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Question: Is there a closed symplectic manifold satisfying the following? $$|Td(M)| > \sum_{i \in \mathbb{Z}} b_{i}(M) $$ here $Td(X)$ is the Todd genus of $M$ i.e. the integral of the Todd class on the fundamental cycle of $M$.

This cannot happen for complex projective manifolds. By Hirzebruch-Riemann-Roch we have $Td(X) = \chi(\mathcal{O}_{X}) = \sum_{i}(-1)^{i}h^{i,0}$. So the inequality follows from the fact that $h^{i,0} \leq b_{i}(X)$ hence we get $$ |\chi(\mathcal{O}_{X})| \leq \sum_{i\in \mathbb{Z}}b_{i}(X).$$

I am interested in this inequality since it is used as the first step to prove some boundedness results for projective manifolds here https://arxiv.org/pdf/alg-geom/9607016.pdf (see proof of 4.2.3).

Many properties of projective manifolds have been found to fail for symplectic manifolds, so I am guessing there is a counterexample.

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    $\begingroup$ The same argument for complex projective manifolds works for complex manifolds where the Frölicher spectral sequence degenerates at the first page. $\endgroup$
    – Michael Albanese
    Commented Feb 23, 2019 at 17:09
  • $\begingroup$ Is it true that HRR holds for any compact complex manifold? $\endgroup$
    – YHBKJ
    Commented Feb 23, 2019 at 17:11
  • $\begingroup$ @YHBKJ I believe that was proved by O'Brien, Toledo, and Tong. $\endgroup$
    – Jason Starr
    Commented Feb 23, 2019 at 22:53
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    $\begingroup$ I apologize. I misspelled the name of O'Brian. Here is the MathSciNet header. MR0790113 (87h:32045); O'Brian, Nigel R.; Toledo, Domingo; Tong, Yue Lin L.; A Grothendieck-Riemann-Roch formula for maps of complex manifolds.; Math. Ann. 271 (1985), no. 4, 493–526. $\endgroup$
    – Jason Starr
    Commented Feb 23, 2019 at 22:57

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