Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck PDF Download

Author: Jean-Michel Bismut
Publisher: Springer Nature
ISBN: 303127234X
Category : Mathematics
Languages : en
Pages : 181

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Book Description
This monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coherent sheaves of a compact complex manifold with values in its Bott-Chern cohomology, and the proof of a corresponding Riemann-Roch-Grothendieck theorem. One main tool used is the equivalence of categories established by Block between the derived category of bounded complexes with coherent cohomology and the homotopy category of antiholomorphic superconnections. Chern-Weil theoretic techniques are then used to construct forms that represent the Chern character. The main theorem is then established using methods of analysis, by combining local index theory with the hypoelliptic Laplacian. Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck is an important contribution to both the geometric and analytic study of complex manifolds and, as such, it will be a valuable resource for many researchers in geometry, analysis, and mathematical physics.

Author: Jean-Michel Bismut
Publisher: Springer Science & Business Media
ISBN: 3319001280
Category : Mathematics
Languages : en
Pages : 211

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Book Description
The book provides the proof of a complex geometric version of a well-known result in algebraic geometry: the theorem of Riemann–Roch–Grothendieck for proper submersions. It gives an equality of cohomology classes in Bott–Chern cohomology, which is a refinement for complex manifolds of de Rham cohomology. When the manifolds are Kähler, our main result is known. A proof can be given using the elliptic Hodge theory of the fibres, its deformation via Quillen's superconnections, and a version in families of the 'fantastic cancellations' of McKean–Singer in local index theory. In the general case, this approach breaks down because the cancellations do not occur any more. One tool used in the book is a deformation of the Hodge theory of the fibres to a hypoelliptic Hodge theory, in such a way that the relevant cohomological information is preserved, and 'fantastic cancellations' do occur for the deformation. The deformed hypoelliptic Laplacian acts on the total space of the relative tangent bundle of the fibres. While the original hypoelliptic Laplacian discovered by the author can be described in terms of the harmonic oscillator along the tangent bundle and of the geodesic flow, here, the harmonic oscillator has to be replaced by a quartic oscillator. Another idea developed in the book is that while classical elliptic Hodge theory is based on the Hermitian product on forms, the hypoelliptic theory involves a Hermitian pairing which is a mild modification of intersection pairing. Probabilistic considerations play an important role, either as a motivation of some constructions, or in the proofs themselves.

Author: Hua Qiang
Publisher:
ISBN:
Category :
Languages : en
Pages : 0

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Book Description
In the paper [Blo10], Block constructed a dg-category P A0,· using cohesive modules which is a dg-enhancement of Db Coh(X), the bounded derived category of complexes of analytic sheaves with coherent cohomology. This enables us to study coherent sheaves using global differential geometric constructions.In the first part of my thesis, we construct natural superconnections in the sense of Quillen [Qui85] on cohesive modules and use them to define characteristic classes with values in Bott-Chern cohomology. In addition, we generalize the double transgression formulas in [BGS88a] [BC65] [Don87] and prove the invariance of these characteristic classes under derived equivalences. This provides an extension of Bott-Chern characteristic classes to coherent sheaves on complex manifolds and answers the question raised in [Bis13].In the second part of my thesis, we define the generalized Dolbeault-Dirac operator on the generalized Dolbeault complex for a cohesive module. We identify it with a generalized Dirac operator in the sense of Clifford modules and Clifford superconnections as in [BGV91]. Applying the heat kernel method and a theorem of Getzler in [Get91], we first derive a generalization of the Hirzebruch-Riemann-Roch formula to compute the Euler characteristic. Then, generalizing Bismut's proof of the family index theorem, we derive a generalization of the classical Grothendieck-Riemann-Roch formula with values in Bott-Chern cohomology in special cases.

Author: Gerd Faltings
Publisher: Princeton University Press
ISBN: 1400882478
Category : Mathematics
Languages : en
Pages : 118

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Book Description
The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.

Author: Minh Kha
Publisher: Springer Nature
ISBN: 3030674282
Category : Mathematics
Languages : en
Pages : 96

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Book Description
This book is devoted to computing the index of elliptic PDEs on non-compact Riemannian manifolds in the presence of local singularities and zeros, as well as polynomial growth at infinity. The classical Riemann–Roch theorem and its generalizations to elliptic equations on bounded domains and compact manifolds, due to Maz’ya, Plameneskii, Nadirashvilli, Gromov and Shubin, account for the contribution to the index due to a divisor of zeros and singularities. On the other hand, the Liouville theorems of Avellaneda, Lin, Li, Moser, Struwe, Kuchment and Pinchover provide the index of periodic elliptic equations on abelian coverings of compact manifolds with polynomial growth at infinity, i.e. in the presence of a "divisor" at infinity. A natural question is whether one can combine the Riemann–Roch and Liouville type results. This monograph shows that this can indeed be done, however the answers are more intricate than one might initially expect. Namely, the interaction between the finite divisor and the point at infinity is non-trivial. The text is targeted towards researchers in PDEs, geometric analysis, and mathematical physics.

Author: Alexander Reznikov
Publisher: Springer Science & Business Media
ISBN: 9780817641153
Category : Mathematics
Languages : en
Pages : 352

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Book Description
"This work is an outgrowth of a conference held at the Hebrew University in Jerusalem on Regulators in Analysis, Geometry and Number Theory, and should appeal to a broad audience of graduate students and research mathematicians."--BOOK JACKET.

Author: Serge Lang
Publisher: Courier Dover Publications
ISBN: 048683980X
Category : Mathematics
Languages : en
Pages : 273

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Book Description
Author Serge Lang defines algebraic geometry as the study of systems of algebraic equations in several variables and of the structure that one can give to the solutions of such equations. The study can be carried out in four ways: analytical, topological, algebraico-geometric, and arithmetic. This volume offers a rapid, concise, and self-contained introductory approach to the algebraic aspects of the third method, the algebraico-geometric. The treatment assumes only familiarity with elementary algebra up to the level of Galois theory. Starting with an opening chapter on the general theory of places, the author advances to examinations of algebraic varieties, the absolute theory of varieties, and products, projections, and correspondences. Subsequent chapters explore normal varieties, divisors and linear systems, differential forms, the theory of simple points, and algebraic groups, concluding with a focus on the Riemann-Roch theorem. All the theorems of a general nature related to the foundations of the theory of algebraic groups are featured.

Author: Nicole Berline
Publisher: Springer Science & Business Media
ISBN: 9783540200628
Category : Mathematics
Languages : en
Pages : 384

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Book Description
In the first edition of this book, simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut) were presented, using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in an attractive paperback.

Author: Frances Clare Kirwan
Publisher: Cambridge University Press
ISBN: 9780521423533
Category : Mathematics
Languages : en
Pages : 278

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Book Description
This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.

Author: Emilio Lluis-Puebla
Publisher: Springer
ISBN: 3540466398
Category : Mathematics
Languages : en
Pages : 172

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Book Description
This book is a general introduction to Higher Algebraic K-groups of rings and algebraic varieties, which were first defined by Quillen at the beginning of the 70's. These K-groups happen to be useful in many different fields, including topology, algebraic geometry, algebra and number theory. The goal of this volume is to provide graduate students, teachers and researchers with basic definitions, concepts and results, and to give a sampling of current directions of research. Written by five specialists of different parts of the subject, each set of lectures reflects the particular perspective ofits author. As such, this volume can serve as a primer (if not as a technical basic textbook) for mathematicians from many different fields of interest.