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Let $\mathbb{K}$ an algebraically closed field of characteristic $0$, let $X$ be a smooth minimal surface of general type.

It is known that surfaces satisfy, among other thing, the (Bogomolov-Miayoka-Yau) inequality $3c_2(X)\geq c_1(X)^2$ [2, Proposition 7.1]. If $X$ saturates this inequality then $\Omega_X^1$ and $K_X$ are ample, and for any curve $C$ on $X$ one has the inequality $\displaystyle2g(C)-2\geq\frac{1}{3}K_X\cdot C$ [1, Corollary 4.3]. Moreover, if $\mathbb{K}=\mathbb{C}$ then these surfaces are all and only quotients of $\mathbb{B}_2\subset\mathbb{C}^2$ (the open complex ball) [3, Theorem 4].

  1. If $\mathbb{K}\neq\mathbb{C}$, under the hypothesis $3c_2(X)=c_1(X)^2$, are known example of such surfaces?
  2. If $\Omega_X^1$ and $K_X$ are ample, and for any curve $C$ on $X$ the inequality $\displaystyle2g(C)-2\geq\frac{1}{3}K_X\cdot C$ holds, is $3c_2(X)=c_1(X)^2$?

Bibliography

[1] U. Bruzzo, A. Capasso, B. Graña Otero - Positivity for Higgs vector bundles. arXiv:2307.16578 [math.AG]

[2] Y. Miyaoka - The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai (1985) 449--476. Adv. Stud. Pure Math. 10, North-Holland Publishing Co. (1987) Amsterdam.

[3] S.-T. Yau - Calabi's conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. 74 (1977) 1798--1799.

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    $\begingroup$ I do not understand your first question. Every such surface over an algebraically closed field $K/\mathbb{Q}$ is the base change of a surface defined over an algebraically closed subextension $\mathbb{Q}\subset L\subset K$ that also admits a field extension $\mathbb{Q}\subset L \subset \mathbb{C}$. $\endgroup$
    – Jason Starr
    Commented Aug 21, 2023 at 11:19
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    $\begingroup$ The inequality $2g(C)-2\geq \frac{1}{3} K\cdot C$ is very weak: it holds as soon as $C^2\geq 0$ and $C\cdot K\geq 0$. Just take a general complete intersection surface of sufficiently high degree. $\endgroup$
    – abx
    Commented Aug 21, 2023 at 11:53
  • $\begingroup$ @JasonStarr If I extend the field, I can use the base change and I understand your comment. But if I restrict the field: how can I do this? $\endgroup$
    – Armando j18eos
    Commented Aug 21, 2023 at 12:10
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    $\begingroup$ This is the "Lefschetz principle": define $L$ to be the algebraic closure of the smallest subextension of $K/\mathbb{Q}$ that contains all coefficients of the defining equations of your surface. This is the algebraic closure of a finitely generated extension of $\mathbb{Q}$, hence $L$ is isomorphic to a subfield of $\mathbb{C}$. $\endgroup$
    – Jason Starr
    Commented Aug 21, 2023 at 15:34

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