Complex Analytic Spaces Research Articles

ACC for generalized log canonical thresholds for complex analytic spaces

ACC for generalized log canonical thresholds for complex analytic spaces

Families and unfoldings of singular holomorphic Lie Algebroids

In this paper, we investigate families of singular holomorphic Lie algebroids on complex analytic spaces. We introduce and study a special type of deformation called unfoldings of Lie algebroids, which generalizes the theory of singular holomorphic foliations developed by T. Suwa. We show that a one to one correspondence between transversal unfoldings and holomorphic flat connections on a natural Lie algebroid on the bases exists.

Brauer groups and étale homotopy type

AbstractExtending a result of Schröer on a Grothendieck question in the context of complex analytic spaces, we prove that the surjectivity of the Brauer map for schemes depends on their étale homotopy type. We use properties of algebraic spaces to apply this to some classes of proper and smooth algebraic schemes. In particular, we recover a result of Hoobler and Berkovich for abelian varieties.

Symplectic reduction along a submanifold

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$-space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$, where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

Deformation of rational singularities and Hodge structure

For a one-parameter degeneration of reduced compact complex analytic spaces of dimension $n$, we prove the invariance of the frontier Hodge numbers $h^{p,q}$ (that is, with $pq(n-p)(n-q)=0$) for the intersection cohomology of the fibers and also for the cohomology of their desingularizations, assuming that the central fiber is reduced, projective, and has only rational singularities. This can be shown to be equivalent to the invariance of the dimension of the cohomology of structure sheaf (which is known in the algebraizable case), since we can prove the Hodge symmetry for all the Hodge numbers $h^{p,q}$ together with $E_1$-degeneration of the Hodge-to-de Rham spectral sequence for nearby fibers, assuming only the projectivity of the central fiber. For the proof of the main theorem, we calculate certain graded pieces of the induced $V$-filtration for the first non-zero member of the Hodge filtration on the intersection complex Hodge module of the total space, which coincides with the direct image of the dualizing sheaf of a desingularization as in Kollar's conjecture. This calculation implies also that the order of nilpotence of the local monodromy is smaller than the general singularity case by 2 in the situation of the main theorem assuming further smoothness of general fibers. We can prove a partial converse of the main theorem under some hypothesis.

Topological Triviality in Families of Functions on Subanalytic Sets and Complex Analytic Spaces

Abstract We study families of analytic functions defined on either subanalytic sets or complex analytic spaces. We give sufficient conditions for a family depending linearly on one parameter to have constant topological type, extending classical results due to Lê and Ramanujam [9] and Parusiński [12]. In the particular case of isolated singularity families defined on an ICIS, we prove that the $\mu $-constancy implies constant topological type.

On the structure of some $p$-adic period domains

We study the geometry of the p-adic analogues of the complex analytic period spaces first introduced by Griffiths. More precisely, we prove the Fargues-Rapoport conjecture for p-adic period domains: for a reductive group G over a p-adic field and a minuscule cocharacter µ of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G, µ) is fully Hodge-Newton decomposable.

Vanishing and comparison theorems in rigid analytic geometry

We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$. Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces.

LCK metrics on complex spaces with quotient singularities

In this article we introduce a generalization of locally conformally Kahler metrics from complex manifolds to complex analytic spaces with singularities and study which properties of locally conformally Kahler manifolds still hold in this new setting. We prove that if a complex analytic space has only quotient singularities, then it admits a locally conformally Kahler metric if and only if its universal cover admits a Kahler metric such that the deck automorphisms act by homotheties of the Kahler metric. We also prove that the blow-up at a point of an LCK complex space is also LCK.

GAGA theorems in derived complex geometry

In this paper, we expand the foundations of derived complex analytic geometry introduced by Jacob Lurie in 2011. We start by studying the analytification functor and its properties. In particular, we prove that for a derived complex scheme locally almost of finite presentation X X , the canonical map X a n → X X^{\mathrm {an}} \to X is flat in the derived sense. Next, we provide a comparison result relating derived complex analytic spaces to geometric stacks. Using these results and building on the previous work of the author and Tony Yue Yu, we prove a derived version of the GAGA theorems. As an application, we prove that the infinitesimal deformation theory of a derived complex analytic moduli problem is governed by a differential graded Lie algebra.

Algebraic intersection spaces

We define a variant of intersection space theory that applies to many compact complex and real analytic spaces [Formula: see text], including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze “local duality obstructions,” which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in [Formula: see text] of a tubular neighborhood of the singular set.

Normalization of Complex Analytic Spaces from a Global Viewpoint

In this work, we study some algebraic and topological properties of the ring of global analytic functions on the normalization of a reduced complex analytic space . If is a Stein space, we characterize in terms of the (topological) completion of the integral closure of the ring of global holomorphic functions on X (inside its total ring of fractions) with respect to the usual Frechet topology of . This shows that not only the Stein space but also its normalization is completely determined by the ring of global analytic functions on X. This result was already proved in 1988 by Hayes–Pourcin when is an irreducible Stein space, whereas in this paper we afford the general case. We also analyze the real underlying structures and of a reduced complex analytic space and its normalization . We prove that the complexification of provides the normalization of the complexification of if and only if is a coherent real analytic space. Roughly speaking, coherence of the real underlying structure is equivalent to the equality of the following two combined operations: (1) normalization + real underlying structure + complexification, and (2) real underlying structure + complexification + normalization.

Functorial desingularization over Q: boundaries and the embedded case

Our main result establishes functorial desingularization of noetherian quasi-excellent schemes over Q with ordered boundaries. A functorial embedded desingularization of quasi-excellent schemes of characteristic zero is deduced. Furthermore, a standard simple argument extends these results to other categories including, in particular, (equivariant) embedded desingularization of the following objects of characteristic zero: qe algebraic stacks, qe formal schemes, complex and non-archimedean analytic spaces. We also obtain a semistable reduction theorem for formal schemes.

Stein domains in Banach algebraic geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry.

Local monomialization of analytic maps

In this paper local monomialization theorems are proven for morphisms of complex and real analytic spaces. This gives the generalization of the local monomialization theorem for morphisms of algebraic varieties over a field of characteristic zero proven in [16] and [18] to analytic spaces.

Étoiles and valuations

We establish some properties of étoiles and associated valuations over complex analytic spaces, establishing that Abhyankar's inequality holds. We give some examples of pathological behavior of these valuations. We prove a regularization theorem for complex analytic morphisms. The property of a morphism being regular and the regularization of a morphism play a major role in this theory.

Lifting free divisors

Let $\varphi:X\to S$ be a morphism between smooth complex analytic spaces, and let $f=0$ define a free divisor on $S$. We prove that if the deformation space $T^1_{X/S}$ of $\varphi$ is a Cohen-Macaulay $\mathcal{O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi$, then $f\circ \varphi=0$ defines a free divisor on $X$; this is generalized in several directions. Among applications we recover a result of Mond-van Straten, generalize a construction of Buchweitz-Conca, and show that a map $\varphi:\mathbb{C}^{n+1}\to \mathbb{C}^n$ with critical set of codimension $2$ has a $T^1_{X/S}$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\varphi$ is the algebraic quotient $X\to S=X// G$ with $X// G$ smooth, we describe sufficient conditions for $T^1_{X/S}$ to be Cohen-Macaulay of codimension $2$. In one such case, a free divisor on $\mathbb{C}^{n+1}$ lifts under the operation of "castling" to a free divisor on $\mathbb{C}^{n(n+1)}$, partially generalizing work of Granger-Mond-Schulze on linear free divisors. We give several other examples of such representations.

On Saito's normal crossing condition

Kyoji Saito defined a residue map from the logarithmic differential 1-forms along a reduced complex analytic hypersurface to the meromorphic functions on the hypersurface. He studied the condition that the image of this map coincides with the weakly holomorphic functions, that is, with the functions on the normalization. With Michel Granger, the author proved that this condition is equivalent to the hypersurface being normal crossing in codimension one. In this article, the condition is given a natural interpretation in terms of regular differential forms beyond the hypersurface case. For reduced equidimensional complex analytic spaces which are free in codimension one, the geometric interpretation of being normal crossing in codimension one is shown to persist.

The hyperbolic Ax-Lindemann-Weierstraß conjecture

1.1. Bi-algebraic geometry and the Ax-Lindemann-Weierstras property. — Let X and S be complex algebraic varieties and suppose π : Xan −→ San is a complex analytic, nonalgebraic, morphism between the associated complex analytic spaces. In this situation the image π(Y) of a generic algebraic subvariety Y ⊂ X is usually highly transcendental and the pairs (Y ⊂ X,V ⊂ S) of irreducible algebraic subvarieties such that π(Y) = V are rare and of particular geometric significance. We will say that an irreducible subvariety Y ⊂ X (resp. V ⊂ S) is bi-algebraic if π(Y) is an algebraic subvariety of S (resp. any analytic irreducible component of π−1(V) is an irreducible algebraic subvariety of X). Notice that V ⊂ S is bi-algebraic if and only if any analytic irreducible component of π−1(V) is bi-algebraic.

On ℓ-independency in families of motivic ℓ-adic representations

Let k be a finitely generated field of characteristic 0 embedded into \({\mathbb{C},\,X}\) a smooth, separated and geometrically connected scheme over k with generic point \({\eta}\) and \({f : Y \rightarrow X}\) a smooth proper morphism. Let \({f^{an}_{\mathbb{C}} : Y^{an}_{\mathbb{C}} \rightarrow X^{an}_{\mathbb{C}}}\) denote the associated morphism of complex analytic spaces. For \({x \in X(\mathbb{C})}\), write H for the Betti cohomology of \({Y^{an}_{\mathbb{C}, x}}\) with coefficients in \({\mathbb{Q}}\) and for \({x \in X}\), write \({{\rm H}_{\ell}}\) for the l-adic cohomology of \({Y_{\overline{x}}}\) with coefficients in \({\mathbb{Q}_{\ell}}\) (Under our assumptions on \({f : Y \rightarrow X,\,{\rm H}}\) and \({{\rm H}_{\ell}}\) are independent of x). For every prime l, let \({X^{ex}_{\ell}}\) be the set of all \({x \in X}\) where the Zariski closure \({G_{\ell, x}}\) of the image of the Galois representation \({\Gamma_{k(x)} \rightarrow {\rm GL}({\rm H}_{\ell})}\) has dimension strictly smaller than the dimension of \({G_{\ell,\eta}}\). By previous works of A. Tamagawa and the author, \({X^{ex}_{\ell}}\) is ‘small’ in the sense that if X is a curve then for every integer \({\delta \geq 1}\) the set of all \({x \in X^{ex}_{\ell}}\) with \({[k(x) : k] \leq \delta}\) is finite. Set \({X^{ex} := {\bigcap}_{\ell} X^{ex}_{\ell}}\). The Tate conjectures predict that for every \({x \in X}\) the \({G_{\ell, x}}\) are defined over \({\mathbb{Q}}\), reductive and independent of l hence, in particular, that the sets \({X^{ex}_{\ell}}\) are independent of l. Let \({\overline{G}}\) denote the Zariski closure of the image of the monodromy representation \({\pi_{1} (X_{\mathbb{C}}^{an}; \, x) \rightarrow {\rm GL}({\rm H})}\). Then \({\overline{G}}\) is a semi-simple algebraic group of rank—say—r. The main result of this note is that for \({x \notin X^{ex},\,G_{\ell, x} \cap \overline{G}_{\mathbb{Q}_{\ell}}}\) is a semi-simple algebraic group of rank r. This implies in particular that: (1) If \({\overline{G}_{\overline{\mathbb{Q}}}}\) has only simple factors of type \({A_{n}}\) then \({X^{ex}_{\ell}}\) is independent of l; (2) For every prime l and \({x \notin X^{ex}}\) the unipotent radical of \({G_{\ell, x}}\) coincides with the unipotent radical of \({G_{\ell, \eta}}\) and, in particular, is independent of \({x \notin X^{ex}}\); (3) For every prime l, if there exists \({x_{\ell} \in X}\) such that \({G_{\ell, x_{\ell}}}\) is reductive then for every \({x \notin X^{ex},\,G_{\ell, x}}\) is reductive. (3) applies in particular when \({{\rm H}}\) is a geometrically irreducible \({\overline{G}}\)-module. This implies, for instance, that apart from a few exceptional cases, for every r-tuple \({\underline{d}=(d_{1}, \dots, d_{r})}\) of integers \({\geq 2}\) there exists a non-singular complete intersection in \({\mathbb{P}^{n+r}_{\mathbb{Q}}}\) with multi-degree \({\underline{d}}\) for which the Tate semi-simplicity conjecture holds (for every prime l).