On E. Verlinde’s Formula in the Context of Stable Bundles

1991 ◽  
Vol 06 (16) ◽  
pp. 2847-2858 ◽  
Author(s):  
Raoul Bott
Keyword(s):  
Algebraic Geometry ◽  
Point Of View ◽  
Stable Bundles ◽  
Quotient Rings

E. Verlinde’s formula for the dimension of the nonabelian θ-functions is discussed from an algebraic geometry point of view and related to certain quotient rings of the representative ring of sums.

Fourier-Mukai Transforms in Algebraic Geometry

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Algebraic Geometry ◽  
Smooth Projective Variety ◽  
Derived Category ◽  
Point Of View ◽  
Basic Knowledge ◽  
Research Directions ◽  
Coherent Sheaves ◽  
Geometric Point ◽  
Singular Cohomology

This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

scholarly journals Application of Quotient Rings for Stability Analysis in Chemical Systems

2011 ◽  
Vol 66 (3-4) ◽  
pp. 231-241
Author(s):  
Sonja Sauerbrei ◽  
Anke Sensse ◽  
Markus Eiswirth
Keyword(s):  
Algebraic Geometry ◽  
Stability Analysis ◽  
Surface Reaction ◽  
Normal Forms ◽  
Jacobian Matrix ◽  
Stationary Solutions ◽  
Polynomial Rings ◽  
Chemical Systems ◽  
The Stability ◽  
Quotient Rings

Concepts from algebraic geometry (polynomial rings) can be used to determine analytically the stationary solutions in chemical reactions systems, more generally, systems of ordinary differential equations of polynomial form. The stability analysis via the Jacobian matrix often leads to complicated expressions which can hardly be analyzed. It is shown that these expressions can be simplified by forming quotient rings of the corresponding polynomial ring. The coefficients in the characteristic equation of the Jacobian can be represented by the normal forms obtained by generating the quotient rings so that their sign changes in dependence of a kinetic parameter and, hence, the stability can be determined. The procedure is illustrated using a well-known surface reaction.

scholarly journals Au-dessous de Specℤ

2008 ◽  
Vol 3 (3) ◽  
pp. 437-500 ◽  
Author(s):  
Bertrand Toën ◽  
Michel Vaquié
Keyword(s):  
Algebraic Geometry ◽  
Linear Group ◽  
Toric Varieties ◽  
Base Change ◽  
Point Of View ◽  
Natural Numbers ◽  
Ring Spectra ◽  
Field With One Element

AbstractIn this article we use the theories of relative algebraic geometry and of homotopical algebraic geometry (cf. [HAGII]) to construct some categories of schemes defined under Specℤ. We define the categories of ℕ-schemes, 1-schemes, -schemes, +-schemes and 1-schemes, where (from an intuitive point of view) ℕ is the semi-ring of natural numbers, 1 is the field with one element, is the ring spectra of integers, + is the semi-ring spectra of natural numbers and 1 is the ring spectra with one element. These categories of schemes are linked together by base change functors, and all of them have a base change functor to the category of ℤ-schemes. We show that the linear group Gln and the toric varieties can be defined as objects in these categories.

scholarly journals Approximations, ghosts and derived equivalences

2019 ◽  
Vol 150 (2) ◽  
pp. 813-840
Author(s):  
Yiping Chen ◽  
Wei Hu
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Noncommutative Geometry ◽  
Endomorphism Rings ◽  
Derived Equivalences ◽  
Symmetric Approximation ◽  
Quotient Rings ◽  
Tilting Objects

AbstractApproximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weakly n-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry, and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals.

scholarly journals Algebraic polymorphisms

2008 ◽  
Vol 28 (2) ◽  
pp. 633-642 ◽  
Author(s):  
KLAUS SCHMIDT ◽  
ANATOLY VERSHIK
Keyword(s):  
Algebraic Geometry ◽  
Invariant Measure ◽  
Special Class ◽  
Markov Operator ◽  
Point Of View ◽  
Compact Groups ◽  
Rational Matrix ◽  
Conjugate Operator ◽  
Markov Operators ◽  
Toral Automorphisms

AbstractIn this paper we consider a special class of polymorphisms with invariant measure, the algebraic polymorphisms of compact groups. A general polymorphism is—by definition—a many-valued map with invariant measure, and the conjugate operator of a polymorphism is a Markov operator (i.e. a positive operator on L2 of norm 1 which preserves the constants). In the algebraic case a polymorphism is a correspondence in the sense of algebraic geometry, but here we investigate it from a dynamical point of view. The most important examples are the algebraic polymorphisms of a torus, where we introduce a parametrization of the semigroup of toral polymorphisms in terms of rational matrices and describe the spectra of the corresponding Markov operators. A toral polymorphism is an automorphism of $\mathbb {T}^m$ if and only if the associated rational matrix lies in $\mathrm {GL}(m,\mathbb {Z})$. We characterize toral polymorphisms which are factors of toral automorphisms.

scholarly journals Questions de corps de définition pour les variétés abéliennes en caractéristique positive

2008 ◽  
Vol 7 (4) ◽  
pp. 623-639 ◽  
Author(s):  
Franck Benoist ◽  
Françoise Delon
Keyword(s):  
Algebraic Geometry ◽  
Abelian Variety ◽  
Positive Characteristic ◽  
Abelian Varieties ◽  
Direct Proof ◽  
Point Of View ◽  
Theoretical Point ◽  
Definition Of

AbstractDichotomies in various conjectures from algebraic geometry are in fact occurrences of the dichotomy among Zariski structures. This is what Hrushovski showed and which enabled him to solve, positively, the geometric Mordell–Lang conjecture in positive characteristic. Are we able now to avoid this use of Zariski structures? Pillay and Ziegler have given a direct proof that works for semi-abelian varieties they called ‘very thin’, which include the ordinary abelian varieties. But it does not apply in all generality: we describe here an abelian variety which is not very thin. More generally, we consider from a model-theoretical point of view several questions about the fields of definition of semi-abelian varieties.

SOME RESULTS AND PROBLEMS RELATED TO UNIVERSAL ALGEBRAIC GEOMETRY

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1133-1164 ◽  
Author(s):  
BORIS PLOTKIN
Keyword(s):  
Algebraic Geometry ◽  
Point Of View ◽  
Variety Of Algebras ◽  
Geometric Properties ◽  
Algebraic Sets ◽  
The Subject ◽  
Universal Algebraic Geometry ◽  
Future Work ◽  
The Given

In universal algebraic geometry (UAG), some primary notions of classical algebraic geometry are applied to an arbitrary variety of algebras Θ and an arbitrary algebra H ∈ Θ. We consider an algebraic geometry in Θ over the distinguished algebra H and we also analyze H from the point of view of its geometric properties. This insight leads to a system of new notions and stimulates a number of new problems. They are new with respect to algebra, algebraic geometry and even with respect to the classical algebraic geometry. In our approach, there are two main aspects: the first one is a study of the algebra H and its geometric properties, while the second is focused on studying algebraic sets and algebraic varieties over a "good", particular algebra H. Considering the subject from the second standpoint, the main goal is to get forward as far as possible in a classification of algebraic sets over the given H. The first approach does not require such a classification which is itself an independent and extremely difficult task. We also consider some geometric relations between different H1 and H2 in Θ. The present paper should be viewed as a brief review of what has been done in universal algebraic geometry. We also give a list of unsolved problems for future work.

scholarly journals Algebraic Deformations of Polarized Varieties

1968 ◽  
Vol 31 ◽  
pp. 185-245 ◽  
Author(s):  
T. Matsusaka
Keyword(s):  
Algebraic Geometry ◽  
Point Of View ◽  
General Point ◽  
Complex Manifolds ◽  
Singular Variety ◽  
Projective Embedding

Let V be a projectively embeddable complete non-singular variety of dimension n > 1. Let f be a projective embedding of V, U a non-singular variety, W a non-singular variety and φ a morphism of W onto U such that φ-1(u0) = f(V) for some point u0 of U. Denote by ∑(V) the set of all those complete non-singular fibres φ-1(u), u ∈ U, as we consider all possible (f, U, W). Suppose that we call members of ∑(V) (algebraic) deformations of V and propose to study ∑(V) from the stand point of algebraic geometry, as a generalization of the case of curves. This has been taken up at least locally by Kodaira, Spencer, Kuranishi and others in the case of characteristic 0 from a little more general point of view of complex manifolds (cf. [9] and references given in [16]).

scholarly journals Symplectic resolutions of quiver varieties

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Gwyn Bellamy ◽  
Travis Schedler
Keyword(s):  
Algebraic Geometry ◽  
Local Structure ◽  
Point Of View ◽  
Interesting Consequence ◽  
Quiver Varieties ◽  
Symplectic Leaves ◽  
To Come ◽  
Symplectic Resolutions ◽  
Smooth Locus

AbstractIn this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. We prove that they are all symplectic singularities in the sense of Beauville and completely classify which admit symplectic resolutions. Moreover we show that the smooth locus coincides with the locus of canonically $$\theta $$ θ -polystable points, generalizing a result of Le Bruyn; we study their étale local structure and find their symplectic leaves. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT.

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