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I remember that some where, I saw an equality like $ c_2-c_1^2 \geq 0$ on surfaces ($c_1$ and $c_2$ are Chen classes), but I don't remember the exact form of inequality neither its name.

Can you help?

What I want is to know for which surfaces $ c_2-c_1^2 \geq 0$ holds?

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Yes. That is what I was looking for. – Mohammad F. Tehrani Jun 8 '11 at 17:57
What is the reference for later claim? – Mohammad F. Tehrani Jun 8 '11 at 18:01
Mohammad, In fact I realised that McQuillan proved a stronger result (I guess the result I have stated was proven by Bogomolov). You can read about this on pages 36-37 of a (very nice) survey of Claire Voisin : . She gives there the reference to McQuillan's articel – Dmitri Jun 8 '11 at 18:54
up vote 4 down vote accepted

The integer $c_1^2(S) - c_2(S)$ is the second Segre class of the surface $S$. For surfaces of general type a Riemann-Roch computation shows that its positivity implies that $$ \lim \sup \frac{\log h^0\big(S,\mathrm{Sym}^i(\Omega^1_S) \otimes \mathcal L\big) }{\log i} =3 $$ for any line-bundle $\mathcal L$.

This fact has been explored by Bogomolov back in the seventies, to prove the finiteness of rational and elliptic curves on surfaces of general type with positive second Segre class, see for instance this Bourbaki seminar. The point of Bogomolov's argument is that a symmetric $1$-form defines a multi-foliation (also called web) on $S$. If $i: \mathbb P^1 \to S$ is a non-trivial morphism and $\omega \in H^0(S,\mathrm{Sym}^i \Omega^1_S)$ then $$i^* \omega \in H^0(\mathbb P^1, \mathrm{Sym}^i \Omega^1_{\mathbb P^1}) = H^0(\mathbb P^1,\mathcal O_{\mathbb P^1}(-2i)) . $$ We deduce that $i^* \omega$ vanishes identically, i.e., the image of $i$ is a leaf of the multi-foliation defined by $\omega$. If there are infinitely many of them, a theorem by Jouanolou implies that we have a $1$-parameter family of rational curves on $S$, thus $S$ is uniruled and cannot be of general type. If we start with a section of $\mathrm{Sym}^i \Omega^1_S \otimes \mathcal L$ with $\mathcal L^*$ ample, the very same argument shows the finiteness of elliptic curves on $S$. A more involved argument, but following the same lines, shows the boundeness of curves of bounded genus.

More recently, McQuillan proved that surfaces of general type with positive sencond Segre class do not admit Zariski dense entire curves in Diophantine approximations and foliations. This work lead to a birational classification of foliations on projective surfaces (by McQuillan, Brunella, and Mendes) very much in the spirit of Enriques-Kodaira classification, see this paper and references therein.

Similar results are not known for surfaces satisfying the inequality you impose: $c_2 -c_1^2\ge 0$. Already the starting point, the existence of symmetric differential forms, is rather non-trivial. Very recently Demailly proved the existence higher order differential equations with coefficients in duals of ample line-bundles.

Concerning surfaces with non-positive second Segre class let me mention that any smooth surface in $\mathbb P^3$ of degree at least $5$ is of general type and satisfies the inequality $c_1^2(S) - c_2(S) \le 0$. As far as I know, even the finiteness of rational curves on a generic quintic surface is unknown.

If, by any chance, one knows that they are minimal then the graph below borrowed from this wikipedia page might tell something. Notice that the line containing the ruled surfaces is $2c_2 = c_1^2$.

Otherwise the situation is even less encouraging. After a blow-up $c_2$ increases by one while $c_1^2$ decreases by one. After sufficiently many blow-ups we always end up with a surface with negative second Segre class.

alt text

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It seems to me that something is missing after the formula - what do you claim about lim sup? – Dmitri Jun 9 '11 at 11:43
Thanks Dmitri. I have just corrected it and added some more details/pointers to the literature. – Jorge Vitório Pereira Jun 9 '11 at 12:37
Thank you very much for expending the answer, and providing the references!!! I deleted my comment (in fact now I see that I thought correctly about Bogomolov+McQuillan's result but for some reason in my head it swapped the sign :) ) It is truly amazing for me that the case of generic quintic in $P^3$ is not settled yet... – Dmitri Jun 9 '11 at 14:52
Motivation for my question: Suppose $X$ is a smooth Calabi-Yau 3-fold and $D\subset X$ a smooth nef (and therefore $K_D$ is nef ) surface in it. Then by adjunction formula and the fact that $c_2(X)\cdot D \geq 0$ we see that the inequality I mentioned above holds. So this put a restriction on the set of possible surfaces and I liked to know if we know such surfaces. – Mohammad F. Tehrani Jun 9 '11 at 17:29

The possible ratios $\frac{c_2}{c_1^2}$ are dense in the admissible interval [1/5,3]: this is proven by A. Sommese in: On the density of ratios of Chern numbers of algebraic surfaces. Math. Ann. 268 (1984), no. 2, 207–221.

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