3. Mukai conjecture and related problems
There are plenty of characterizations of complex projective spaces \mathbb{P}^n_\mathbb{C} in algebraic geometry. These characterizations are related to the positivity of the tangent bundle, the negativity of the canonical line bundle, its cone of curves, Seshadri constants, etc.In the 1988, Mukai asked whether it is possible to produce characterizations of products of projective spaces. Mukai proposed the following conjecture. Let X be a smooth Fano variety of dimension n, write -K_X\sim iL where L is an ample Cartier divisor and i is the largest integer for which the linear equivalence holds (i.e., the so-called index of the Fano variety). Then, we have that \rho(X)\leq \dim(X)/(i-1) and the equality holds if and only if X\simeq \mathbb{P}^{i-1}.
Mukai’s conjecture is known up to dimension 5 due to the work of Andreatta, Chierici, and Occhetta. For spherical varieties, Mukai’s conjecture is known due to the work of Gagliardi and Hofscheier. Reinece has proved Mukai’s conjecture for some Fano quiver moduli. Generalizations of Mukai’s conjecture for singular varieties are known for toric varieties due to the work of Fujita.
So far, the only characterization of products of projective spaces in the literature is due to Ochetta and it is related to the existence of many unsplit covering families of rational curves. Below, we propose some new characterizations of products of projective spaces and related problems.
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For a log Calabi–Yau pair (X,B), we know that the sum of the coefficients of B is bounded above by \dim(X)+\rho(X). Furthermore, if the upper bound is attained, then X is a toric variety. The previous result is due to Brown, McKernan, Svaldi, and Zong. The previous statement motivates the next question in which we replace the boundary divisor with a nef decomposition of -K_X. The following is known as "Mukai type conjecture".
Conjecture 3.1.
Let X be a klt Fano variety. Write -K_X\equiv \sum_{i=1}^k a_i M_i where each a_i \in \mathbb{R}_{>0} and each M_i is a nef Cartier divisor which is not numerically trivial. Then, we have \sum_{i=1}^k a_i\leq \dim(X)+\rho(X) and the equality holds if and only if X is a product of projective spaces. -
The following problem is a slight improvement of the previous one in which we give an absolute upper bound for the sum of the coefficients of the nef decomposition which is independent of the Picard rank. A positive answer would give a characterization of products of projective lines.
Problem 3.2.
Let X be a klt Fano variety of dimension n. Write -K_X\equiv \sum_{i=1}^k a_i M_i where each a_i\in \mathbb{R}_{>0} and each M_i is a nef Cartier divisor which is not numerically trivial. Then, we have \sum_{i=1}^k a_i\leq 2n. Furthermore, the equality holds if and only if X\simeq (\mathbb{P}^1)^n. -
The following problem is the most ambitious question in this direction and mixes the concept of complexity of log Calabi–Yau pairs with generalized pairs.
Problem 3.3.
Let (X,B,\mathbf{M}) be a generalized log Calabi–Yau pair. Let B=\sum_{i=1}^k a_i B_i be its decomposition into prime components. Let X' be the model where the b-nef divisor \mathbf{M} descends. Assume we can write \mathbf{M}_{X'}\equiv \sum_{i=1}^j b_i M_i where each b_i\geq 0 and each M_i is a nef Cartier divisor which is not numerically trivial. Then, we have \sum_{i=1}^k a_i + \sum_{i=1}^j b_i \leq \dim X + \rho(X).Furthermore, if the equality holds, then X is a toric variety. -
The study of Mukai’s conjecture and the Mukai type conjecture brings us to the topic of smooth Fano varieties all whose extremal contractions are fibratrions. The following is central to the previous problems.
Problem 3.4.
Let X be a smooth Fano variety all whose extremal contractions are fibrations. Prove that {\rm Nef}(X) is a simplicial cone.
Cite this as: AimPL: Higher-dimensional log Calabi-Yau pairs, available at http://aimpl.org/higherdimlogcy.