Gregory Sankaran: Research

Department of Mathematical Sciences, University of Bath

Picture of me, which you can't see for some reason

This page lists and describes my research activities. For undergraduate-level and other mathematics, see my teaching page; for general interests, my personal page; or for directions, my home page.

Mathematicians' personal web pages are expected to have a picture of them in Oberwolfach. This one was taken by Slava Nikulin.

  1. Papers
  2. Research students
  3. Other research activities


Most of my papers since 1992 can be found on the arXiv: all of them are at least cross-posted to math.AG. The list below is more comprehensive, though. It includes some papers too old for arXiv, some that are ineligible because they are purely surveys or in the nature of technical reports, and some that I chose not to publicise that way.
  1. Rationality and arithmetic of the moduli of abelian varieties [ps] [pdf]
    Daniel Loughran and Gregory Sankaran.
    Establishes the unirationality (for g=4,5) and stable rationality (for g=3) over the rational numbers of the moduli space of principally polarised abelian g-folds. This has entertaining consequences for arithmetic.
  2. Slopes of Siegel cusp forms and geometry of compactified Kuga varieties [ps] [pdf]
    F. Poon, R. Salvati Manni and G.K. Sankaran.
    Determines the Kodaira dimension of all the n-fold families of Kummer varieties over the moduli space of principally polarised abelian g-folds.
  3. Lazard-style CAD and Equational Constraints [ps] [pdf]
    James Davenport, Akshar Nair, Gregory Sankaran and Ali Uncu. ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation, July 2023, pp 218--226. doi: 10.1145/3597066.3597090
  4. The CAD Conundrum: Lex-Least vs Order [ps] [pdf]
    S. McCallum, A. Nair, J. Davenport and G. Sankaran. 2020 22nd International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), Timisoara, Romania, 2020, pp. 32-35. doi: 10.1109/SYNASC51798.2020.00017
  5. A supersingular coincidence [ps] [pdf]
    G.K. Sankaran. Ramanujan J. 59 (2022) 609--613. doi: 10.1007/s11139-021-00526-4
    Explains why the fifteen supersingular primes are the ones for which it is known that the moduli space of (1,p)-polarised abelian surfaces is of general type.
  6. Blowups with log canonical singularities [ps] [pdf]
    G.K. Sankaran and F. Santos. Geometry & Topology 25 (2021), 2145--2166.
    Proves a conjecture of Caucher Birkar, using toric geometry and lattice polytopes.
  7. Curtains in CAD: why are they a problem and how do we fix them? [ps] [pdf]
    A.S. Nair, J.H. Davenport and G.K. Sankaran.
    Fuller version of a decomposition algorithm for a hypersurface.
  8. Locally symmetric varieties and holomorphic symplectic manifolds [ps] [pdf]
    G.K. Sankaran.
    Notes from a survey talk in Kinosaki, October 2019.
  9. Saturated and primitive smooth compactifications of ball quotients [ps] [pdf]
    P.G. Beshkov, A.K. Kasparian and G.K. Sankaran. Ann. Sofia Univ. Fac. Math. and Inf. 106 (2019), 53--77.
    A progress report: contains some partial results, published at this stage mainly to comply with Bulgarian PhD regulations.
  10. Lazard’s CAD exploiting equality constraints [ps] [pdf]
    A. Nair, J.H. Davenport, G.K. Sankaran and S. McCallum. ACM Communications in Computer Algebra 53 No. 3 (2019), 138-141.
    Modifies a standard algorithm for decomposing real affine space so as to decompose a hypersurface instead.
  11. On benefits of equality constraints in lex-least invariant CAD (extended abstract) [ps] [pdf]
    A.S. Nair, J.H. Davenport and G.K. Sankaran. In: Proceedings SC2 2019.
    Provides some more detail of the decomposition algorithm.
  12. Toroidal compactification: the generalised ball case [ps] [pdf]
    A.K. Kasparian and G.K. Sankaran. In: Moduli Spaces and Locally Symmetric Spaces (Lizhen Ji and Shing-Tung Yau, Eds.), Surveys in Modern Mathematics 16, Ch. 3, pp. 107-133, Higher Education Press, Beijing 2021.
    Largely a survey: a worked example of toroidal compactification, with little claim to originality but perhaps useful.
  13. Fast Matrix Operations in Computer Algebra [ps] [pdf]
    Z. Tonks, J.H. Davenport and G.K. Sankaran. In: T. Jebelean, V. Negru, D. Petcu, D. Zaharie, T. Ida and S. Watt, Eds., Proceedings 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2017), Timisoara, Romania, 21-24 September 2017, pp. 67-70.
    A little elementary algebra applied to a small practical computing problem. Notable mainly for citing Dodgson (that is, Lewis Carroll) in a genuine mathematical context.
  14. Regular Cylindrical Algebraic Decomposition [ps] [pdf]
    J.H. Davenport, A.F. Locatelli and G.K. Sankaran. J. London Math. Soc. 101 No. 1 (2020), 43-59.
    Topology and real semi-algebraic geometry: a harmless problem coming from computer science meets the h-cobordism theorem.
  15. Weighted Castelnuovo-Mumford regularity and weighted global generation [ps] [pdf]
    F. Malaspina and G.K. Sankaran. J. Algebra Appl. 18 No. 2 (2019), 1950028.
    A failure report on an attempt to define a notion of regularity adapted to weighted projective spaces. It does work, but not very well.
  16. Genus 4 curves on abelian surfaces [ps] [pdf]
    P. Borówka and G.K. Sankaran. Proc. Amer. Math. Soc. 145 No. 12 (2017), 5023-5034.
    Surprisingly little is known about this sort of question. The hyperelliptic curves are particularly interesting.
  17. Fundamental groups of toroidal compactifications [ps] [pdf]
    A.K. Kasparian and G.K. Sankaran. Asian J. Math. 22 No. 5 (2018), 941-954.
    We tell you what the fundamental group of a toroidal compactification is, in terms of the arithmetic group involved.
  18. Moduli of K3 surfaces and irreducible symplectic manifolds [ps] [pdf]
    V. Gritsenko, K. Hulek and G.K. Sankaran. In: G. Farkas & I. Morrison (Eds.), Handbook of Moduli, Vol I, Adv. Lect. Math. (ALM) 24 Int. Press, Somerville, MA (2013), 459-526.
    A survey article, fairly thorough and including some new material, on moduli of polarised hyperkähler manifolds.
  19. On some lattice computations related to moduli problems [ps] [pdf]
    A. Peterson and G.K. Sankaran, with an appendix by V. Gritsenko. Rend. Sem. Mat. Univ. Pol. Torino 68 No. 3 (2010), 289-304.
    This fills a small gap in the paper on K3 surfaces: it settles the case d=52 which should have been mentioned there, but wasn't.
    There is a supplement containing the computer programs used in this paper.
  20. Moduli spaces of polarised symplectic O'Grady varieties and Borcherds products [ps] [pdf]
    V. Gritsenko, K. Hulek and G.K. Sankaran. J. Differential Geometry 88 No. 1 (2011), 61-85.
    Handles one of the known types of hyperkähler manifold (the OG10 case).
  21. Smooth rationally connected threefolds contain all smooth curves [ps] [pdf]
    G.K. Sankaran. In: W. Ebeling, K. Hulek & K. Smoczyk (Eds.), Complex and Differential Geometry, Springer Proceedings in Mathematics 8 (2011), 393-402.
    This "cool result" (Richard Thomas) answers a question of Eisenbud and Harris but does not appear to be of any actual use.
  22. Boundedness for surfaces in weighted projective 4-spaces [ps] [pdf]
    L.V. Rammea and G.K. Sankaran. Proc. Amer. Math. Soc. 139 (2011), 3393-3403.
    Any quasi-smooth surface in any weighted projective space of sufficiently high degree is of general type.
  23. Abelianisation of orthogonal groups and the fundamental group of modular varieties [ps] [pdf]
    V. Gritsenko, K. Hulek and G.K. Sankaran. J. Algebra 322 (2009), 463-478.
    Technicalities about the structure of arithmetic subgroups of O(2,n).
  24. Moduli spaces of irreducible symplectic manifolds [ps] [pdf]
    V. Gritsenko, K. Hulek and G.K. Sankaran. Compositio Math. 146 (2010) 404-434.
    Foundational paper on the moduli of polarised hyperkähler manifolds.
  25. The moduli space of étale double covers of genus 5 curves is unirational [ps] [pdf]
    E. Izadi, M. Lo Giudice and G.K. Sankaran. Pacific J. Math. 239 (2009), 39-52.
    Surprisingly this was not known.
  26. Numerical obstructions to abelian surfaces in toric Fano 4-folds [ps] [pdf]
    G.K. Sankaran. Kodai Math. J. 31 (2008), 1-20.
    This corrects the error in Abelian surfaces in toric 4-folds, below, and locates the cohomology classes in smooth toric Fano 4-folds that could be represented by an abelian surface.
  27. Hirzebruch-Mumford proportionality and locally symmetric varieties of orthogonal type [ps] [pdf]
    V. Gritsenko, K. Hulek and G.K. Sankaran. Documenta Math. 13 (2008), 1-19.
    Systematically exploits the asymptotic approach to estimating plurigenera of orthogonal locally symmetric varieties.
  28. Cusp forms: a clarification [ps] [pdf]
    G.K. Sankaran. Unpublished note, 2008.
    This fills a small gap, pointed out by R. Salvati Manni, in The Kodaira dimension of the moduli of K3 surfaces below, and is put here for reference. The matter is more fully addressed in later papers.
  29. The Kodaira dimension of the moduli of K3 surfaces [ps] [pdf]
    V. Gritsenko, K. Hulek and G.K. Sankaran. Invent. Math. 167 (2007), 519-567.
    The original publication is available at
    Almost all moduli spaces of polarised K3 surfaces are of general type.
  30. The Hirzebruch-Mumford volume for the orthogonal group and applications [ps] [pdf]
    V. Gritsenko, K. Hulek and G.K. Sankaran. Documenta Math. 12 (2007), 215-241.
    Number-theoretic technicalities associated with O(2,n)
  31. The nef cone of toroidal compactifications of A4 [ps] [pdf]
    K. Hulek and G.K. Sankaran. Proc. London Math. Soc. 88 (2004), 659-704.
    A technical look at the geometry of the 2nd Voronoi compactification of the moduli of principally polarised abelian 4-folds.
  32. Abelian surfaces with odd bilevel structure [ps] [pdf]
    G.K. Sankaran. In: M. Reid, A. Skorobogatov (Eds.), Number theory and algebraic geometry, London Math. Soc. Lecture Note Series, 303 (2003), 279-300.
    A study of a special moduli space of abelian surfaces introduced by Mukai.
  33. The moduli space of bilevel-6 abelian surfaces [ps] [pdf]
    G.K. Sankaran and J. Spandaw. Nagoya Math. J. 168 (2002), 113-125.
    A special case with some extra geometry.
  34. Das Titsgebäude von Siegelschen Modulgruppen vom Geschlecht 2 [ps] [pdf]
    M. Friedland and G.K. Sankaran. Abh. Math. Sem. Univ. Hamburg 71 (2001), 49-68.
    A tedious but at the time useful computation, written German because half of it was taken from Friedland's Diplomarbeit.
    There is a supplement with an English summary [ps] [pdf] and some pictures. The pictures are quite pretty.
  35. Algebraic construction of normalized coprime factors for delay systems [ps] [pdf]
    J.R. Partington and G.K. Sankaran. Math. Control Signals Systems 15 (2002), 1-12.
    I know nothing about this subject.
  36. The geometry of Siegel modular varieties [ps] [pdf]
    K. Hulek and G.K. Sankaran. In: S. Mori, Y. Miyaoka (Eds.), Higher Dimensional Birational Geometry, Advanced Studies in Pure Mathematics 35 (2002), 89-156.
    A survey article, with some new material: apparently still not out of date.
  37. Abelian surfaces in toric 4-folds [ps] [pdf]
    G.K. Sankaran. Math. Ann. 313 (1999), 409-427.
    An attempt to understand this problem, but the example in the last section is completely wrong because of a simple miscalculation. The idea is all right.
  38. Degenerations of (1,3) abelian surfaces and Kummer surfaces [ps] [pdf]
    K. Hulek, I. Nieto and G.K. Sankaran. In: P. Pragacz, M. Szurek, J. Wisniewski (Eds.), Algebraic Geometry: Hirzebruch 70, AMS Contemporary Mathematics 241 (1999), 177-192.
    Details of how polarised abelian surfaces degenerate.
  39. Heisenberg-invariant Kummer surfaces [ps] [pdf]
    K. Hulek, I. Nieto and G.K. Sankaran. Proc. Edin. Math. Soc. 43 (2000), 425-439.
    Some special geometry, eventually reaching back into early 20th century English projective geometry.
  40. Recent results on moduli of abelian surfaces [ps] [pdf]
    G.K. Sankaran.
    Notes from a survey talk in Kinosaki, November 1993.

Research students


I have two current research students:


Eleven people have so far completed a PhD under my supervision.

Other research students

Some other people should be mentioned here. Atika Ahmed started a PhD with me in 2011, but for health reasons she was not able to complete it. Timothy Logvinenko studied with Alastair King and graduated in 2004, but for a while I was officially also supervising him, for technical reasons. I don't claim any credit, but I want to acknowledge the link.

Other Research Activities

Here are some other mathematical things that I do or have done. They may give you some idea of what I can be persuaded to do.