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Klt Varieties With Conjecturally Minimal Volume
Abstract
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 -invariant) exactly; it is extremely large, roughly in dimension n. These examples give improved lower bounds in Birkar’s theorem on boundedness of complements for Fano varieties.
... 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. ...
... This shows that the constant (4) obtained by Birkar in [3] is at least 1799233. More generally, [42,Theorem 8.1] shows that ( ) grows at least doubly exponentially with d. In contrast to this, our statement shows that a Fano variety of absolute coregularity 0 either admits a 1-complement or a 2-complement. ...
... However, it is expected that we can take 2 = 66. By the work of Totaro [42], we know that grows at least doubly exponentially with d. ...
We study the relation between the coregularity, the index of log Calabi–Yau pairs and the complements of Fano varieties. We show that the index of a log Calabi–Yau pair (X,B) of coregularity 1 is at most , where is the Weil index of . This extends a recent result due to Filipazzi, Mauri and Moraga. We prove that a Fano variety of absolute coregularity 0 admits either a 1 -complement or a 2 -complement. In the case of Fano varieties of absolute coregularity 1 , we show that they admit an N -complement with N at most 6. Applying the previous results, we prove that a klt singularity of absolute coregularity 0 admits either a 1 -complement or 2 -complement. Furthermore, a klt singularity of absolute coregularity 1 admits an N -complement with N at most 6. This extends the classic classification of A,D,E -type klt surface singularities to arbitrary dimensions. Similar results are proved in the case of coregularity 2 . In the course of the proof, we prove a novel canonical bundle formula for pairs with bounded relative coregularity. In the case of coregularity at least 3 , we establish analogous statements under the assumption of the index conjecture and the boundedness of B-representations.
... (See also Kollár's example in the broader class of log canonical pairs with standard coefficients, Remark 2.3.) 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, X 42 ⊂ P 3 (21, 14,6,11), with B the curve {x 3 ...
... Totaro found that their surface is a non-quasi-smooth hypersurface in a weighted projective space, X 438 ⊂ P 3 (219, 146, 61, 11). Generalizing that construction, he produced a klt variety of each dimension with ample canonical class and conjecturally minimal volume [14,Theorem 2.1]. ...
... The numerology here is similar but not identical to that of the klt variety with ample canonical class and conjecturally minimal volume. In particular, the latter example involves the same weight a n+1 [14,Theorem 2.1]. For comparison, the volume of K X in that example is asymptotic to 2 2n+2 /s 4n n , which is much smaller than the volume of K X + B above. ...
We construct log canonical pairs (X, B) with B a nonzero reduced divisor and KX+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).
... Traditionally, the focus of explicit birational geometry has been on varieties of dimension at most 3 with at worst canonical singularities. Recent research, however, has expanded this focus to include higher-dimensional varieties, those with singularities worse than canonical, and new invariants such as Tian's α-invariant (also known as the global log canonical threshold in [28]), minimal log discrepancies, and the value N in the boundedness of N -complements. ...
... 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. ...
... • Proving that the varieties are exceptional: The difficulty here is that there is no straightforward theoretical formula to control the lower bound of the α-invariant, so the explicit structure of the variety needs to be used. A natural idea here is to use similar arguments as the weighted cone construction in [28,Section 4]. At the end of the day, this idea works, and the proof is completed in Theorems 6.5 and 6.6. ...
We construct exceptional Fano varieties with the smallest known minimal log discrepancies in all dimensions. These varieties are well-formed hypersurfaces in weighted projective space. Their minimal log discrepancies decay doubly exponentially with dimension, and achieve the optimal value in dimension 2.
... (See also Kollár's example in the broader class of log canonical pairs with standard coefficients, Remark 2.3.) In this paper, we give a simpler description of Alexeev-Liu's example: it is a nonquasi-smooth hypersurface in a weighted projective space, X 42 ⊂ P 3 (21, 14,6,11), with B the curve {x 3 = 0} ∩ X. (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 B a nonzero reduced divisor such that K X + B is ample and has extremely small volume. ...
... Totaro found that their surface is a non-quasi-smooth hypersurface in a weighted projective space, X 438 ⊂ P 3 (219, 146, 61, 11). Generalizing that construction, he produced a klt variety of each dimension with ample canonical class and conjecturally minimal volume [14,Theorem 2.1]. ...
... The numerology here is similar but not identical to that of the klt variety with ample canonical class and conjecturally minimal volume. In particular, the latter example involves the same weight a n+1 [14,Theorem 2.1]. For comparison, the volume of K X in that example is asymptotic to 2 2n+2 /s 4n n , which is much smaller than the volume of K X + B above. ...
We construct log canonical pairs (X,B) with B a nonzero reduced divisor and 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).
... However, by existing examples, the absolute minimum of K 2 is attained by a surface with vanishing geometric genus. So far, the smallest known volume is 1 48983 , found by Alexeev and the second author [AL19a], and later in [Tot23] using a different method. On the other hand, the current proved lower bound for K 2 is: ...
... It provides an embedding of X into the weighted projective space ProjR(X, K X ), called the canonical embedding in this paper. Drawing inspiration from the works of Totaro and his collaborators [ET23,ETW23,Tot23], we utilized the Riemann-Roch theorem for singular surfaces, as presented by Blache in [Bla95], to compute the plurigenera of V . From our calculations, we figure out that V is possibly a hypersurface of degree 86 in the weighted projective space P := P (6, 11, 25, 43). ...
We show that the minimal volume of surfaces of log general type, with non-empty non-klt locus on the ample model, is . Furthermore, the ample model V achieving the minimal volume is determined uniquely up to isomorphism. The canonical embedding presents V as a degree 86 hypersurface of . This motivates a one-parameter deformation of V to klt stable surfaces within the weighted projective space. Consequently, we identify a rational curve in the corresponding moduli space . As an important application, we deduce that the smallest accumulation point of the set of volumes for projective log canonical surfaces equals .
... The smallest positive volume for klt projective surfaces is expected to be 1 48983 which was first given by V. Alexeev and W. Liu [AL19a,Theorem 1.4]. It is shown very recently by B. Totaro that the surface is a degree 438 hypersurface in P(219, 146, 61, 11) that is not quasi-smooth [Tot23,Introduction]. However, we don't know whether this value is optimal or not. ...
... The anti-canonical volume of the exceptional Fano surface with mld = 3 35 (Case 11 of Table 8), 1 21385 , is the second smallest known volume of exceptional del Pezzo surfaces. It is smaller than the volume of any exceptional del Pezzo surface in [CPS10] but is larger than Totaro's latest example of a degree 354 non-quasi-smooth hypersurface in P(177, 118, 49, 11) which has anti-canonical volume 1 31801 [Tot23]. Remark 8.13. ...
We prove that the first gap of -complementary thresholds of surfaces is . More precisely, the largest -complementary threshold for surfaces that is strictly less than 1 is . This result has many applications in explicit birational geometry of surfaces and threefolds and allows us to find several other optimal bounds on surfaces. We show that the first gap of global log canonical threshold for surfaces is , answering a question of V. Alexeev and W. Liu. We show that the minimal volume of log surfaces with reduced boundary and ample log canonical divisor is , answering a question of J. Koll\'ar. We show that the smallest minimal log discrepancy (mld) of exceptional surfaces is . As a special case, we show that the smallest mld of klt Calabi-Yau surfaces is , reproving a recent result of L. Esser, B. Totaro, and C. Wang. After a more detailed classification, we classify all exceptional del Pezzo surfaces that are not -lt, and show that the smallest mld of exceptional del Pezzo surfaces is . We also get better upper bounds of n-complements and Tian's -invariants for surfaces. Finally, as an analogue of our main theorem in high dimensions, we propose a question associating the gaps of -complementary thresholds with the gaps of mld's and study some special cases of this question.
... In particular, the linear system |´2K X | is non-empty. In [39, §8], Totaro investigates Fano varieties with large bottom weight, which is the smallest positive integer m for which H 0 pX,´mK X q ‰ 0. In particular, [39,Theorem 8.1] implies the existence of a Fano 4-fold that does not admit an m-complement for m ď 1799233. This shows that the constant N p4q obtained by Birkar in [3] is at least 1799233. ...
... This shows that the constant N p4q obtained by Birkar in [3] is at least 1799233. More generally, [39,Theorem 8.1] shows that N pnq grows at least doubly exponentially with n. In contrast to this, our statement shows that a Fano variety of absolute coregularity 0 either admits a 1-complement or a 2-complement. ...
We study the relation between the coregularity, the index of log Calabi-Yau pairs, and the complements of Fano varieties. We show that the index of a log Calabi-Yau pair (X,B) of coregularity 1 is at most , where is the Weil index of . This extends a recent result due to Filipazzi, Mauri, and Moraga. We prove that a Fano variety of absolute coregularity 0 admits either a 1-complement or a 2-complement. In the case of Fano varieties of absolute coregularity 1, we show that they admit an N-complement with N at most 6. Applying the previous results, we prove that a klt singularity of absolute coregularity 0 admits either a 1-complement or 2-complement. Furthermore, a klt singularity of absolute coregularity 1 admits an N-complement with N at most 6. This extends the classic classification of A,D,E-type klt surface singularities to arbitrary dimensions. Similar results are proved in the case of coregularity 2. In the course of the proof, we prove a novel canonical bundle formula for pairs with bounded relative coregularity. In the case of coregularity at least 3, we establish analogous statements under the assumption of the index conjecture and the boundedness of B-representations.
... 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. ...
We show Kawamata's effective nonvanishing conjecture (also known as the Ambro--Kawamata nonvanishing conjecture) holds for quasismooth weighted complete intersections of codimension 2. Namely, for a quasismooth weighted complete intersection X of codimension 2 and an ample Cartier divisor H on X such that is ample, the linear system is nonempty.
... Moreover, the volume vol(X) equals the intersection number (−K X ) n . Varieties of general type, Calabi-Yau varieties and Fano varities with various singularities and very small volume are studied in [6,24,25]. Balletti, Kasprzyk, and Nill prove the weighted projective space P n (1, 1, 2(s n − 1)/s n−1 , . . . , 2(s n − 1)/s 1 ) has the largest volume 2(s n − 1) 2 among all n−dimensional canonical toric Fano varieties for n ≥ 4 [5, Corollary 1.3]. ...
For Fano varieties of various singularities such as canonical and terminal, we construct examples with large Fano index. By low-dimensional evidence, we conjecture that our examples have the largest Fano index for all dimensions.
Given positive integers and a subset , let denote the set of Iitaka volumes of ‐dimensional projective log canonical pairs such that the Iitaka–Kodaira dimension and the coefficients of come from . In this paper, we show that, if satisfies the descending chain condition, then so does for . In case and , and are shown to share more topological properties, such as closedness in and local finiteness of accumulation complexity. In higher dimensions, we show that the set of Iitaka volumes for ‐dimensional klt pairs with Iitaka dimension satisfies the DCC, partially confirming a conjecture of Zhan Li. We give a more detailed description of the sets of Iitaka volumes for the following classes of projective log canonical surfaces: (1) smooth properly elliptic surfaces, (2) projective log canonical surfaces with coefficients from or . In particular, the minima as well as the minimal accumulation points are found in these cases.
For Fano varieties of various singularities such as canonical and terminal, we construct examples with large Fano index growing doubly exponentially with dimension. By low-dimensional evidence, we conjecture that our examples have the largest Fano index for all dimensions.
The absolute regularity of a Fano variety, denoted by , is the largest dimension of the dual complex of a log Calabi–Yau structure on X. The absolute coregularity is defined to be
The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of X. In this note, we review the history of Fano varieties, give some examples, survey some theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.
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.
We establish the Noether inequality for projective 3-folds. More precisely, we prove that the inequality holds for all projective 3-folds X of general type with either or , where is the geometric genus and is the canonical volume. This inequality is optimal due to known examples found by M. Kobayashi in 1992.
We propose a problem of minimizing normalized volumes of valuations. For a
-Gorenstein klt singularity, by proving a key properness estimate,
we prove that the set of valuations with uniformly bounded normalized volumes
is compact. Under a continuity hypothesis, the normalized volume functional can
thus be minimized at a real valuation. We calculate several examples to show
that this is an interesting problem. We also point out its relation to Fujita's
work related to K-semistability.
We classify four-dimensional quasismooth weighted hypersurfaces with small
canonical class, and verify a conjecture of Johnson and Kollar on infinite
series of quasismooth hypersurfaces with anticanonical hyperplane section in
the case of fourfolds. By considering the quotient singularities that arise, we
classify those weighted hypersurfaces that are canonical, Calabi-Yau, and Fano
fourfolds. We also consider other classes of hypersurfaces, including Fano
hypersurfaces of index greater than 1 in dimensions 3 and 4.
We study degenerate complex Monge-Ampere equations of the form (omega+dd^c\varphi)^n = e^{t \varphi}mu where omega is a big semi-positive form on a compact Kaehler manifold X of dimension n , t in {R}^+ , and mu=fomega^n is a positive measure with density fin L^p(X,omega^n) , p>1 . We prove the existence and unicity of bounded omega -plurisubharmonic solutions. We also prove that the solution is continuous under a further technical condition. In case X is projective and omega=psi^*omega' , where psi:Xto V is a proper birational morphism to a normal projective variety, [omega']in NS_{{R}} (V) is an ample class and mu has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation. We use these results to construct singular Kaehler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.
We show that log canonical thresholds satisfy the ACC
We show that the number of birational automorphisms of a variety of general
type X is bounded by c \cdot \vol(X,K_X), where c is a constant which only
depends on the dimension of X.
We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in ℙ(a
0,a
1,a
2,a
3). As a corollary we obtain the existence of orbifold Kähler-Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-formed del Pezzo weighted hypersurfaces in ℙ(a
0,a
1,a
2,a
3).
This paper studies hypersurface exceptional singularities in defined by non-degenerate function. For each canonical hypersurface singularity, there exists a weighted homogeneous singularity such that the former is exceptional if and only if the latter is exceptional. So we study the weighted homogeneous case and prove that the number of weights of weighted homogeneous exceptional singularities are finite. Then we determine all exceptional singularities of the Brieskorn type of dimension 3.
One of the main achievements of algebraic geometry over the last 30 years is the work of Mori and others extending minimal models and the Enriques-Kodaira classification to 3-folds. This book, first published in 2000, is an integrated suite of papers centred around applications of Mori theory to birational geometry. Four of the papers (those by Pukhlikov, Fletcher, Corti, and the long joint paper Corti, Pukhlikov and Reid) work out in detail the theory of birational rigidity of Fano 3-folds; these contributions work for the first time with a representative class of Fano varieties, 3-fold hypersurfaces in weighted projective space, and include an attractive introductory treatment and a wealth of detailed computation of special cases.
We construct a surface with log terminal singularities and ample canonical class that has and a log canonical pair (X,B) with a nonempty reduced divisor B and ample that has . Both examples significantly improve known records.
We study log canonical thresholds (also called global log canonical threshold or -invariant) of -linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds, that is, given a natural number d and a positive real number , the set of Fano varieties of dimension d with -log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a question of Serre. Next we show that if the log canonical threshold of the anti-canonical system of a Fano variety is at most one, then it is computed by some divisor, answering a question of Tian in this case.
In this paper, we study the linear systems on Fano varieties X with klt singularities. In a given dimension d, we prove is non-empty and contains an element with "good singularities" for some number m depending only on d; if in addition X is -lc for some , then we show that we can choose m depending only on d and so that defines a birational map. Further, we prove Shokurov's conjecture on boundedness of complements, and show that certain classes of Fano varieties form bounded families.
We compute the global log canonical thresholds of quasi-smooth well-formed complete intersection log del Pezzo surfaces of amplitude 1 in weighted projective spaces. As a corollary we show the existence of orbifold Kähler—Einstein metrics on many of them.
These are the revised notes of my lectures at the 1995 Santa Cruz summer
institute. Changes from the Jan.26,1996 version: Major: 7.1--2; Medium:
3.7, 3.11, 5.3.3, 7.9.2, 9.8.
We determine the full list of anticanonically embedded quasismooth Fano hypersurfaces in weighted projective 4-spaces. There are 48 infinite series and 4442 sporadic examples. In particular, the Reid-Fletcher list of 95 types of anticanonically embedded quasismooth terminal Fano threefolds in weighted projective 4-spaces is complete. We also prove that many of these Fano hypersurfaces admit a Kähler-Einstein metric, and study the nonexistence of tigers on these Fano 3-folds. Finally, we prove that there are only finitely many families of quasismooth Calabi-Yau hypersurfaces in weighted projective spaces of any given dimension. This implies finiteness for various families of general type hypersurfaces.
This book gives a comprehensive treatment of the singularities that appear in the minimal model program and in the moduli problem for varieties. The study of these singularities and the development of Mori's program have been deeply intertwined. Early work on minimal models relied on detailed study of terminal and canonical singularities but many later results on log terminal singularities were obtained as consequences of the minimal model program. Recent work on the abundance conjecture and on moduli of varieties of general type relies on subtle properties of log canonical singularities and conversely, the sharpest theorems about these singularities use newly developed special cases of the abundance problem. This book untangles these interwoven threads, presenting a self-contained and complete theory of these singularities, including many previously unpublished results.
this paper is to give simpler proofs for several theoremscontained in articles [2], [3] of one of the authors, and at the sametime make these proofs effective. We give explicit formulas for the boundsin the theorems. It has to be admitted, however, that these bounds areunrealistically high.
We give a purely algebro-geometric proof that if the alpha-invariant of a
Q-Fano variety X is greater than dim X/(dim X+1), then (X,O(-K_X)) is K-stable.
The key of our proof is a relation among the Seshadri constants, the
alpha-invariant and K-stability. It also gives applications concerning the
automorphism group.
We verify a conjecture of Miyanishi about smooth quasi-projective surfaces over the complex numbers, namely, either such a surface has a pluricanonical log form, or a general point of the surface lies on a one dimensional image of the affine line. Applying standard techniques of the minimal model program, we reduce the problem to a detailed study of log del Pezzo surfaces. Log del Pezzo surfaces are normal projective surfaces, with ample anticanonical divisor and quotient singularities. We also give a partial classification of log del Pezzo surfaces of Picard number one.
These are the revised notes of my lectures at the 1995 Santa Cruz summer institute. Changes from the Jan.26,1996 version: Major: 7.1--2; Medium: 3.7, 3.11, 5.3.3, 7.9.2, 9.8.
We find the closest approximation to 1 from below using a sum of n Egyptian fractions.
We determine the complete list of anticanonically embedded quasi smooth log del Pezzo surfaces in weighted projective 3-spaces. We prove that many of these admit a K\"ahler-Einstein metric and most of them do not have tigers.
The Noether inequality for algebraic threefolds (with an appendix by J. Kollár)
- Chen