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

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Publications (87)

Torsion algebraic cycles and complex cobordism
  • Article

October 1996

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17 Reads

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120 Citations

Journal of the American Mathematical Society

Burt Totaro

We show that the cycle map on a variety X, from algebraic cycles modulo algebraic equivalence to integer cohomology, lifts canonically to a topologically defined quotient of the complex cobordism ring of X. This more refined cycle map gives a topological proof that the Griffiths group is nonzero for some varieties X, without any use of Hodge theory. We also use this more refined cycle map to give examples of torsion algebraic cycles which map to 0 in Deligne cohomology but are not algebraically equivalent to 0, thus answering some questions by Colliot-Thelene and Schoen.

Configuration spaces of algebraic varieties

October 1996

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23 Reads

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213 Citations

Topology

Burt Totaro

this paper Determines the rational cohomology ring of the configuration space of n-tuples of distinct points in a smooth complex projective variety X. The answer depends only on the cohomology ring of X.

Topological invariants of singular spaces

8 Reads

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1 Citation

Burt Totaro

Modern geometry is primarily the geometry of manifolds, that is, spaces which look like Euclidean space locally but perhaps not globally. I will talk today about a step beyond manifolds: singular spaces. These are spaces which look like Euclidean space at most points, but not all. It is reasonable to study manifolds first, but singular spaces also come up naturally in examples. The zero set of a polynomial function may be a manifold, but in general it may have singularities. For instance, the zero set of the polynomial x 2 + y 2 − z 2 , pictured below, has a singularity at the origin. {(x, y, z) ∈ R 3 | x 2 + y 2 − z 2 = 0} In many applications, the singularities are the most important points. One may think of black holes in relativity, or shock waves in fluid mechanics. I will talk about singularities defined by analytic equations (for example, polynomial equations). One can ask about more general singularities, but these analytic singularities are the simplest step beyond manifolds.

Citations (62)

... (1) Almost homogeneous spaces (see [26], [23]); (2) Smooth hypersurfaces of a projective space (see [26], [7]; cf. [42]); (3) Fano threefolds (see [2], [21]; cf. [42]); (4) Fano manifolds containing a rational curve with trivial normal bundle (see [21]); (5) Fano fourfolds with Fano index ≥ 2 (see [41]; cf. ...

Reference:

Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number 1 (with an appendix by Jie Liu)
Endomorphisms of varieties and Bott vanishing
  • Citing Article

  • November 2024

Journal of Algebraic Geometry

Tatsuro Kawakami

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Burt Totaro

... Thanks to the first author, the third author, B. Totaro, and others, numerous examples of varieties with extreme invariants have been established in arbitrary dimensions. These extreme values often show doubly exponential growth or decay with respect to the dimension of the ambient variety [9,10,11,12,28,29,31]. This paper continues this series of studies by focusing on the minimal log discrepancy of exceptional Fano varieties. ...

Reference:

Exceptional Fano varieties with small minimal log discrepancy
Log canonical pairs with conjecturally minimal volume
  • Citing Article
  • Publisher preview available

  • August 2024

manuscripta mathematica

Louis Esser

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Burt Totaro

... In particular, the linear system | − 2 | is nonempty. In [42, §8], Totaro investigates Fano varieties with large bottom weight, which is the smallest positive integer m for which 0 ( , − ) ≠ 0. In particular, [42,Theorem 8.1] implies the existence of a Fano 4-fold that does not admit an m-complement for ≤ 1799233. This shows that the constant (4) obtained by Birkar in [3] is at least 1799233. ...

Reference:

Complements and coregularity of Fano varieties
Klt Varieties With Conjecturally Minimal Volume
  • Citing Article

  • March 2023

International Mathematics Research Notices

Burt Totaro

... Many important examples come from weighted complete intersections in weighted projective spaces, see for example [9,5,7,14,13]. Pizzato, Sano, and Tasin confirmed Conjecture 1.1 for weighted complete intersections which are Fano or Calabi-Yau or which are of codimension 1. ...

Reference:

Effective nonvanishing for weighted complete intersections of codimension two
Varieties of general type with doubly exponential asymptotics

Transactions of the American Mathematical Society Series B

Louis Esser

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Burt Totaro

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Chengxi Wang

... Many important examples come from weighted complete intersections in weighted projective spaces, see for example [9,5,7,14,13]. Pizzato, Sano, and Tasin confirmed Conjecture 1.1 for weighted complete intersections which are Fano or Calabi-Yau or which are of codimension 1. ...

Reference:

Effective nonvanishing for weighted complete intersections of codimension two
Klt Varieties of General Type with Small Volume
  • Citing Article

  • April 2022

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE

Burt Totaro

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Chengxi Wang

... In the following theorem, we obtain the fractional analogue of the floor and ceiling formulae (2) and (3). Esser, Tao, Totaro, and Wang [1] considered signed fractional function g(x) = x + 1 2 − x which takes values in − 1 2 , 1 2 , and proved that r j=0 g k 2 j + 2g ...

Reference:

Some refinements of formulae involving floor and ceiling functions
Optimal Sine and Sawtooth Inequalities

Journal of Fourier Analysis and Applications

Louis Esser

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Terence Tao

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Burt Totaro

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Chengxi Wang

... When the discretely valued field is henselian of characteristic > 0, Kato [Kat89, Page 110] and Izhboldin [Izh96] analyzed the wild quotient of +1, ( ) = +1 ( , Z/ ( )) = 1 ( , Ω ,log ) (Section 2.3), which is the quotient of this group by its tamely ramified part (defined below). Totaro [Tot22] generalizes the result to arbitrary discrete valuation fields. Kato defined an increasing filtration of +1, ( ) as follows: For ≥ 0, let be the subgroup of +1, ( ) generated by elements of the form 1 1 ∧ · · · ∧ with ∈ , 1 , . . . ...

Reference:

Brauer $p$-dimension and Kato's Swan Conductor
Cohomological Invariants in Positive Characteristic
  • Citing Article

  • January 2021

International Mathematics Research Notices

Burt Totaro

... The latter sits in the long exact exponential sequenc먨/ (i) The cohomology of the Hilbert scheme S rns of a K3 surface is torsion free, cf. [44,56]. In fact, H 3 pS rns , Zq " 0. (ii) The cohomology of the generalised Kummer variety K 2 pAq of dimension four is torsion free, cf. ...

Reference:

The period-index problem for hyperk\"ahler manifolds
The integral cohomology of the Hilbert scheme of points on a surface

Forum of Mathematics Sigma

Burt Totaro

... It is thus natural to ask whether the Fourier transform on rational Chow groups preserves integral cycles modulo torsion or, more generally, lifts to a homomorphism between integral Chow groups. This question was raised by Moonen and Polishchuk [MP10] and Totaro [Tot21]. More precisely, Moonen and Polishchuk gave a counterexample for abelian varieties over non-closed fields and asked about the case of algebraically closed fields. ...

Reference:

Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties
THE INTEGRAL HODGE CONJECTURE FOR 3-FOLDS OF KODAIRA DIMENSION ZERO
  • Citing Article

  • February 2020

Journal of the Institute of Mathematics of Jussieu

Burt Totaro