ATMW The Grothendieck-Riemann-Roch Theorem (2015)

Venue: School of Mathematics & Statistics University of Hyderabad
Dates:  7th - 18th Dec , 2015

 

Convener(s) Speakers, Syllabus and Time table Applicants/Participants

 

School Convener(s)

Name  Dr. Archana S. Morye

 Dr. Anilatmaja Aryasomayajula

Dr. Tathagata Sengupta

Mailing Address  School of Mathematics & Statistics,
University of Hyderabad,
Hyderabad – 500 046, India.
 School of Mathematics & Statistics,
University of Hyderabad,
Hyderabad – 500 046, India.
School of Mathematics & Statistics,
University of Hyderabad,
Hyderabad – 500 046, India

 

Speakers and Syllabus 

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

Date 09:30–11:00 11:30–13:00 14:00–15:30 16:00-17:00
Dec. Lecture 1 Lecture 2 Lecture 3 Tutorial
07 Ananyo Dan Alok Maharana Alok Maharana Module 1
08 Ananyo Dan Alok Maharana Alok Maharana Revised schedule†
09 Ananyo Dan Alok Maharana Vijayalaxmi Trivedi Module 1
10 Ananyo Dan Vijayalaxmi Trivedi Vijayalaxmi Trivedi Module 3
11 Vijayalaxmi Trivedi Vijayalaxmi Trivedi Ananyo Dan Module 2
14 D. S. Nagaraj R. R. Simha R. R. Simha Module 4
15 D. S. Nagaraj R. R. Simha D. S. Nagaraj Revised schedule‡
16 D. S. Nagaraj R. R. Simha V. Srinivas Module 6
17 D. S. Nagaraj V. Srinivas V. Srinivas Module 4
18 V. Srinivas V. Srinivas Revised schedule§  
  •  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.

Actual Participants

 

Sr. SID Full Name Gender Affiliation Position in College/ University University/ Institute M.Sc./ M.A. Year of Passing M.Sc./ M.A Ph.D. Degree Date
1 4152 Mr. Nikhilesh Dasgupta Male Indian Statistical Institute PhD Indian Statistical Institute, Kolkata 2014  
2 4347 Mr. Anoop Singh Male Harish-Chandra Research Institute PhD Student Allahabad University 2013  
3 4349 Mr Pradeep Das Male Harish-Chandra Research Institute PhD Student Indian Institute of Technology, Kanpur 2011  
4 4358 Mr Sourav Sen Male Ramakrishna Mission Vidyamandira PhD Ramakrishna Mission Vivekananda University 2013  
5 4373 Dr. Safdar Quddus Male NISER Research Associate Washington University 2008 5/7/2013
6 4411 Mr Manikandan S Male Harish Chandra Research Institute PhD Student Bharathidasan University 2011  
7 4453 Ms Samarpita Ray Female Indian Institute of Science PhD Jadavpur University, Kolkata 2013  
8 4771 Mr. Arijit Mukherjee Male Hyderabad Central University PhD University of Calcutta 2014  
9 4784 Ms. Jyoti Dasgupta Female IIT Madras Ph.D IIT Bombay 2014  
10 4785 Mr. Bivas Khan Male IIT Madras Ph.D IIT Bombay 2014  
11 4794 Mr Samik Mitra Male University of Hyderabad MSc Student 2nd Year Chennai Mathematical Institute 2014  
12 4824 Miss Inder Kaur Female Freie Universitaet Berlin PhD Humboldt Universitaet Berlin 2012  
13 4845 Mr. Praveen Kumar Roy Male Chennai Mathematical Institute PhD University of Delhi 2014  
14 4873 Ms Mitra Koley Female Chennai Mathematical Institute PhD IIT Guwahati 2011  
15 4886 Mr. Navaneeth C C Male Chennai Mathematical Institute MSc Student Chennai Mathematical Institute Appeared / Awaiting Result  
16 4900 Mr. Rajib Sarkar Male Chennai Mathematical Institute PhD IIT Madras 2014  
17 4079 Mr. Sourav Das Male Chennai Mathematical Institute PHD IIT Guwahati 2012  
18 4934 Mr. Debayudh Das Male Chennai Mathematical Institute PhD IIT Madras 2013  
19   Anik Rai   TIFR, Mumbai Ph. D. student      
20   Chandranandan   TIFR, Mumbai Ph. D. student,      
21   Mr. Plawan Das            

 


 How to reach

1. By air:
  •  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). 
 
2.  By train:
  • From both Secunderabad and Nampally railway stations, it is best to take a local train, called MMTS service to Lingampally. The journey lasts approximately half an hour to 45 minutes. From Lingampally station, University of Hyderabad is 3-4 kms away, and one can take an auto, which costs around 50 rs, or take bus number 216, 217 (or 216 D/K ) to reach the University.