ATMW The Grothendieck-Riemann-Roch Theorem (2015)

Academic content
As mentioned earlier, the workshop comprised six modules. Each module consisted of five lectures
with each lecture being of 1.5 hrs duration. The details regarding the modules are as follows:
Module 1: Introduction to scheme theory.
Module 2: Introduction to sheaf theory, Riemann-Roch theorem for curves and surfaces.
Module 3: Sheaf cohomology.
Module 4: Chow groups and intersection theory.
Module 5: Chern classes, Todd classes and the Grothendieck-Riemann-Roch theorem
Module 6: The analytic aspects of Grothendieck-Riemann-Roch theorem.

 Instructors
Module 1: Dr. Ananyo Dan, Humboldt University, Germany.
Module 2: Dr. Alok Maharana, IISER, Mohali.
Module 3: Prof. Vijayalaxmi Trivedi, TIFR Mumbai.
Module 4: Prof. D. S. Nagaraj, IMSC, Chennai.
Module 5: Prof. V. Srinivas, TIFR Mumbai.
Module 6: Prof. R. R. Simha, TIFR, Mumbai (retired).

Tutors
Module 1, 3, 6:
Ananyo Dan (Humboldt University)
Anilatmaja Aryasomayajula (University
of Hyderabad).
Module 2, 4, 5:
Alok Maharana (IISER, Mohali)
Tathagata Sengupta (University of Hyderabad)

Time table

•  Prof. Indranil Biswas of TIFR, Mumbai delivered the UPE distinguished lecture at 16:00–17:00 on the 8th.
  The title of a talk was Isomonodromic deformations of logarithmic connections and stability.
• To accommodate the early departure of an instructor on the 16th, Prof. R. R. Simha lectured from 16:00–17:30
  on the 15th. The tutorial scheduled for 16:00–17:00 on the 15th was moved to 14:30–15:30 on the 18th.
• 14:30–15:30 Tutorial for Module 5.

Academic reports of the Resource Persons
1) Report of Dr. Ananyo Dan, Module 1
Lecture 1. Introduction to sheaf theory as mentioned in [1, Chapter 2, section 1], definitions of presheaves, sheaves, morphism of sheaves, kernel, image and cokernel of sheaves, subsheaf, inverse image sheaf and direct image sheaf. Introduction to scheme theory as in [1, Chapter 2 section 1], definition of locally ringed spaces, morphsims of locally ringed spaces, affine and projective schemes and Zariski topology.
Lecture 2. Theory of Hilbert polynomials, detailed proofs of [1, Chapter 1, Proposition 7.1, 7.3, 7.4 and Theorem 7.5], definition of Hilbert polynomial and multiplicities of graded modules.
Lecture 3. Study of Bezout’s theorem, detailed proof of [1, Chapter 1, Proposition 7.6, Theorem 7.7 and Corollary 7.8]. Introduction to the theory of divisors, proof of [1, Chapter 2, Lemma 6.1].
Lecture 4. Continuation of study of divisors, detailed proof of [1, Chapter 2, Proposition 6.2],definition of Cartier divisors, proof of [1, Chapter 2, Proposition 6.11], introduction to invertible sheaves and Picard groups, proof of [1, Proposition 6.13, Corollary 6.14 and Proposition 6.15].
Lecture 5. Definition of flat morphisms of schemes, proof of deformation invariance of Hilbert polynomial (see [1, Chapter 3, Theorem 9.9]).
Tutorial 1. Addressed student doubts and question on sheaf theory and scheme theory.
Tutorial 2. Cup-product map between cohomology groups.
Tutorial 3. Problems on Euler-Characteristic suggested by Prof. R.R. Simha.
Tutorial 4. Introduction to Koszul complex.
References
[1] Robin Hartshorne, Algebraic geometry, volume 52, Springer Science & Business Media,
1977.

2) Report of Dr. Alok Maharana, Module 2
A report on five lectures of one and half hours each, delivered during 7-9 December, 2015, in the workshop on Grothendieck-Riemann-Roch theorem held at Hyderabad university.The topic of my lectures was Riemann-Roch theorem for non-singular projective curves including a discussion of basics on sheaves, line bundles, divisors etc. The contents of the lectures was as follows:
Lecture 1. Various examples explaining sheafification, direct sum and tensor products of sheaves mainly using the structure sheaf of a curve and its various twists by O(1) with special emphasis on the projective line. Skyscraper sheaves.
Lecture 2. Lecture 2: Sheaves of modules, quasi-coherent and coherent sheaves and their basic properties. Ideal sheaves, locally free sheaves, vector bundles over schemes and the 1-1 correspondence between vector bundles of rank r and locally free sheaves of rank r on a scheme. Sheaf of differentials. Theorem that a scheme of finite type over an algebraically closed field is non-singular if and only if its sheaf of differentials is locally free sheaf of rank equal to dimension of the scheme. Cotangent and tangent bundle and the Euler sequence.
Lecture 3. Definition of canonical bundle, calculation of canonical bundle of the projective space and any smooth hypersurface of the projective space. Divisors on a smooth curve, linear equivalence, divisor class group and Picard group. Calculation of class group on some basic examples.
Lecture 4. Isomorphism of divisor class group and Picard group for a smooth projective curve.
Lecture 5. Finite dimensionality of global sections of a line bundle on a smooth projective curve.Riemann-Roch theorem for curves. Various corollaries of the Riemann-Roch theorem.
Tutorial. Relevant problems were discussed in the tutorial sessions.

3)Report of Prof. Vijayalaxmi Trivedi, Module 3
This is a report on my lectures in the workshop titled The Grothendieck-Riemann-Roch Theorem which was held at University of Hyderabad from 7 Dec. to 18 Dec. 2015.I lectured on the topic cohomology of coherent sheaves. This topic was covered in 5 lectures of one and half hour each, and followed by tutorials conducted by Ananyo Dan and Alok Maharana.I sent in advance typed notes on the relevant material based on the book Algebraic Geometry by R. Hartshorne.
The material covered here included cohomology of affine schemes, Cech cohomology and its applications to compute cohomologies of line bundles on Projective spaces. Using this we proved the finite generation of cohomology and Serre vanishing Theorem for coherent sheaves on a projective scheme over a Noetherian scheme. This led us to a well defined notion of the Euler Characteristic of coherent sheaves with respect to a very ample line bundle on a Projective scheme.In the last lecture of this series, we introduced the notion of higher direct image sheaf and along with the proof of other properties, we also proved the coherence theorem for higher direct image sheaf where the morphism is a projective morphism of Noetherian schemes.Overall, it was a well structured workshop with the content suited to the level of the audience.

4)Report of Prof. D. S. Nagaraj, Module 4
Cycles on an algebraic scheme over a field were defined. Rational equivalence on cycles were defined and chow group was defined and several examples were given. For a regular algebraic varieties intersection of cycles were defined and Chow moving Lemma was stated. Using these it was shown that the Chow group of a regular quasi-projective varieties has graded commutative ring structure known as Chow ring. Some examples of these rings are considered and few applications were mentioned.

5)Report of Prof. V. Srinivas, Module 5
The main aim of this module was to prove the Grothendieck Riemann-Roch theorem for quasi projective varieties defined over an algebraically closed field. In a series of five lectures, we introduced Segre classes, Chern classes, Chern character, and Todd classes for vector bundles defined over quasi projective varieties, and developed all the machinery to prove the Theorem. We proved all the relevant details, and used it to prove the Grothendieck Riemann-Roch theorem.

 6)Report of Prof. R. R. Simha, Module 6
Here is a list of the topics surveyed in my five talks at the GRR workshop: Review of the Riemann-Roch for compact Riemann surfaces. The Riemann-Roch Problem: express the arithmetic genus (= χ of the structure sheaf) of a compact complex manifold in terms of topological data. Example of the two-dimensional case. Chern numbers. The multiplicative property of the arithmetic genus and the uniqueness of the Todd polynomials. Axiomatic description of the Chern classes. The Splitting Principle. Kaehler manifolds and the Hodge Decomposition. The Hodge Index Theorem and Thom’s Theorem. Generalised Todd Index, and the main inductive step in the proof of Riemann-Roch for smooth complex projective varieties. The Riemann-Roch theorem for line bundles and vector bundles of higher rank.

 How to reach

•  Hire a "Radio cab" to reach the university directly, and it may cost around 800-1000 rs.  
•  Take a shuttle coach from the airport (which costs 200 rs) towards JNTU and get down at Gachibowli flyover. From Gachibowli flyover  University of Hyderabad is 3 kms away, and one can take an auto (which should cost around 50 rs) or take any city bus towards Lingampally (University is second stop from Gachibowli flyover).