arXiv subjects

Burt Totaro

Publications and source records attributed to Burt Totaro.

At least 19 recordsLinked to original sources

arXiv2025

Endomorphisms of singular del Pezzo surfaces

A natural problem of algebraic dynamics is to classify the complex projective varieties that admit an endomorphism of degree greater than 1. Joshi solved the problem for all canonical del Pezzo surfaces with Picard number 1 except one, a surface with a du Val singularity of type $E_8$. The method of Bott vanishing does not resolve this case. We show here that the $E_8$ surface has no endomorphism of degree greater than 1. For the proof, we extend the method of Amerik-Rovinsky-Van de Ven, involving Chern number inequalities, from varieties to Deligne-Mumford stacks. This approach should be useful for other hard cases in the classification of varieties with endomorphisms.

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arXiv2025

Chow groups with twisted coefficients

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.

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arXiv2024

Terminal 3-folds that are not Cohen-Macaulay

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.

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arXiv2024

Endomorphisms of varieties and Bott vanishing

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 $F$-regularity.

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arXiv2023

Divided powers in the Witt ring of symmetric bilinear forms

The Witt ring of symmetric bilinear forms over a field has divided power operations. On the other hand, it follows from Garibaldi-Merkurjev-Serre's work on cohomological invariants that all operations on the Witt ring are essentially linear combinations of exterior powers. We find the explicit formula for the divided powers as a linear combination of exterior powers. The coefficients involve the ``tangent numbers'', related to Bernoulli numbers. The divided powers on the Witt ring give another construction of the divided powers on Milnor K-theory modulo 2.

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arXiv2023

Log canonical pairs with conjecturally minimal volume

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).

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arXiv2023

Endomorphisms of Fano 3-folds and log Bott vanishing

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.

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arXiv2023

Bott vanishing for Fano 3-folds

Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that $H^j(X,Ω^i_X\otimes L)=0$ for every $j>0$, $i\geq 0$, and $L$ ample. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano 3-folds that satisfy Bott vanishing. There are many more than expected. Along the way, we conjecture that for every projective birational morphism $π\colon X\to Y$ of smooth varieties, and every line bundle $A$ on $X$ that is ample over $Y$, the higher direct image sheaf $R^jπ_*(Ω^i_X\otimes A)$ is zero for every $j>0$ and $i\geq 0$.

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arXiv2022

Klt varieties of conjecturally minimal volume

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 anticanonical 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 $α$-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.

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arXiv2022

Calabi-Yau varieties of large index

Call a projective variety $X$ Calabi-Yau if its canonical divisor is ${\bf Q}$-linearly equivalent to zero. The smallest positive integer $m$ with $mK_X$ linearly equivalent to zero is called the index of $X$. We construct Calabi-Yau varieties with the largest known index in high dimensions. In our examples, the index grows doubly exponentially with dimension. We conjecture that our examples have the largest possible index, with supporting evidence in low dimensions. The examples are obtained by mirror symmetry from our Calabi-Yau varieties with an ample Weil divisor of small volume. We also give examples for several related problems, including Calabi-Yau varieties with large orbifold Betti numbers or small minimal log discrepancy.

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arXiv2022

Varieties of general type with doubly exponential asymptotics

We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior.

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arXiv2021

Klt varieties of general type with small volume

By Hacon-McKernan-Xu, there is a positive lower bound in each dimension for the volume of all klt varieties with ample canonical class. We show that these bounds must go to zero extremely fast as the dimension increases, by constructing a klt $n$-fold with ample canonical class whose volume is less than $1/2^{2^n}$. These examples should be close to optimal. We also construct a klt Fano variety of each dimension $n$ such that $H^0(X,-mK_X)=0$ for all $1\leq m < b$ with $b$ roughly $2^{2^n}$. Here again there is some bound in each dimension, by Birkar's theorem on boundedness of complements, and we are showing that the bound must increase extremely fast with the dimension.

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arXiv2021

Optimal sine and sawtooth inequalities

We determine the optimal inequality of the form $\sum_{k=1}^m a_k\sin kx\leq 1$, in the sense that $\sum_{k=1}^m a_k$ is maximal. We also solve exactly the analogous problem for the sawtooth (or signed fractional part) function. Equivalently, we solve exactly an optimization problem about equidistribution on the unit circle.

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arXiv2021

The Hilbert scheme of infinite affine space and algebraic K-theory

We study the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ from an $\mathbb{A}^1$-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ is $\mathbb{A}^1$-equivalent to the Grassmannian of $(d-1)$-planes in $\mathbb{A}^\infty$. We then describe the $\mathbb{A}^1$-homotopy type of $\mathrm{Hilb}_d(\mathbb{A}^n)$ in a range, for $n$ large compared to $d$. For example, we compute the integral cohomology of $\mathrm{Hilb}_d(\mathbb{A}^n)(\mathbb{C})$ in a range. We also deduce that the forgetful map $\mathrm{FFlat}\to\mathrm{Vect}$ from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an $\mathbb{A}^1$-equivalence after group completion. This implies that the moduli stack $\mathrm{FFlat}$, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum $\mathrm{kgl}$ representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the $\mathrm{kgl}$-homology of smooth proper schemes over a perfect field.

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arXiv2020

Torus actions, Morse homology, and the Hilbert scheme of points on affine space

We formulate a conjecture on actions of the multiplicative group in motivic homotopy theory. In short, if the multiplicative group G_m acts on a quasi-projective scheme U such that U is attracted as t approaches 0 in G_m to a closed subset Y in U, then the inclusion from Y to U should be an A^1-homotopy equivalence. We prove several partial results. In particular, over the complex numbers, the inclusion is a homotopy equivalence on complex points. The proofs use an analog of Morse theory for singular varieties. Application: the Hilbert scheme of points on affine n-space is homotopy equivalent to the subspace consisting of schemes supported at the origin.

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arXiv2020

Cohomological invariants in positive characteristic

We determine the mod p cohomological invariants for several affine group schemes G in chararacteristic p. These are invariants of G-torsors with values in etale motivic cohomology, or equivalently in Kato's version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod p etale motivic cohomology of fields, extending Vial's computations of the operations on the mod p Milnor K-theory of fields.

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arXiv2019

The integral cohomology of the Hilbert scheme of points on a surface

We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme X^[n] has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas's reduced obstruction theory for nested Hilbert schemes of surfaces.

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arXiv2019

The integral Hodge conjecture for 3-folds of Kodaira dimension zero

We prove the integral Hodge conjecture for all 3-folds X of Kodaira dimension zero with H^0(X, K_X) not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic.

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