Haochen Cheng 程昊辰

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

I am helping 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 subtle relation between the positivity of Hodge bundles and local systems in positive characteristic.

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. Positivity of Hodge bundles and rigidity of modular morphisms in positive characteristic. In progress, contact me for further information.
   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.
3. On the generic semi-positivity of canonically polarized varieties in positive characteristic. In progress, contact me for further information.
   Abstract

   We study the generic semi-positivity of canonically polarized varieties $X$ in positive characteristic. In particular, we give a counter-example of Rössler's conjecture on the positivity of $\Omega_X$ and propose an updated conjecture.

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.