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2021•arXiv (Cornell University)Open access

On a numerical criterion for Fano fourfolds

Haidong Liu

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Abstract

In this paper, we prove a special case of Campana--Peternell's conjecture in dimension 4. Specifically, we show that a projective smooth fourfold $X$ with $c^2_1(X)\cdot c_2(X)\neq 0$ and strictly nef anti-canonical divisor $-K_X$ is a Fano fourfold. To this aim, we completely solve the non-vanishing conjecture for strictly nef anti-canonical divisors in dimension 4.

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What this paper is about

In this paper, we prove a special case of Campana--Peternell's conjecture in dimension 4. Specifically, we show that a projective smooth fourfold $X$ with $c^2_1(X)\cdot c_2(X)\neq 0$ and strictly nef anti-canonical divisor $-K_X$ is a Fano fourfold. To this aim, we completely solve the non-vanishing conjecture for strictly nef anti-canonical divisors in dimension 4.

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Available abstract

In this paper, we prove a special case of Campana--Peternell's conjecture in dimension 4. Specifically, we show that a projective smooth fourfold $X$ with $c^2_1(X)\cdot c_2(X)\neq 0$ and strictly nef anti-canonical divisor $-K_X$ is a Fano fourfold. To this aim, we completely solve the non-vanishing conjecture for strictly nef anti-canonical divisors in dimension 4.

Key concepts: Fano plane, Conjecture, Divisor (algebraic geometry), Dimension (graph theory), Mathematics, Pure mathematics, Combinatorics

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