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Log canonical pairs with conjecturally minimal volume

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Louis Esser
Louis Esser
Louis Esser
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Burt Totaro
Burt Totaro
Burt Totaro
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Abstract

We construct log canonical pairs (X, B) with B a nonzero reduced divisor and KX+BKX+BK_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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manuscripta math. 175, 865–895 (2024)
© The Author(s), under exclusive licence to Springer-Verlag
GmbH Germany, part of Springer Nature 2024
Louis Esser ·Burt Totaro
Log canonical pairs with conjecturally minimal volume
Received: 23 June 2024 / Accepted: 29 July 2024 / Published online: 19 August 2024
Abstract. We construct log canonical pairs (X,B)with Ba nonzero reduced divisor and
KX+Bample 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).
Consider the problem of finding a complex projective log canonical pair (X,B)
with Ba nonzero reduced divisor and KX+Bample such that the volume of
KX+Bis as small as possible. This problem arises naturally in attempts to classify
stable varieties of general type [13, Remark 7.10]. We know that there is some
positive lower bound for the volume in each dimension, by Hacon–McKernan–Xu
[8, Theorem 1.6].
V. Alexeev and W. Liu constructed a log canonical pair (X,B)of dimension 2
with Ba nonzero reduced divisor and KX+Bample such that KX+Bhas volume
1/462 [1, Theorem 1.4]. J. Liu and V. Shokurov showed that this example is not
at all arbitrary: it has the smallest possible volume in dimension 2, under the given
conditions [13, Theorem 1.4]. (See also Kollár’s example in the broader class of
log canonical pairs with standard coefficients, Remark2.3.)
In this paper, we give a simpler description of Alexeev-Liu’s example: it
is a non-quasi-smooth hypersurface in a weighted projective space, X42 ⊂
P3(21,14,6,11), with Bthe curve {x3=0}∩ X(in coordinates [x0,x1,x2,x3]).
(This fits into a remarkable number of classification problems in algebraic geometry
for which the extreme case is known or conjectured to be a weighted hypersurface
[4,5].) We generalize that construction to produce a log canonical pair (X,B)of
any dimension with Ba nonzero reduced divisor such that KX+Bis ample and has
L. Esser: Department of Mathematics, Princeton University, Fine Hall, Washington Road,
Princeton, NJ 08544-1000, USA. e-mail: esserl@math.princeton.edu
B. Totaro (B
): UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-
1555, USA.e-mail: totaro@math.ucla.edu
Mathematics Subject Classification: 14J40 ·14B05 ·14J32
https://doi.org/10.1007/s00229-024-01588-6
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... For instance, we write down examples of klt Calabi-Yau varieties (rather than pairs) with the smallest known mld. Some of the properties of these examples are explicated in a separate paper [10]. Their mld decreases doubly exponentially with dimension, and we expect that they achieve the smallest mld of any klt Calabi-Yau variety in each dimension (Conjecture 4.4). ...
... The construction shares some similarities with the large index Calabi-Yau example due to Totaro, Wang and the author [12,Section 7]. The intricate identities relating the constants defining both of these examples are worked out fully in [10]. Therefore, we only define the examples below and sketch proofs of their properties. ...
... Theorem 4.3 [10,Theorem 5.1]. In each dimension ≥ 2, the hypersurface V defined above is wellformed, Calabi-Yau and quasismooth of degree D. The hypersurface V carries an action of the cyclic group such that / is a complex klt Calabi-Yau variety with mld 1/ . ...
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