Boundedness of Complements for Log Calabi–Yau Threefolds

Chen, Guodu; Han, Jingjun; Xue, Qingyuan

doi:10.1007/s42543-022-00057-x

Cited by 5 publications

(6 citation statements)

References 37 publications

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“…This is considered to be much more difficult. See [10] for a similar result. Remark 6.3 (Explicit bound of threefold mlds).…”

Section: Further Remarksmentioning

confidence: 58%

Remark on complements on surfaces

Liu

2023

Forum of Mathematics, Sigma

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We give an explicit characterization on the singularities of exceptional pairs in any dimension. In particular, we show that any exceptional Fano surface is $\frac {1}{42}$ -lc. As corollaries, we show that any $\mathbb R$ -complementary surface X has an n-complement for some integer $n\leq 192\cdot 84^{128\cdot 42^5}\approx 10^{10^{10.5}}$ , and Tian’s alpha invariant for any surface is $\leq 3\sqrt {2}\cdot 84^{64\cdot 42^5}\approx 10^{10^{10.2}}$ . Although the latter two values are expected to be far from being optimal, they are the first explicit upper bounds of these two algebraic invariants for surfaces.

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“…This is considered to be much more difficult. See [10] for a similar result. Remark 6.3 (Explicit bound of threefold mlds).…”

Section: Further Remarksmentioning

confidence: 58%

Remark on complements on surfaces

Liu

2023

Forum of Mathematics, Sigma

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“…According to the thermodynamic principle, during the solidification process of molten steel, the precipitation of carbides was accompanied by a decrease in its solubility, so the precipitation condition is that the actual solubility product of the elements generating carbides was greater than the equilibrium solubility product. [ 30,31 ] Therefore, the formation of carbides was judged by the calculation of solubility product and reaction Gibbs free energy change. The reaction formula of metal element M and carbon element C forming carbide M x C y in molten steel is

$$x \left[\right.

…”

Section: Resultsmentioning

confidence: 99%

Carbide Precipitation Behavior of High‐Alloy Steel During Solidification and Cooling Process

He,

Li,

Liu

et al. 2024

steel research int.

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The precipitation, growth, and transformation of carbides rich in Cr and Mo during solidification and cooling process of high‐alloy steel for cold working dies are investigated. The Fe–C pseudobinary phase diagram of high‐alloy steel is calculated by Thermo‐Calc software to ascertain the critical phase transformation temperatures. The temperatures of phase transformation during the solidification process are determined by a differential scanning calorimeter. Then the microstructure of the samples at these characteristic temperatures is observed in situ by high‐temperature confocal microscope. The results show that the characteristic temperatures for phase transformation and carbide precipitation are 1380, 1315, 1140, and 980 °C, respectively. At 1315 °C, thick rod‐like or island‐like M7C3 carbides enriched in Cr and Fe are formed. At 1140 °C, secondary M7C3, adhering to the primary eutectic M7C3, grows rapidly in a rod‐like morphology and maintains a similar composition. The secondary M6C carbides of segregated Mo elements precipitate in fan‐shaped clusters near M7C3 through the dissolution of M7C3 and the conversion of M2C. Thermodynamic calculation further indicates that the eutectic Cr7C3 precipitates at 1213 °C, and secondary Cr7C3 remains under conditions favorable for precipitation below the solidus, preceding the precipitation of secondary carbides Cr23C6 and Mo2C.

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“…It is also compelling to consider if explicit boundedness of N -complements can be established for varieties with more general coefficients. In fact, even for klt R-complementary pairs with arbitrary coefficients, it has been amazingly shown that the boundedness of complements still holds for Fano varieties [Sho20] (even for generalized pairs [CHHX23]), surfaces [CHX23] (see also [Zen23]), and threefolds [CHX23]. These results are expected to play a vital role in future birational geometry research.…”

Section: Proof Of the Main Theoremsmentioning

confidence: 98%

“…These results are expected to play a vital role in future birational geometry research. In particular, from the standpoint of explicit birational geometry, the results in [CHX23] led to question if explicit boundedness of N -complements for klt Rcomplementary threefolds with arbitrary, or at least DCC boundary coefficients, can be achieved. However, this appears to be a more challenging question, even in dimensions 2 and 3, and alternative approaches are required.…”

Section: Proof Of the Main Theoremsmentioning

confidence: 99%

Anticanonical Volumes of Fano 4-Folds

Birkar

2023

Springer Proceedings in Mathematics &Amp; Statistics

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In this paper, we study the explicit geometry of threefolds, in particular, Fano varieties. We find an explicitly computable positive integer N , such that all but a bounded family of Fano threefolds have N -complements. This result has many applications on finding explicit bounds of algebraic invariants for threefolds. We provide explicit lower bounds for the first gap of the R-complementary thresholds for threefolds, the first gap of the global lc thresholds, the smallest minimal log discrepancy of exceptional threefolds, and the volume of log threefolds with reduced boundary and ample log canonical divisor. We also provide an explicit upper bound of the anti-canonical volume of exceptional threefolds. While the bounds in this paper may not and are not expected to be optimal, they are the first explicit bounds of these invariants in dimension three. Contents 1. Introduction 2. Preliminaries 2.1. Sets 2.2. Pairs and singularities 2.3. Complements 2.4. Stein degree 2.5. Generalized pairs 3. Explicit bounds of invariants and their general behavior 3.1. Some bounds of algebraic invariants 3.2. Global lc threshold and mld of exceptional pairs 3.3. Canonical bundle formulas 3.4. A lemma on Cartier index 4. Curves 4.1. Bound of the lc threshold 4.2. Bound of the global lc threshold and exceptional mlds 4.3. Bound of complements and indices 5. Explicit bounds on surfaces 5.1. Bound of the lc threshold 5.2. Bound of complements and indices: the standard coefficient case 5.3. Lower bound of volumes 6. Proof of Theorem 1.1 6.1. Reduction of the theorem 6.2. The plt case 6.3. The non-plt case 6.4. The general case 7. Proof of the main theorems 7.1.

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Boundedness of Complements for Log Calabi–Yau Threefolds

Cited by 5 publications

References 37 publications

Remark on complements on surfaces

Remark on complements on surfaces

Carbide Precipitation Behavior of High‐Alloy Steel During Solidification and Cooling Process

Anticanonical Volumes of Fano 4-Folds

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