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The generalised Mukai conjecture for spherical varieties

Giuliano Gagliardi, Johannes Hofscheier, Heath Pearson

TL;DR

The paper tackles the generalized Mukai conjecture for $\mathbb{Q}$-factorial spherical Fano varieties by introducing the $\\widetilde{\\wp}$ function and linking it to the absolute complexity $\\gamma(X)$ of log Calabi–Yau pairs. It proves $\\widetilde{\\wp}(X) \ge \gamma(X)$ and shows that $\\widetilde{\\wp}(X)<1$ forces $X$ to be toric, enabling a bound $$(\\iota_X-1)\\rho_X \le \dim X-\\widetilde{\\wp}(X) \le \dim X-\\gamma(X) \le \dim X.$$ Equality then characterizes toric products of projective spaces. An explicit example is analyzed, and the work discusses the Mukai-type conjecture, providing a spherical-variety perspective that recovers and extends prior results in toric, horospherical, and symmetric cases. The results create a unified, combinatorial-geometric framework connecting Mukai-type inequalities to toric degenerations and log Calabi–Yau complexities, with broader implications for understanding Fano varieties.

Abstract

We prove the generalised Mukai conjecture for $\mathbb{Q}$-factorial spherical Fano varieties. In this case, a stronger inequality holds featuring an extra term - the minimum absolute complexity of a log Calabi-Yau pair - which measures how close the Fano variety is to being toric.

The generalised Mukai conjecture for spherical varieties

TL;DR

The paper tackles the generalized Mukai conjecture for -factorial spherical Fano varieties by introducing the function and linking it to the absolute complexity of log Calabi–Yau pairs. It proves and shows that forces to be toric, enabling a bound Equality then characterizes toric products of projective spaces. An explicit example is analyzed, and the work discusses the Mukai-type conjecture, providing a spherical-variety perspective that recovers and extends prior results in toric, horospherical, and symmetric cases. The results create a unified, combinatorial-geometric framework connecting Mukai-type inequalities to toric degenerations and log Calabi–Yau complexities, with broader implications for understanding Fano varieties.

Abstract

We prove the generalised Mukai conjecture for -factorial spherical Fano varieties. In this case, a stronger inequality holds featuring an extra term - the minimum absolute complexity of a log Calabi-Yau pair - which measures how close the Fano variety is to being toric.

Paper Structure

This paper contains 10 sections, 9 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Mukai-type conjecture
  3. Defining the p-tilde function
  4. The generalised Mukai conjecture for spherical varieties
  5. Interpreting the p-bar function
  6. Constructing a log canonical pair
  7. Proof of the spherical generalised Mukai conjecture
  8. Some remarks on the Mukai conjecture
  9. An example
  10. A note on the Mukai-type conjecture

Key Result

Theorem 1.2

Let $X$ be a $\mathbb{Q}$-factorial spherical Fano variety, then with equality if and only if $X\cong{(\mathbb{P}^{\iota_X-1})}^{\rho_X}$.

Figures (2)

  • Figure 1: Reflexive polytope yielding a non-$\mathbb{Q}$-factorial toric Fano threefold.
  • Figure 2: The valuation cone and colors of $G/H$ in $\mathcal{N}_\mathbb{Q}$; the colored fan of $\mathbb{P}^5\hookleftarrow G/H$; and the polytope $Q^*$ with $\mathcal{T}$ shaded.

Theorems & Definitions (30)

  • Conjecture 1.1: BCDD, The generalised Mukai conjecture
  • Theorem 1.2: The spherical generalised Mukai conjecture
  • Theorem 1.3: GH15
  • Remark 1.4
  • Theorem 1.5: Geomcharoftoric
  • Theorem 1.6: $=$ \ref{['P function theorem']}
  • Definition 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • ...and 20 more