Fano, Calabi–Yau, and general type are distinguished by the positivity behavior of the canonical divisor.
Fano, Calabi--Yau, and general type (Tag D-VARFANO)
Let \(X\) be a smooth complete variety with canonical divisor \(K_X\).
\(X\) is Fano if \(-K_X\) is ample.
\(X\) is Calabi–Yau if \(K_X \sim_\QQ 0\); the stricter convention demands \(\omega_X \cong \OO_X\) together with \(h^j(\OO_X) = 0\) for \(1 \leq j \leq \dim X - 1\).
\(X\) is of general type if \(K_X\) is big.
A del Pezzo surface is a Fano variety of dimension two.
A smooth projective curve of genus \(g\) is Fano if and only if \(g = 0\), Calabi–Yau if and only if \(g = 1\), and of general type if and only if \(g \geq 2\), since \(\deg K=2g-2\). A complete variety with an ample divisor is projective, so every Fano variety is projective.
The two conventions for Calabi–Yau are not equivalent: abelian varieties have \(K_X=0\) but do not satisfy the intermediate-cohomology vanishing, so they satisfy the first convention and not the second. The del Pezzo surfaces are \(\PP^1\times\PP^1\) and \(\PP^2\) blown up at \(r\le8\) general points, with \(K^2=9-r\); the cubic surface is the case \(r=6\) and contains \(27\) lines.
Hypersurfaces
Hypersurfaces in $\PP^n$, classified by adjunction (Tag FE-VARADJ)
\(K_{\PP^n} = -(n+1)H\), so \(-K_{\PP^n} = \OO(n+1)\) is very ample and \(\PP^n\) is Fano. For \(D \subseteq \PP^n\) a smooth hypersurface of degree \(d\), adjunction gives \[ K_D = \restrictionof{ (K_{\PP^n} + D) }{D} = \restrictionof{ \qty{ -(n+1)H + dH } }{D} = (d - n - 1) \restrictionof{H}{D} . \] So \(D\) is Fano for \(d < n+1\), Calabi–Yau for \(d = n+1\), and of general type for \(d > n+1\).
This computation supplies examples in either class on request, and a single adjunction line covers both. The quadric surface in \(\PP^3\) is \(d = 2 < 4\), hence del Pezzo, and indeed \(\PP^1 \times \PP^1\); the quartic surface in \(\PP^3\) is \(d = 4 = n+1\), the K3 surface; the quintic threefold in \(\PP^4\) is the Calabi–Yau of mirror symmetry. For \(n = 2\) the same computation is the plane-curve genus formula \(g = \binom{d-1}{2}\), with \(d \leq 2\) rational, \(d = 3\) elliptic, \(d \geq 4\) general type.
The vanishing \(h^j(\OO_D) = 0\) demanded by the strict Calabi–Yau convention is not part of the adjunction computation; it comes from Lefschetz, which says the middle-dimension cohomology is the only new one.
For a smooth hypersurface \(X\subset\PP^n\) of degree \(d\), \(n\ge2\), adjunction and \(K_{\PP^n}=-(n+1)H\) give \(K_X=(d-n-1)H|_X\). So \(X\) is Fano for \(d<n+1\), has \(K_X\sim0\) for \(d=n+1\), and has ample \(K_X\) for \(d>n+1\). In \(\PP^3\), smooth quadric and cubic surfaces are del Pezzo surfaces, and smooth quartic surfaces are K3 surfaces.
Where completeness enters
Fano varieties are complete by definition.
Complete varieties and projectivity (Tag D-VARCOMP)
A variety \(X\) over \(k\) is complete, equivalently proper, if the structure morphism \(X \to \Spec k\) is proper: separated, of finite type, and universally closed.
(Consequences of completeness)
Let \(k\) be algebraically closed and \(X\) a complete variety over \(k\). Then \(\OO_X(X) = k\), the image of \(X\) under any morphism of varieties is closed, and any morphism from \(X\) to an affine variety is constant. Every projective variety is complete; over \(\CC\), a variety is complete if and only if it is compact in the Euclidean topology.
\(\AA^1\) is not complete, although \(\AA^1\to\Spec k\) is a closed map: after base change, the projection \(\AA^1 \times \AA^1 \to \AA^1\) sends the closed hyperbola \(V(xy-1)\) to \(\AA^1 \sm \ts{0}\), which is not closed.
Complete varieties need not be projective: Hironaka constructed a nonsingular complete threefold that is not projective. Every complete curve and every nonsingular complete surface is projective, while there are complete normal surfaces that are not projective.
A complete variety with an ample line bundle is projective, so every Fano variety is projective.