A complex surface is a complex manifold of complex dimension two, hence real dimension four. Compact complex surfaces need not admit a Kähler metric; a Hopf surface is an example. Every holomorphic one-form on a compact complex surface is closed, and the Betti and Hodge bounds for compact complex surfaces still apply without Kähler assumptions.

An analytic subvariety is a closed subset locally defined by finitely many holomorphic functions. It is irreducible if it is not the union of two proper closed analytic subsets. An irreducible analytic subvariety of complex codimension $k$ in an $n$-dimensional complex manifold has pure complex dimension $n-k$. Global irreducibility does not forbid multiple local branches at a singular point.

A proper holomorphic submersion has compact smooth fibres, and the Ehresmann fibration theorem gives differentiable local trivializations. Their integral cohomology, modulo torsion, forms a local system. If the fibres are Kähler manifolds, their Hodge filtrations form holomorphic subbundles and give a variation of Hodge structure. Properness is needed for this usual conclusion.

A map between complex manifolds is holomorphic when its local coordinate expression has holomorphic scalar components. For a smooth map, this is equivalent to its differential being complex-linear at every point. These maps compose by the chain rule. For one-dimensional manifolds this reduces to a holomorphic map between Riemann surfaces; the general definition also applies to multiplication and inversion on a complex Lie group.

This normalization of the first Kronecker limit formula follows from the Fourier expansion of the completed nonholomorphic Eisenstein series. At $s=1$, $K_{1/2}(t)=\sqrt{\pi /(2t)}{e}^{-t}$, so the nonconstant terms sum to $-2\log |{\prod }_{m\ge 1}(1-q^m)|$. The constant terms supply $\pi y/6-(\log y)/2+(\gamma -\log (4\pi ))/2$. Combining them gives the Dedekind eta function expression, with $\gamma$ the Euler--Mascheroni constant.

In a complete $n$-period binomial market with riskless gross return $R$, pricing density $Z$ and utility function $U(x)=\gamma x^{1/\gamma }$, assume $\gamma >1$ or $\gamma <0$ and put $\eta =\gamma /(\gamma -1)$. The optimal terminal payoff is $X^{\ast }=R^n{w}_0{Z}^{-\eta }/\mathbb{E}[Z^{1-\eta }]$. Its marginal utility is proportional to $Z$; the supporting-line inequality for the strictly concave utility function proves optimality using the pricing budget constraint.

This parameter restriction makes the utility increasing and strictly concave on positive wealth: $U\prime (x)=x^{1/\gamma -1}>0$ and $U\prime \prime (x)=(1/\gamma -1)x^{1/\gamma -2}<0$. For $0<\gamma <1$ the curvature reverses, and the supporting-tangent maximum argument no longer applies.

Let $X$ have nested compact sets $K_n$ covering it, with $K_n\subseteq \mathrm{int}{K}_{n+1}$. Then every point has a neighborhood in some $K_n$, and every compact subset is contained in one $K_n$. For continuous scalar functions, take positive weights with ${\sum }_n{w}_n<\infty$ and defineThe bounded metric transform makes the sum finite and gives its triangle inequality; the covering property gives definiteness. On $\mathbb{R}$, one may use $K_n=[-n,n]$ and $w_n=2^{-n}$.

If $\sigma (f_j,f)\to 0$, each fixed compact supremum norm tends to zero, since its summand is bounded by the whole metric. Conversely, control finitely many compact supremum norms and then the summable weight tail. This proves that $\sigma$ realizes locally uniform convergence under the stated exhaustion conditions.

Merely writing a space as a union of compact sets is insufficient. Enumerate the points of the compact space $\{0\}\cup \{1/n:n\ge 1\}$ and take finite initial subsets as $K_n$. The resulting metric detects pointwise control on those finite sets, not uniform control on the entire compact space. The neighborhood and compact-containment conditions explain the difference.

A parallel spinor has zero spin curvature. Contracting the spinorial Ricci identity with a gamma matrix gives $R_{\mu \nu }{\gamma }^{\nu }\psi =0$. For each fixed $\mu$, squaring Clifford multiplication by that Ricci row gives $|R_{\mu \cdot }{|}^2\psi =0$. Riemannian positivity and a nonzero spinor force every row to vanish. This implication is local and does not itself require compactness; the compact hypothesis is used to obtain parallelism from harmonicity.

For bounded linear operators, or formally in an associative algebra,Differentiate the two exponentials using the product rule: the result is $e^{\lambda A}[A,B]e^{-\lambda A}$. Induction gives the $n$th derivative $e^{\lambda A}{\mathrm{ad}}_A^n(B)e^{-\lambda A}$, and the Taylor series at zero proves the formula. For bounded operators convergence follows from $\Vert {\mathrm{ad}}_A^n(B)\Vert \le (2\Vert A\Vert {)}^n\Vert B\Vert$. For unbounded quantum operators, use a common invariant domain and justify the series there, or use the finite-dimensional system of commutators when their span closes.

For real-linear operators $[A,B]=I$ and a nonzero vector $y$ with $Ay=0$, induction gives $[A,B^i]=i{B}^{i-1}$, hence the displayed formula. Applying $A^n$ to a nontrivial finite dependence with highest term $c_n{B}^{n}y$ leaves $n!c_{n}y$, a contradiction. Thus $y,By,B^{2}y,\dots$ are linearly independent, and their span is invariant under both operators. Characteristic zero is essential to the factorial argument.

Over a field $k$, the trace-zero $2\times 2$ matrices form the special linear Lie algebra ${\mathfrak{sl}}_2(k)$. The matrix trace of a commutator is zero, so this is a Lie subalgebra of the general linear Lie algebra. A basis isIn fact $\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)=aH+bE+cF$, uniquely. The individual products areSubtracting in the stated order givesFor characteristic different from two, this is the standard sl2 triple; see Etingof, Section 11.4. The products remain valid in characteristic two, but there $H=I$ is central and $2E=2F=0$, so the familiar characteristic zero structure changes.

The following table gives all nine basis brackets. Each entry is $[\text{row},\text{column}]$; the diagonal vanishes by alternation, and exchanging row and column reverses the sign.

Bilinearity now computes any commutator, for exampleThe three structure constants of a Lie algebra relations encode all of this multiplication. They also show why the sl2 triple is useful in Lie algebra representations: the actions of $E$ and $F$ move between eigenvalues of the action of $H$ in steps of two.

Weak combinatory reduction closes the displayed contractions under application contexts. Weak equivalence is its reflexive, symmetric, transitive closure. The adjective distinguishes this syntactic reduction from an extensional rule equating functions that agree after an extra argument. A weak normal form has no subexpression matching a contraction rule; a partially applied combinator can be such a normal form.

For $s$ collocation nodes, the residual of the collocation polynomial vanishes at each node and contains the factor $\omega (\tau )={\prod }_i(\tau -c_i)$. Quadrature exactness through degree $q-1$ implies ${\int }_0^1\omega (\tau )p(\tau )d\tau =0$ for every polynomial of degree at most $q-s-1$. Expand the smooth vector-field interpolation error and the derivative of the exact flow in powers of the step. Every endpoint-error term of degree below $q$ has this vanishing weighted moment, so the endpoint local error is $O(h^{q+1})$, although the interior stage accuracy is generally only $s$. A step Lipschitz estimate and discrete Gronwall inequality then give global order $q$. Gaussian nodes attain $q=2s$, while the first nonexact quadrature moment bounds the order from above.

Collisionless free streaming is the ballistic motion of particles out of an overdensity. It erases perturbations below the comoving distance travelled by particles while their thermal velocities are appreciable.

A short exact sequence of cochain complexes of sheaves $0\to E\to F\to G\to 0$ induces $\cdots \to {\mathcal{H}}^i(E)\to {\mathcal{H}}^i(F)\to {\mathcal{H}}^i(G)\to {\mathcal{H}}^{i+1}(E)\to \cdots$. Taking stalks is exact, so the construction and exactness follow from the ordinary long exact sequence of cohomology of complexes at every stalk. The stalkwise connecting maps come from the lift-and-differentiate construction and form a sheaf morphism.

Modulo two the antipodal map acts trivially on fiber cohomology. The Wang sequence supplies a class $y$ restricting to the generator of $H^2(S^2;{\mathbb{F}}_2)$. The Leray-Hirsch theorem with classes $1,y$ gives a basis $1,t,y,ty$, so $ty\ne 0$. Pullback from the circle gives $t^2=0$, while dimension gives $y^2=0$. This cohomology ring agrees with that of $S^1\times S^2$, although the original manifold is nonorientable.

For positive unequal dimensions $p,q$, let $u,v$ be the pullbacks of the integral orientation classes of the factors. The Künneth theorem gives the free abelian basis $1,u,v,uv$, with degrees $0,p,q,p+q$. The products satisfy $u^2=v^2=0$ and $vu=(-1{)}^{pq}uv$, by naturality and graded commutativity of the cup product. In particular, for $p=2,q=4$, the cohomology ring is $\mathbb{Z}[u,v]/(u^2,v^2)$, with degrees two and four.

On ${\mathbb{P}}_k^1$, $\mathcal{O}$ and $\mathcal{O}(-1)$ are line bundles, so each stalk of either is a free rank-one module over the local ring. Nevertheless their spaces of global sections are $k$ and zero. Indeed a section of $\mathcal{O}(-1)$ on the two standard charts would require a polynomial $a(t)$ to equal $t^{-1}b(t^{-1})$, which has only strictly negative powers. Only the zero section can satisfy this. Pointwise existence of unrelated stalk isomorphisms does not supply a morphism of sheaves.

A sheaf of modules is globally generated when the evaluation map from its global sections is surjective. Equivalently, the germs of global sections span each stalk as a module over the local ring. For a coherent sheaf on a projective variety, every sufficiently positive twist is globally generated: embed the variety in projective space, present its direct-image sheaf as a quotient of a finite sum of $\mathcal{O}(-a)$, twist until those line bundles are globally generated, and restrict their generating sections to the variety.

For a coherent sheaf $\mathcal{F}$ on a projective variety over a field, every $H^q(X,\mathcal{F})$ is finite-dimensional, and Grothendieck vanishing gives zero for $q>\dim X$. A closed embedding in projective space and a finite twisting resolution of a coherent sheaf on projective space reduce finiteness to the explicit cohomology of twisting sheaves on projective space. The same resolution proves Serre vanishing after sufficiently positive twisting.