodromy action of the fundamental groups of Y (n) −∆ and (Y − y0) (n) −∆ on the fibers of the above topological coverings (see Theorem 2.8 and Theorem 2.10). The connected (=irreducible) components of H n (Y ) and H G n (Y, y0) are in one-to-one correspondence with the orbits of these monodromy actions. The case Y = P, G = Sd is classical. Hurwitz, using results of Clebsch [Cl], proved in [Hu] the connectedness of the space which parameterizes equivalence classes of simple coverings of P branched in n points (see [Vo2, Lemma 10.15] for a modern account). The Hurwitz spaces of Galois coverings of P were first introduced and studied by Fried in [Fr] in connection with the inverse Galois problem. Fried and Volklein prove in [FV] that H n (P ) has a structure of algebraic variety over Q and if furthermore the center of G is trivial they relate the solution of the inverse Galois problem to the existence of Q-rational points of H n (P ). They also address the problem of determining the connected components of the complex variety H n (P ). Berstein and Edmonds study in [BE] the Hurwitz spaces of simply branched coverings X → Y and address the problem of its connectedness in relation to the topological classification of the generic branched coverings between

MONADS FOR FRAMED TORSION-FREE SHEAVES ON MULTI-BLOW-UPS OF THE PROJECTIVE PLANE

We construct monads for framed torsion-free sheaves on blow-ups of the complex projective plane at finitely many distinct points. Using these monads we prove that the moduli space of such sheaves is a smooth algebraic variety. Moreover, we construct monads for families of such sheaves parametrized by a noetherian scheme S of finite type. A universal monad on the moduli space is introduced and used to prove that the moduli space is fine.

Kontsevich's conjecture on an algebraic formula for vanishing cycles of local systems

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the corresponding locally free sheaf with integrable connection having regular singularity at innity. We also prove its local version, which may be viewed as a natural generalization of a result of E. Brieskorn in the isolated singularity case. We then generalize these to the case of the de Rham complexes of regular holonomic D-modules where we have to use the tensor product with a certain sheaf of formal microlocal dierential operators instead of the formal completion.

On the algebraicity of the zero locus of an admissible normal function

AbstractWe show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic. In Part II of the paper, which is an appendix, we compute the Tannakian Galois group of the category of one-variable admissible real nilpotent orbits with split limit. We then use the answer to recover an unpublished theorem of Deligne, which characterizes the ${\mathrm{sl} }_{2} $-splitting of a real mixed Hodge structure.

A defect relation for meromorphic maps on generalized p-parabolic manifolds intersecting hypersurfaces in complex projective algebraic varieties

AbstractWe establish a defect relation for algebraically non-degenerate meromorphic maps over generalized p-parabolic manifolds that intersect hypersurfaces in smooth projective algebraic varieties, extending certain results of H. Cartan, L. Ahlfors, W. Stoll, M. Ru, P. M. Wong and Philip P. W. Wong and others.

Deformations of affine varieties and the Deligne crossed groupoid

Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R=K〚ℏ〛). We consider associative (resp. Poisson) R-deformations of the structure sheaf OX. The set of R-deformations has a crossed groupoid (i.e. strict 2-groupoid) structure. Our main result is that there is a canonical equivalence of crossed groupoids from the Deligne crossed groupoid of normalized polydifferential operators (resp. polyderivations) of X to the crossed groupoid of associative (resp. Poisson) R-deformations of OX. The proof relies on a careful study of adically complete sheaves. In the associative case we also have to use ring theory (Ore localizations) and the properties of the Hochschild cochain complex.The results of this paper extend previous work by various authors. They are needed for our work on twisted deformation quantization of algebraic varieties.