smooth projective variety with arbitary large degree irreducible divisors Does every  projective complex two dimensinal manifold have divisors $D$ of arbitary large degree (integral of chern class of $[D]$  with a fixed Kahler form) which cannot be written as $D= dD_1$ where $D_1$  is  a divisor of lower
degree?
Example: Take $\mathbb{CP}^{2}$ and divisors which are zero sets of homogenous polynomials
 $(x_1)^{d} + (x_2)^{d} + (x_3)^{d}$ where $d$ varies up to infinity.
If so a reference
 A: I think the usual definition of the degree of a divisor class $D$ on a smooth complex projective variety $X$ is that ${\rm deg}(D)=\int_X c_1(D).\omega$, where $\omega := c_1(H)$ is the specific Kahler form corresponding to the hyperplane class $H := \mathcal O_X(1) = i^\star \mathcal O_{\mathbb {CP}^n}(1)$ induced by the your chosen embedding $i : X \hookrightarrow \mathbb{CP}^n$ of $X$ into projective space. (This way, the degree is guaranteed to be an integer.)
Notice that we can construct divisor classes $D$ of arbitrarily high degree by taking $D = nH$ for arbitrarily large $n \in \mathbb N$. Indeed, ${\rm deg}(nH) = n \int_X \omega . \omega = nd_X$, where $d_X := \int_X \omega . \omega > 0$ is the degree of $X$ with respect to its embedding in projective space, and so the quantity ${\rm deg}(nH) = nd_X$ can be made arbitrarily large by making $n$ arbitrarily large.
Of course, in my example, $D = nH$ is a multiple of $H$ as a divisor class up to linear equivalence. But judging from way you discussed the example of $\mathbb {CP}^2$ in your original post, I believe what you're really interested in knowing whether we can find an individual element $E$ in the linear system $|D|=|nH|$ that doesn't decompose into a sum $E = C_1 + \dots + C_k$ where $C_1, \dots  C_k$ are of lower degree. The fact that it's possible to find such an element $E$ in $|D|$ where this kind of decomposition doesn't happen is a consequence of Bertini's theorem (see part 1 of the statement, about general elements in the linear system being irreducible and reduced, and note that Bertini really is applicable in this case because $nH$ is very ample, ensuring that the complete linear system $|nH|$ has no fixed components).
What if we ask the question of whether we can find divisor classes $D$ of arbitrarily high degree such that $D$ is not linearly equivalent to a sum of divisor classes of lower degree (or, if you wish,  whether we can find $D$ of arbitrarily high degree such that $D$ is not linearly equivalent to a multiple of a single divisor class of lower degree)? Then $X=\mathbb{CP}^2$ provides a counter-example (to both questions), because the group of divisor classes on $\mathbb {CP}^2$ is generated by a single element, the hyperplane class $H$.
