|INDIAN INSTITUTE OF SCIENCE EDUCATION & RESEARCH, MOHALI|
INSPIRE Faculty (Mathematics)
Knowledge city, Sector 81, SAS Nagar,
Manauli PO 140306
Web: My webpage
My interests are in the fields of Algebraic Geometry and Algebraic K-Theory. I focus on the study of algebraic cycles, triangulated categories of motives, algebraic cobordism and oriented cohomology theories.
In the eighties, Beilinson and Deligne independently described a conjectural abelian tensor category of mixed motives containing Grothendieck's category of pure motives as the full subcategory of semi-simple objects. Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many expected properties of the derived category of the conjectural abelian category of motives. The construction has been extended by Cisinski-Deglise to motives over a base-scheme. Levine constructed a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme S generated by the motives of smooth projective S-schemes. I have provided a tensor structure on the homotopy category mentioned above, when S is semi-local and essentially smooth over a field of characteristic zero. This is done by defining a pseudo-tensor structure on Levine's DG category of motives, which induces a tensor structure on its homotopy category. The tensor structure thus defined on the category of smooth motives matches the one defined by Levine working with rational coefficients. I also showed that with the same conditions on S, the fully faithful functor from the category of smooth motives on S to Cisinski-Deglise's category of motives over a base, is a tensor functor. As a corollary, we also get exact duality on the category of smooth motives.
The theory of algebraic cobordism as pioneered by Levine and Morel allows one to study algebraic cycles and motives from a more general perspective. It is an interesting question how various adequate equivalences for algebraic cycles can be lifted up to the level of algebraic cobordism cycles. Jointly with Park, we define and study the notion of numerical equivalence for algebraic cobordism cycles. For cobordism cycles, the naive notion of counting the number of intersection points of two varieties does not work and the ring of integers Z has to be replaced by the graded ring L (called the Lazard ring), which is the cobordism ring of a point. Using the product in cobordism rings with values in L, we define the notion of numerical equivalence. For a smooth projective variety X over a field k of characteristic zero, we show that algebraic cobordism modulo numerical equivalence is a finitely generated L-module and going modulo the coefficients of the Lazard ring, is isomorphic to the Chow group of X modulo numerical equivalence. We establish algebraic cobordism modulo numerical equivalence as an oriented cohomology theory.