Burt Totaro’s research while affiliated with University of California, Los Angeles and other places

Rost defined the Chow group of algebraic cycles with coefficients in a locally constant torsion etale sheaf. We generalize the definition to allow non-torsion coefficients. Chow groups with twisted coefficients are related to Serre's notion of "negligible cohomology" for finite groups. We generalize a computation by Merkurjev and Scavia of negligible cohomology, in terms of twisted Chow groups. We compute the Chow groups of the classifying space BG with coefficients in an arbitrary G-module, for several finite groups G (cyclic, quaternion, ${\bf Z}/2\times {\bf Z}/2$). There are connections with the theory of algebraic tori, notably the concept of coflasque resolutions. We compare twisted Chow groups with twisted motivic cohomology as defined by Heller-Voineagu-Ostvaer. Surprisingly, there is a surjection from twisted motivic cohomology to twisted Chow groups, but it is not always an isomorphism.

We show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some classification results on varieties admitting endomorphisms (for Fano threefolds of Picard number one and several other cases) to any characteristic. The classification results in characteristic zero are due to Amerik–Rovinsky–Van de Ven, Hwang–Mok, Paranjape–Srinivas, Beauville, and Shao–Zhong. Our method also bounds the degree of morphisms into a given variety. Finally, we relate endomorphisms to global F F -regularity.

We construct log canonical pairs (X, B) with B a nonzero reduced divisor and KX+B$K_X+B$ ample that have the smallest known volume. We conjecture that our examples have the smallest volume in each dimension. The conjecture is true in dimension 2, by Liu and Shokurov. The examples are weighted projective hypersurfaces that are not quasi-smooth. We also develop an example for a related extremal problem. Esser constructed a klt Calabi–Yau variety which conjecturally has the smallest mld in each dimension (for example, mld 1/13 in dimension 2 and 1/311 in dimension 3). However, the example was only worked out completely in dimensions at most 18. We now prove the desired properties of Esser’s example in all dimensions (in particular, determining its mld).

An important local vanishing theorem for the minimal model program is the fact that klt singularities in characteristic zero are Cohen-Macaulay. In contrast, even in the narrow setting of terminal singularities of dimension 3, we show that Cohen-Macaulayness can fail in characteristic p or mixed characteristic (0,p) for p equal to 2, 3, or 5. This is optimal, by work of Arvidsson-Bernasconi-Lacini. The examples are quotients of regular schemes by the cyclic group G of order p. In characteristic p or mixed characteristic, such quotients can exhibit a wide range of behavior. Our key technical tool is a sufficient condition for quotients by G to have only toric singularities.

Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that Hj(X,ΩXi⊗L)=0$H^j(X,\Omega ^i_X\otimes L)=0$ for j>0$j>0$, i≥0$i\ge 0$, and L ample. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano threefolds that satisfy Bott vanishing. There are many more than expected.

We construct log canonical pairs (X,B) with B a nonzero reduced divisor and $K_X+B$ ample that have the smallest known volume. We conjecture that our examples have the smallest volume in each dimension. The conjecture is true in dimension 2, by Liu and Shokurov. The examples are weighted projective hypersurfaces that are not quasi-smooth. We also develop an example for a related extremal problem. Esser constructed a klt Calabi-Yau variety which conjecturally has the smallest mld in each dimension (for example, mld 1/13 in dimension 2 and 1/311 in dimension 3). However, the example was only worked out completely in dimensions at most 18. We now prove the desired properties of Esser's example in all dimensions (in particular, determining its mld).

Kawakami and the author showed that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. That was a new way to analyze which varieties have nontrivial endomorphisms. In this paper, we extend that result to a logarithmic version of Bott vanishing for an endomorphism with a totally invariant divisor. We apply this to Fano 3-folds. Meng-Zhang-Zhong showed that the only smooth complex Fano 3-folds that admit an int-amplified endomorphism are the toric ones. Also, Achinger-Witaszek-Zdanowicz showed that the only smooth complex Fano 3-folds that are images of toric varieties are the toric ones. Using log Bott vanishing, we reprove both results and extend them to characteristic p, for morphisms of degree prime to p.

We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anti-canonical volume. We conjecture that our examples have the smallest volume in every dimension, and we give low-dimensional evidence for that. In order to improve on earlier examples, we are forced to consider weighted hypersurfaces that are not quasi-smooth. We show that our Fano varieties are exceptional by computing their global log canonical threshold (or $\alpha$-invariant) exactly; it is extremely large, roughly $2^{2^n}$ in dimension n. These examples give improved lower bounds in Birkar’s theorem on boundedness of complements for Fano varieties.