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I suggest you: Eisenbud, Harris - The Geometry of Schemes, (2000) Springer Verlag, paragraph III.2.7; Eisenbud, Harris - 3264 & All That, Intersection Theory in Algebraic Geometry, chapters 3 and 4 …

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- …

I am working on the complex numbers field $\mathbb{C}$, for simplicity. However you can relax this assumption if you want. Let $X$ be an elliptic surface, id est there is a proper morphism $\pi\colon …

$\DeclareMathOperator\BMY{BMY}$Let $(X,H)$ be a smooth, complex, projective, polarized variety of dimension $n\geq3$, whose canonical bundle $K_X$ is big and nef. Is it know whether the inequality $\ …

Since $X$ admits a Kähler-Einstein metric [Theorem 2,3] and [Theorem 1,2], if $K_X$ is ample the inequality holds with $H=K_X$, by [Theorem IV.4.16,1]. [1] S. Kobayashi (1987) Differential Geometry o …

Let $X$ be a complex minimal surface of general type, id est $K_X$ is big and nef. It is well-known that $\displaystyle\int_X3c_2(X)-c_1(X)^2\geq0$, and the equality holds if and only if $X$ is unifor …

Let $X$ be a(n even dimensional) smooth complex projective variety in $\mathbb{P}^N$, and let $X^{\vee}$ be its dual variety; up to an higher degree Veronese embedding of $X$, I assume that $X^{\vee}$ …

Let $X$ be a smooth complex projective variety of dimension $n\geq2$, let $\mathfrak{E}=(E,\varphi)$ be a Higgs bundle over $X$ of rank $r\geq2$. Does exists a Higgs quotient sheaf $\mathcal{Q}$ of $ …

In M. Aprodu, G. Farkas - Koszul Cohomology and Applications to Moduli, arXiv:0811.3117 [math.AG], proof of Theorem(s) 4.5 (and 4.12), the authors constructed a semistable nodal curve $C^{\prime}$ of …

Lemma. Let $E$ be a rank $r$ nef vector bundle over a polarized smooth complex projective variety $(X,H)$ of dimension $n\leq r$. Then for any $t\in\mathbb{R}_{\geq0}$: $$ \sum_{k=0}^nt^{n-k}\int_Xc_k …

In [1,2] the authors proved the Hard Lefschetz theorem in intersection cohomology: Let $Z$ be a complex projective variety of pure complex dimension $d$, with $\xi\in H^2(Z,\mathbb{Q})$ the first Che …

I'm studying M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772. The premises are the following …

Let $f\in\ker(\exp)$; by definition: \begin{equation} \forall x\in X,\,\exp_x(f_x)=1\in\mathcal{O}_{X,x}^{\times}, \end{equation} in other words: \begin{equation} \exp_x(f_x)=\exp_x(Re(f_x))[\cos(Im(f …

An open subset $U$ of a projective surface $Z$ is big if $\mathrm{codim}_Z(Z\setminus U)\geq2$. Let $X$ and $Y$ be smooth complex projective surface. If there exists a birational map $f:X\dashrightarr …

Premise Let $X$ be a projective variety of dimension $n\geq1$ over an algebraically closed field of characteristic $0$. A Higgs sheaf $\mathfrak{E}$ is a pair $(E,\varphi)$ where $E$ is a $\mathcal …