Haochen Cheng 程昊辰

I am currently a third-year PhD student at Northwestern University, Department of Mathematics. My advisor is Ananth Shankar.

I am organizing the graduate algebraic geometry/arithmetic geometry/number theory seminar in Northwestern: GAGA_NU.

Email: haochencheng2028@u.northwestern.edu

Here is my CV.

I work in arithmetic geometry and number theory. Explicitly, I am mainly working on abelian varieties and Shimura varieties in positive or mixed characteristics, with possible intuition coming from complex geometry. I am also interested in non-abelian Hodge theory, (integral) $p$-adic Hodge theory and Galois representation, etc.

I am currently focusing on the relation between the positivity of Hodge bundles and local systems.

Sacred Valley, Peru, 2026

Papers and Preprints:

1. Non-liftability of Families of Abelian Varieties with Small $\ell$-adic Local System. Submitted.
   Abstract

   We study families of abelian varieties over smooth proper curves with small $\ell$-adic local system over characteristic $p$. We show that such abelian schemes have a non-nef Hodge bundle and cannot be lifted to $W_2(k)$. We also establish an Arakelov-type inequality for families of principally polarized abelian varieties over smooth proper curves in characteristic $p$, assuming $W_2(k)$-liftability. This shows that there are only finitely many isomorphism classes of abelian schemes defined over a fixed smooth proper curve over finite field $k$ that can be lifted to $W_2(k)$.
2. Rigidity of modular morphisms in positive characteristic. In preparation.
   Abstract

   We study the infintiesimal and local rigidity for modular morphisms from smooth proper curves with absolutely irreducible ($\ell$-adic) local system to the moduli space of principally polarized abelian varieties over positive characteristic. We show such map is local rigid, and give counter-examples of infinitesimal rigid maps. As an application, we show the Torelli morphism in positive characterisctic is infinitesimally rigid.

Random writings:

1. A survey on constant abelian subschemes detected by the Hodge bundle in characteristics 0 and $p$: Maximal variation of abelian schemes .
2. An introduction to Higgs semistable bundles over curves: Semistable Higgs.
3. An introduction of restricted Lie algebras, with emphasis on finite flat group schemes: Restricted Lie algebra.
4. A proof of Chebotarev density theorem over function fields: Chebotarev.
5. An introduction to semistable reductions, semistable representations, etc (in progress):Semistable.