scholarly journals GEOMETRY AND TOPOLOGY OF EXTERNAL AND SYMMETRIC PRODUCTS OF VARIETIES

2021 ◽  
pp. 3-18
Author(s):  
Laurentiu George Maxim
Keyword(s):  
Symmetric Products ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
And Topology ◽  
Generating Series ◽  
Recent Developments ◽  
Mixed Hodge Modules

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.

scholarly journals Characteristic classes of symmetric products of complex quasi-projective varieties

2017 ◽  
Vol 2017 (728) ◽  
Author(s):  
Sylvain E. Cappell ◽  
Laurentiu Maxim ◽  
Jörg Schürmann ◽  
Julius L. Shaneson ◽  
Shoji Yokura
Keyword(s):  
Characteristic Classes ◽  
Symmetric Products ◽  
Projective Varieties ◽  
Generating Series

AbstractWe prove generating series formulae for suitable twisted characteristic classes of symmetric products of a

scholarly journals A Riemann–Hurwitz Theorem for the Algebraic Euler Characteristic

2017 ◽  
Vol 60 (3) ◽  
pp. 490-509
Author(s):  
Andrew Fiori
Keyword(s):  
Euler Characteristic ◽  
Euler Characteristics ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Irreducible Components ◽  
Arbitrary Dimensions ◽  
Smooth Projective Varieties ◽  
Ramification Locus ◽  
Hurwitz Theorem

AbstractWe prove an analogue of the Riemann–Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

scholarly journals GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

2013 ◽  
Vol 1 ◽  
Author(s):  
MIHNEA POPA ◽  
CHRISTIAN SCHNELL
Keyword(s):  
Abelian Varieties ◽  
Local Systems ◽  
Coherent Sheaves ◽  
Natural Categories ◽  
Rank One ◽  
Generic Vanishing ◽  
Mixed Hodge Modules ◽  
Irregular Varieties

AbstractWe extend most of the results of generic vanishing theory to bundles of holomorphic forms and rank-one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated with irregular varieties. Our main tools are Saito’s mixed Hodge modules, the Fourier–Mukai transform for $\mathscr{D}$-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson’s harmonic theory for flat bundles. In the process, we also discover two natural categories of perverse coherent sheaves.

scholarly journals Symmetric products of mixed Hodge modules

2011 ◽  
Vol 96 (5) ◽  
pp. 462-483 ◽  
Author(s):  
Laurentiu Maxim ◽  
Morihiko Saito ◽  
Jörg Schürmann
Keyword(s):  
Symmetric Products ◽  
Mixed Hodge Modules

scholarly journals HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES

2010 ◽  
Vol 02 (01) ◽  
pp. 1-55 ◽  
Author(s):  
JEAN-PAUL BRASSELET ◽  
JÖRG SCHÜRMANN ◽  
SHOJI YOKURA
Keyword(s):  
Chern Class ◽  
Blow Up ◽  
Homology Class ◽  
Grothendieck Group ◽  
Characteristic Classes ◽  
Chern Classes ◽  
Singular Spaces ◽  
Coherent Sheaves ◽  
Roch Theorem ◽  
Mixed Hodge Modules

In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformationmCy: K0( var /X) → G0(X) ⊗ ℤ[y], which generalizes the total λ-class λy(T*X) of the cotangent bundle to singular spaces. Here K0( var /X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G0(X) is the Grothendieck group of coherent sheaves of [Formula: see text]-modules. A first construction of mCy is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCy is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito. We define a natural transformation Ty* : K0( var /X) → H*(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty* is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = -1), the Todd class transformation in the singular Riemann-Roch theorem of Baum–Fulton–MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1). We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov–Libgober and the stringy Chern classes of Aluffi and De Fernex–Lupercio–Nevins–Uribe. All our results can be extended to varieties over a base field k of characteristic 0.

scholarly journals Which abelian tensor categories are geometric?

2018 ◽  
Vol 2018 (734) ◽  
pp. 145-186 ◽  
Author(s):  
Daniel Schäppi
Keyword(s):  
Characteristic Zero ◽  
Intrinsic Properties ◽  
Tensor Categories ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Geometric Objects ◽  
Necessary And Sufficient ◽  
Tannaka Duality ◽  
Tannakian Categories

AbstractFor a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding.However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties—expressible entirely within the language of tensor categories—which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne’s result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero.Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products.

scholarly journals A RECOLLEMENT APPROACH TO GEIGLE–LENZING WEIGHTED PROJECTIVE VARIETIES

2016 ◽  
Vol 226 ◽  
pp. 71-105 ◽  
Author(s):  
BORIS LERNER ◽  
STEFFEN OPPERMANN
Keyword(s):  
Algebraic Geometry ◽  
Abelian Category ◽  
New Method ◽  
Projective Varieties ◽  
Noncommutative Algebraic Geometry ◽  
Coherent Sheaves ◽  
Endomorphism Algebras ◽  
Projective Lines

We introduce a new method for expanding an abelian category and study it using recollements. In particular, we give a criterion for the existence of cotilting objects. We show, using techniques from noncommutative algebraic geometry, that our construction encompasses the category of coherent sheaves on Geigle–Lenzing weighted projective lines. We apply our construction to some concrete examples and obtain new weighted projective varieties, and analyze the endomorphism algebras of their tilting bundles.

scholarly journals Derived moduli of schemes and sheaves

2011 ◽  
Vol 10 (1) ◽  
pp. 41-85 ◽  
Author(s):  
J.P. Pridham
Keyword(s):  
Vector Bundles ◽  
Abelian Varieties ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Moduli Of Vector Bundles

AbstractWe describe derived moduli functors for a range of problems involving schemes and quasi-coherent sheaves, and give cohomological conditions for them to be representable by derived geometric n-stacks. Examples of problems represented by derived geometric 1-stacks are derived moduli of polarised projective varieties, derived moduli of vector bundles, and derived moduli of abelian varieties.

scholarly journals Calabi–Yau and fractional Calabi–Yau categories

2019 ◽  
Vol 2019 (753) ◽  
pp. 239-267 ◽  
Author(s):  
Alexander Kuznetsov
Keyword(s):  
Derived Categories ◽  
General Construction ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
General Properties ◽  
Lefschetz Decomposition ◽  
Smooth Projective Varieties

AbstractWe discuss Calabi–Yau and fractional Calabi–Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi–Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of applications of this construction and discuss some general properties of Calabi–Yau categories.

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