We prove a generalization of the algebraic version of Tian conjecture. Precisely, for any smooth strictly increasing function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$g:\mathbb {R}\rightarrow \mathbb {R}_{>0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>:</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">→</mo> <msub> <mi mathvariant="double-struck">R</mi> <mrow> <mo>></mo> <mn>0</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{log}\circ g$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>log</mtext> <mo>∘</mo> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation> convex, we define the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{H}^g$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">H</mi> <mi>g</mi> </msup> </math></EquationSource> </InlineEquation>-invariant on a Fano variety X generalizing the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">H</mi> </math></EquationSource> </InlineEquation>-invariant introduced by Tian–Zhang–Zhang–Zhu and show that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textbf{H}^g$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">H</mi> <mi>g</mi> </msup> </math></EquationSource> </InlineEquation> admits a unique minimizer. Such a minimizer will induce the g-optimal degeneration of the Fano variety X, whose limit space admits a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$g'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>g</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>-soliton. We present an example of Fano threefold which has the same g-optimal degenerations for any g.

Generalized Optimal Degenerations of Fano Varieties

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发表日期：

2026年1月8日

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Linsheng Wang

摘要

We prove a generalization of the algebraic version of Tian conjecture. Precisely, for any smooth strictly increasing function $g:\mathbb {R}\rightarrow \mathbb {R}_{>0}$ $g : R \to R_{> 0}$ with $\textrm{log}\circ g$ $log \circ g$ convex, we define the $\textbf{H}^g$ $H^{g}$ -invariant on a Fano variety X generalizing the $\textbf{H}$ $H$ -invariant introduced by Tian–Zhang–Zhang–Zhu and show that $\textbf{H}^g$ $H^{g}$ admits a unique minimizer. Such a minimizer will induce the g-optimal degeneration of the Fano variety X, whose limit space admits a $g'$ $g^{'}$ -soliton. We present an example of Fano threefold which has the same g-optimal degenerations for any g.