Introduction to lie algebras and representation theory

Cohen‐macaulay modules over cohen‐macaulay rings

G-flux in F-theory and algebraic cycles

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Lectures On Vanishing Theorems

We show the existence of Bridgeland stability conditions on all Fano
threefolds, by proving a modified version of a conjecture by Bayer, Toda, and
the second author. The key technical ingredient is a strong Bogomolov
inequality, proved recently by Chunyi Li. Additionally, we prove the original
conjecture for some toric threefolds by using the toric Frobenius morphism.

Bridgeland Stability Conditions on Fano Threefolds

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Some elementary examples of non-liftable varieties

We generalize and simplify Langer's results concerning Frobenius direct images of line bundles on quadrics, describing explicitly the decompositions of higher Frobenius push-forwards of arithmetically Cohen–Macaulay (ACM) bundles into indecomposables, with an additional emphasis on the case of characteristic two. These results are applied to check which Frobenius push-forwards of the structure sheaf are tilting.

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Frobenius Push-Forwards on Quadrics

We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite etale cover, admit a Zariski-locally trivial fibration with toric fibers over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wiśniewski, which states that a smooth image of a projective toric variety is a toric variety. In order to deal with an important special case, we develop a logarithmic variant of the characterization of ordinary varieties with trivial tangent bundle due to Mehta and Srinivas. Furthermore, we verify our conjecture for surfaces, Fano threefolds, and homogeneous spaces (answering a question posed by Buch-Thomsen-Lauritzen-Mehta). Our proofs are based on a comprehensive theory of Frobenius liftings together with a variety of other techniques including deformation theory of rational curves and Frobenius splittings.

Liftability of the Frobenius morphism and images of toric varieties

A technical ingredient in Faltings' original approach to p-adic comparison theorems involves the construction of $K(\pi, 1)$-neighborhoods for a smooth scheme X over a mixed characteristic dvr with a perfect residue field: every point of X has an open neighborhood whose general fiber is a $K(\pi, 1)$ scheme (a notion analogous to having a contractible universal cover) We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in p-adic Hodge theory The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether Normalization Lemma

$K(\pi, 1)$-neighborhoods and comparison theorems

If $X$ is a smooth toric variety over an algebraically closed field of positive characteristic and $L$ is an invertible sheaf on $X$, it is known that $F_* L$, the push-forward of $L$ along the Frobenius morphism of $X$, is a direct sum of invertible sheaves. We show that this property characterizes smooth projective toric varieties.

Piotr Achinger

Papers

Some elementary examples of non-liftable varieties

Frobenius Push-Forwards on Quadrics

Liftability of the Frobenius morphism and images of toric varieties

$K(\pi, 1)$-neighborhoods and comparison theorems

A characterization of toric varieties in characteristic p