Kapil Hari Paranjape

PIC

Subject Area: Algebraic Geometry; Differential Geometry; Topology

Background and Research Interests

My interests are centred on Geometry, more specifically, on Algebraic Geometry and the study of Algebraic Cycles. Such studies have led me further afield to areas such as Homological Algebra, Algebraic Topology, Differential Geometry and Number Theory. In a different direction, problems of writing computer programs to carry out mathematical tasks as well as questions of a more theoretical nature about computation have also grabbed my interest from time to time.

The primary questions on algebraic cycles centre around the Hodge Conjecture, Grothendieck‚Äôs Standard Conjectures and their expansion in the work of Bloch and Beilinson. In my thesis I proved a case of the Hodge conjecture for a correspondence between K3 surfaces and abelian varieties. Developing some ideas of M. V. Nori, I was able to make concrete questions about hypersurfaces out of the general conjectures of Bloch and Beilinson; moreover, I could prove these conjectures for varieties of degree much lower than the dimension. In joint work with M. Green and P. Griffiths, we were able to show how the arithmetical aspects of the conjectures of Bloch and Beilinson are sharp.

In collaboration with D. Ramakrishnan, there is an attempt to go beyond the conjectures regarding the structure of Galois representations that arise in the cohomology of algebraic varieties. We have conjectured that certain representations naturally arise in a special class of Calabi-Yau varieties and provided some evidence for this conjecture.

Select Publications

  1. Cohomological and Cycle-theoretic Connectivity, Annals of Math. 140 (1994) pp 641-660.
  2. (with D. Ramakrishnan), Calabi-Yau varieties associated with Modular Forms, Proceedings of the Conference on Arithmetic, Algebra and Geometry, University of Hyderabad, 2003, ed. R. Tandon, Hindustan Book Agency, 2005.
  3. (with I. Biswas), The Hodge Conjecture for a general Prym variety, Journal of Algebraic Geometry 11 (2002) pp 33-39.
  4. (with M. Green and P. A. Griffiths), Cycles over fields of transcendence degree one, Michigan Mathematical Journal 52 (2004).
  5. A Geometric characterization of Arithmetic Varieties, Proc. Indian Acad. Sci. (Math) 112 (2002) pp 1-9.