Comparing the minimal Chern number and the cup-length of a symplectic manifold

Question

Let $(M,\omega)$ be a symplectic manifold. One can define its minimal Chern number $N_M$ as: $$ N_M := \text{inf} \lbrace k > 0 \ |\ \exists A \in H_2(M; \mathbb{Z}), \langle c_1, A \rangle = k \rbrace, $$ or alternatively, as the positive generator of $\langle c_1, H_2(M; \mathbb{Z}) \rangle$, where $c_1 \in H^2(M; \mathbb{Z})$ is the first Chern class of the symplectic manifold $(M, \omega)$, and $\langle ., . \rangle$ is the natural pairing between cohomology and homology groups.

On the other hand, one has the cuplength $cl(M)$ of $M$, defined as the minimal positive integer $k$ such that any cup-product $a_1 \cup ... \cup a_k$ of cohomology classes $a_j \in H^*(M; \mathbb{Z})$ of degree greater or equal to $1$ vanishes.

I have seen in several papers (for instance in the introduction of the paper "a fixed point theorem for toric manifolds", by A. Givental), that $$ N_M \leq cl(M), $$ with equality only if $M = \mathbb{C} P^n$ endowed with the Fubini-Study form, in which case both quantities equal $n+1$.

However, I have never seen a proof of this statement. Does someone have an idea of how to prove this fact?

Thank you in advance!

Answer 1

I had a look at the paper of Givental, https://math.berkeley.edu/~giventh/papers/tor.pdf and don't see this statement... If this statement were true, Conjecture 6.1 of Eliashberg from 2015 would be wrong.

Conjecture, Eliashberg. On any manifold $M$ of dimension $n = 2k > 4$ with a cohomology class $\eta \in H^2(M)$ with $\eta^k\ne 0$ and a non-degenerate 2-form $\omega_0$, there exists a symplectic form $ω$ homotopic to $\omega_0$ through non-degenerate forms, whose cohomology class $[\omega]$ can be deformed to $\eta$ keeping its $k$-th power non-vanishing.

Here is the paper of Eliashberg: https://www.ams.org/journals/bull/2015-52-01/S0273-0979-2014-01470-3/S0273-0979-2014-01470-3.pdf

So, why the inequality $N_M\le cl(M)$ contradicts the conjecture of Eliashberg? This is because on $\mathbb CP^3$ there exist almost complex structures with $c_1$ as large as you want. However $cl(\mathbb CP^3)=4$. Now, for $n>4$ take an almost complex structure $J$ on $\mathbb CP^3$ with $c_1(J)=n\in \mathbb Z=H^2(\mathbb CP^3,\mathbb Z)$, choose a (non-closed) two-form $\omega_0$ defining the same $J$. Applying then the conjecture of Eliashberg, we get $\omega$, contradicting the inequality $N_{\mathbb CP^3}\le 4$.

So, my guess is that the statement $N_M\le cl(M)$ was never proven for the class of all symplectic manifolds (otherwise Eliashberg would not make the conjecture)

However, something analogous is known for smooth complex projective varieties. Namely, if you have a smooth Fano variety $X$ with $c_1$ divisible by $\dim X+1$, then $X$ is $\mathbb CP^n$. And $c_1$ can not be divisible by $\dim X+k$ for $k\ge 2$.