KERALA SCHOOL OF MATHEMATICS

Parabolic vector bundles and Lie algebroid connections

Speaker: Prof. Anoop Singh, IIT BHU

Abstract: Given a holomorphic Lie algebroid on an m–pointed Riemann surface, we define parabolic Lie algebroid connections on any parabolic vector bundle equipped with parabolic structure over the marked points. An analogue of the Atiyah exact sequence for parabolic Lie algebroids is constructed. For any Lie algebroid whose underlying holomorphic vector bundle is stable, we give a complete characterization of all the parabolic vector bundles that admit a parabolic Lie algebroid connection. This is a joint work with David Alfaya, Indranil Biswas, and Pradip Kumar.

Click here for talk video.

Collision of singularities

Speaker: Dr. Ritwik Mukherjee, NISER Bhubaneswar

Abstract: In this talk, we will address the following fundamental question: if a curve has two singular points (of a certain type), what happens in the limit, when the two singularities collide? For example, what sort of a singularity do we get, when two nodal points collide (a nodal point locally looks like x2-y2 =0)? The answer is that we get a curve with a tacnodal singularity (a tacnodal point is locally given by y2-x4 =0). This is a believable statement if one looks at a picture of a parabola and a line and sees what happens in the limit when the two points of intersection coincide (the two points of intersection are the nodal points of the underlying cubic, and in the limit, we get a parabola tangent to the line which is a tacnodal point of the cubic).  Although the picture makes our assertion believable, it is far from being proof. We will explain how to convert this picture into a rigorous proof. Similarly, one can ask what happens when three nodal points collide. Looking at the picture of three straight lines colliding into three concurrent lines suggests that we should get a triple point. In this case, our picture has misled us, because there is a second thing that can happen; we can also get an A5 singularity (it is locally given by y2-x6 =0). We will explain how to justify rigorously that these are the only two phenomena that can occur. We will then go on to explain what happens when multiple nodal points collide. If time permits, we will discuss applications of these facts to the question of enumerative geometry of singular curves. This is a work in progress.

Click here for talk video.

On ℓ-away ACM bundles: towards a non-ACM version of Horrocks theorem

Speaker: Dr. Debojyoti Bhattacharya, TIFR Mumbai

Abstract: ACM bundles on a smooth, projective variety can be considered as vector bundles having the simplest possible cohomology (to be more precise, they are vector bundles without intermediate cohomology). The origin of these bundles dates back to the famous splitting theorems by Grothendieck and Horrocks for vector bundles over ℙ1 (and ℙn respectively). In this talk, we will first recall a brief history of ACM and Ulrich (a particular kind of ACM bundles) bundles, followed by their importance and some interesting questions related to them. As a natural continuation of this study in the non-ACM direction, we will then move on to discuss the ‘ℓ-away ACM’ bundles which were recently introduced by Gawron and Genc. After briefly mentioning the existing literature, we will discuss certain results on l-away ACM line bundles over nonsingular cubic surfaces and ‘ℓ-away ACM bundles on ℙ2, especially a view towards a non-ACM version of Horrocks theorem in this setting. This is based on the joint work with A.J. Parameswaran and Jagadish Pine.

Click here for talk video.

Bounded negativity conjecture

Speaker: Prof. Krishna Hanumanthu, CMI

Abstract: Let X be a nonsingular projective surface over an algebraically closed field k of characteristic zero. The Bounded Negativity Conjecture (BNC) says that there exists a positive integer B(X), depending only on X, such that C2 is at least -B(X) for every irreducible and reduced curve C on X. This statement is false if k has positive characteristic but open in general when the characteristic is zero. This conjecture has attracted a lot of attention in recent years and some weaker statements have been proved. We will discuss the research surrounding BNC and talk about some interesting new questions that arose out of this research.

Click here for talk video.

Brauer group of moduli space (moduli stack) of bundles over algebraic curve

Speaker: Arijit Dey, IIT Madras

Abstract: Brauer group of moduli spaces of stable vector bundles with fixed determinant over a smooth algebraic curve was first computed by Balaji-Biswas-Gabber-Nagaraj. Later this result got generalized by Biswas and his collaborators for various moduli spaces, which includes the moduli space (moduli stack) of stable G-bundles (Biswas-Holla). In this talk, we will talk about the general strategy of computing Brauer group of various moduli space (moduli stack) of stable parabolic G-bundles for various reductive groups G. This is a joint work with Sujoy Chakraborty and Indranil Biswas.

Click here for talk video.

On Endomorphism of the moduli space of Principle G-bundles.

Speaker: Prof. Sarbeswar Pal, IISER TVM

Abstract: Let C be a smooth projective curve of genus g3 and MG(C)ss be the moduli space of semistable principal G-bundle, where G is a simple, simply connected linear algebraic group over the complex numbers. In this article, we will show that any non constant endomorphism of MG(C)ss, which preserves the smooth locus, is an isomorphism.

Click here for talk video.

Harder-Narasimhan Stacks of Principal Bundles

Speaker: Prof. Sudarshan Gurjar, IIT Bombay

Let G be a connected split reductive group over a field k of characteristic zero. Let       X → S be a smooth projective morphism of Noetherian k-schemes, with geometrically connected fibers. We formulate a natural definition of a relative canonical reduction, under which families of principal G-bundles of a given Harder-Narasimhan type τ on the fibers of X/S form an Artin algebraic stack Bunτ X/S(G) over S, and as τ varies, these stacks define a stratification of the stack BunX/S(G) by locally-closed substacks. If char   k = p > 0, a weaker property holds.

This is based on joint work with Nitin Nitsure.

Click here for talk video.

Projective Bundle and Blow-up 

Speaker: Shivam Vats, IISER Tirupati

In this talk, we explore the restriction of the null-correlation bundle E on $ \mathbb{P}^3$  to a hyperplane, showing that the projective bundle $\mathbbP(E$) is isomorphic to the blow-up of a smooth quadric in $\mathbb{P}^4$ along a line.

We also demonstrate that for each degree d >1, there exist hypersurfaces of degree 'd' in $\mathbb{P}^4$ containing a line, whose blow-up along the line is isomorphic to the projective bundle over $\mathbb{P}^2$. Finally, we will see some analogous results in higher dimensions.

Click here for talk video.

A question on injective self-maps of algebraic varieties 

Speaker: Nilkantha Das, ISI Kolkata

A famous theorem of Ax says that an injective self map of an algebraic over an algebraically closed field of characteristic zero is necessarily an isomorphism. M. Miyanishi (2005) proposed a generalization of the above result. He conjectured that an endomorphism of an algebraic variety, defined over an algebraically closed field of characteristic zero, is an automorphism

if the endomorphism is injective outside a closed subset of codimension at least 2. We talk about recent progress of Miyanishi’s conjectural generalization of the theorem of Ax. This is based on a joint work with I. Biswas.

Click here for talk video.

Rank-level duality of conformal blocks 

Speaker:  Prof. Swarnava Mukhopadhyay , TIFR Mumbai

Rank-level duality connects the space of global sections of line bundles on moduli space of principal G-bundles on a curve for various different groups. Examples being the pair of groups (SL_m,SL_n), (Sp_{2m}, Sp_{2n}), (G_2, F_4) and many others. These dualities have their origins in conformal field theory, as dualities between WZW models of conformal blocks as well as in representation theory arising from the duality of the Grassmannians Gr(m,m+n) and Gr(n,m+n). 

In this talk, we survey known results, origins and some applications of these rank-level dualities. If time permits, we will discuss some results in the set-up of twisted theta functions or twisted conformal blocks. 

Click here for talk video.

Relationship between stratified and etale fundamental group 

Speaker:  Prof. Manish Kumar  , ISI Bengaluru

Gieseker observed that there is a relationship between etale and stratified fundamental group even over an algebraically closed field of positive characteristic. We will explore the work done in this direction by various people and also talk about a joint work with Indranil Biswas and AJ Parameswaran which says that if a finite generically smooth morphism induces isomorphism of etale fundamental groups then the induced map of the stratified fundamental groups is also an isomorphism. 

Click here for talk video.

Equivariant Complete Minimal Immersions with Complete Ends in Complex Hyperbolic Space and Their Associated Parabolic Higgs Bundles 

Speaker:  Dr. Pradip Kumar, Shiv Nadar University 

We discuss the correspondence between complete -equivariant conformal minimal immersions of punctured Riemann surfaces into the complex hyperbolic plane with finite energy and stable parabolic Higgs bundles. We prove that such an immersion is complete then  the monodromy around each puncture is parabolic, and it helps us to write the parabolic higgs bundle explicitly.    This is an ongoing work with Indranil Biswas and John Loftin.  

Click here for talk video.