BOCHNER–MARTINELLI FORMULAS ON SINGULAR COMPLEX SPACES

2010 ◽  
Vol 21 (02) ◽  
pp. 225-253 ◽  
Author(s):  
VINCENZO ANCONA ◽  
BERNARD GAVEAU
Keyword(s):  
Projective Variety ◽  
Complex Space ◽  
Closed Subspace ◽  
Hodge Structure ◽  
Integral Formula ◽  
Mixed Hodge Structure ◽  
Complex Spaces ◽  
Residue Formula ◽  
Complex Dimension ◽  
Singular Complex Spaces

Let U be a complex space of complex dimension n ≥ 2, P a point of U, π : Ũ → U a modification such that Ũ is nonsingular and D = π-1 (P) is a divisor with normal crossings. A Bochner–Martinelli form on U\{P} is a [Formula: see text]-closed differential form ω on Ũ\D, of pure type (n,n - 1), logarithmic along D. Such form detects a cohomology class of H2n - 1 (U\{P},ℂ) on the singular space U\{P}. Thanks to a general residue formula we prove that the forms ω give rise to an integral formula of Bochner–Martinelli type for holomorphic functions. If U satisfies the following assumption that {there exists a compact complex space X bimeromorphic to a Kähler manifold, and a closed subspace T ⊂ X, such that X\T = U (an affine, or a quasi-projective variety satisfies the above property), we relate Bochner–Martinelli forms to the mixed Hodge structure carried by H2n-1 (U\{P},ℂ). Most of our results hold for complex spaces which are not Stein.

scholarly journals Purity for graded potentials and quantum cluster positivity

2015 ◽  
Vol 151 (10) ◽  
pp. 1913-1944 ◽  
Author(s):  
Ben Davison ◽  
Davesh Maulik ◽  
Jörg Schürmann ◽  
Balázs Szendrői
Keyword(s):  
Projective Variety ◽  
Hodge Structure ◽  
Regular Function ◽  
Mixed Hodge Structure ◽  
Positive Weight ◽  
Quantum Cluster ◽  
Vanishing Cycle ◽  
Marked Points ◽  
Quivers With Potential ◽  
Hard Lefschetz Theorem

Consider a smooth quasi-projective variety $X$ equipped with a $\mathbb{C}^{\ast }$-action, and a regular function $f:X\rightarrow \mathbb{C}$ which is $\mathbb{C}^{\ast }$-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of $f$ on proper components of the critical locus of $f$, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work by Kontsevich and Soibelman, Nagao, and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.

scholarly journals Log Picard Algebroids and Meromorphic Line Bundles

2019 ◽  
Author(s):  
Marco Gualtieri ◽  
Kevin Luk
Keyword(s):  
Line Bundle ◽  
Projective Variety ◽  
Hodge Structure ◽  
Smooth Projective Variety ◽  
Line Bundles ◽  
Mixed Hodge Structure ◽  
Natural Class ◽  
De Rham ◽  
Integrality Condition ◽  
Holomorphic Line Bundles

Abstract We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve the prequantization problem, showing that under the appropriate integrality condition, a log Picard algebroid is the Lie algebroid of symmetries of what is called a meromorphic line bundle, a generalization of the usual notion of line bundle in which the fibers degenerate along the divisor. We give a geometric description of such bundles and establish a classification theorem for them, showing that they correspond to a subgroup of the holomorphic line bundles on the divisor complement. Importantly, these holomorphic line bundles need not be algebraic. Finally, we provide concrete methods for explicitly constructing examples of meromorphic line bundles, such as for a smooth cubic divisor in the projective plane.

scholarly journals Tropical geometry and the motivic nearby fiber

2011 ◽  
Vol 148 (1) ◽  
pp. 269-294 ◽  
Author(s):  
Eric Katz ◽  
Alan Stapledon
Keyword(s):  
Euler Characteristic ◽  
Hodge Structure ◽  
Tropical Geometry ◽  
Mixed Hodge Structure ◽  
General Fiber ◽  
Algebraic Torus ◽  
Tropical Varieties

AbstractWe construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the ‘tropical motivic nearby fiber’. This invariant specializes in the schön case to the Hodge–Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge–Deligne polynomial in the cases of schön hypersurfaces and matroidal tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.

scholarly journals Tautological rings of spaces of pointed genus two curves of compact type

2016 ◽  
Vol 152 (7) ◽  
pp. 1398-1420 ◽  
Author(s):  
Dan Petersen
Keyword(s):  
Hodge Structure ◽  
Galois Representation ◽  
Mixed Hodge Structure ◽  
Poincaré Duality ◽  
General Study ◽  
Compact Type ◽  
Poincare Duality ◽  
Tautological Ring ◽  
Genus Two ◽  
Tautological Rings

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$, the moduli space of $n$-pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$. This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$. This improves and simplifies results of the author and Orsola Tommasi.

scholarly journals A smoothness criterion for complex spaces in terms of differential forms

2020 ◽  
Vol 126 (2) ◽  
pp. 221-228
Author(s):  
Håkan Samuelsson Kalm ◽  
Martin Sera
Keyword(s):  
Complex Space ◽  
Differential Forms ◽  
Complex Spaces ◽  
Smoothness Criterion

For a reduced pure dimensional complex space $X$, we show that if Barlet's recently introduced sheaf $\alpha _X^1$ of holomorphic $1$-forms or the sheaf of germs of weakly holomorphic $1$-forms is locally free, then $X$ is smooth. Moreover, we discuss the connection to Barlet's well-known sheaf $\omega _X^1$.

scholarly journals Goresky–Pardon lifts of Chern classes and associated Tate extensions

2017 ◽  
Vol 153 (7) ◽  
pp. 1349-1371 ◽  
Author(s):  
Eduard Looijenga
Keyword(s):  
Vector Bundle ◽  
Hodge Structure ◽  
Holomorphic Vector Bundle ◽  
Mixed Hodge Structure ◽  
Chern Classes ◽  
Analytic Variety ◽  
Algebraic Setting ◽  
Geometric Conditions ◽  
Stable Cohomology ◽  
Complex Analytic Variety

Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SLn

2006 ◽  
Vol 58 (5) ◽  
pp. 1000-1025 ◽  
Author(s):  
Ajneet Dhillon
Keyword(s):  
Vector Bundles ◽  
Hodge Structure ◽  
Betti Numbers ◽  
Mixed Hodge Structure ◽  
Projective Curve ◽  
Ground Field ◽  
Smooth Projective Curve ◽  
Poincaré Polynomial ◽  
Moduli Stack ◽  
Moduli Of Vector Bundles

AbstractWe compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SLn is one.

scholarly journals On The Holomorphic Automorphism Groups of Complex Spaces

1968 ◽  
Vol 33 ◽  
pp. 85-106 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Lie Group ◽  
Complex Space ◽  
Automorphism Groups ◽  
Analogous Result ◽  
Holomorphic Mapping ◽  
Compact Complex Manifold ◽  
Holomorphic Automorphism ◽  
Complex Spaces ◽  
Weaker Conditions ◽  
Holomorphic Automorphism Groups

For a complex space X we consider the group Aut (X) of all automorphisms of X, where an automorphism means a holomorphic automorphism, i.e. an injective holomorphic mapping of X onto X itself with the holomorphic inverse. In 1935, H. Cartan showed that Aut (X) has a structure of a real Lie group if X is a bounded domain in CN([7]) and, in 1946, S. Bochner and D. Montgomery got the analogous result for a compact complex manifold X ([2] and [3]). Afterwards, the latter was generalized by R.C. Gunning ([11]) and H. Kerner ([16]), and the former by W. Kaup ([14]), to complex spaces. The purpose of this paper is to generalize these results to the case of complex spaces with weaker conditions. For brevity, we restrict ourselves to the study of σ-compact irreducible complex spaces only.

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