# Dingxin Zhang’s homepage

I am interested in algebra, algebraic topology, algebraic number theory and algebraic geometry.

## Some papers

- Degeneration of slopes (pdf)
- with Xi Chen and Xuanyu Pan) Automorphism and Cohomology II: Complete Intersections. 1511.07906
- A Lefschetz (1,1) type theorem for homology groups of nonsingular varieties (whoops, not available yet)

## Some notes

Here are some random notes of mine that might be useful for other people. Apart from a few exceptions they don’t contain original ideas.

- Dwork’s analytic continuation theorem revisited.
- The de Rham ohomology, modulo torsion, of the ring of formal power series with integral coefficients, is very, very large. Here is a note on this phenomenon.
- A brief introduction to B. Dwork’s conjecture (now Daqing Wan theorem). (pdf) — Do allow me to describe Dwork’s trick as a “臭名昭著” one.
- A trivial talk on BBD at the “Hodge seminar” (which is actually on Weil II) of Columbia. (Incidentally, I found an old incarnation of the §2 of the present note.)
- A note on de Rham cohomology of manifolds.
- A proof of the Poincaré lemma in differential geometry.

Essentially what this proof does is to utilize the Leray spectral sequence in the context of de Rham theory. - A proof of the fact the absoulte Galois group of ℂ((t)) is procyclic.

Of course this is rather easy if one knows some ramification theory of local fields. Without that, it’s still very easy :) - Negativity of the horizontal subbundle

The proof presented in this note bypasses the Lie-theoretic calculations appeared in the classical literature; instead I use the curvature formula for Hodge bundles (used classically by Griffiths in many places). - Thomason-Trobaugh via Neeman

The title is a bit misleading, this is really an exposition of some useful techniques exploited by A. Neeman in the study of triangulated categories (including: Brown representation, Bousfield localization, compact objects in Verdier quotients ––– the last being a result of the legendary paper of Thomason-Trobaugh). - An example of a smooth surface with nonreduced Picard

Basically one just builds an elliptic surface with a wild fiber (thanks to Liedtke); then uses some base change to cook up more wild fibers; then applies some simple yoga. - Baby trinity: Simpson’s theory for line bundles
- A proof of Kodaira vanishing theorem

Yet another (pretentious) proof of the Kodaira vanishing theorem. Instead of using a mysterious exact sequences of log differentials, I tend to use another exact sequence (a variant of the Schur complex if you wish to call it such) plus some simple spectral sequences.

## Other stuff

Séminaire Cocktail de S.-C. Wong at ZJU • ??? • D-modules • Frobenii • Some papers • Graduate students • Mental disorder • Happy me • Professor J. M. Starr (My Ph.D advisor) • Hodge theory seminar with Professor C. Schnell (2016) • TeX Frequently Asked Questions • Defect of Monotype’s Baskerville

## Coordinates

Dingxin Zhang

Email me

208 Goldsmith Building

Brandeis University

Waltham, MA 02453