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For an invertible operator $\beta$ on $V$, the mapis a linear isomorphism. Conjugate choices of $\beta$ induce conjugate operators ${\varphi }_{\beta }$.

For a complex submanifold $Y\subseteq X$, the holomorphic normal bundle is the quotient holomorphic vector bundle

Convexity is essential. In $\mathbb{R}$, letThis set is closed, symmetric, and has gaps arbitrarily near zero. Given $x\ne 0$, choose $k$ so small-scale that the interval
$[|x|/(2\cdot 4^{-k}),|x|/4^{-k}]$ has length at least one; it contains an integer $n$, and then $x/n\in S$. Thus ${\bigcup }_{n}nS=\mathbb{R}$, although $S$ is not a neighbourhood of zero.

For a rank-$e+1$ vector bundle $E\to X$ and $p:\mathbb{P}(E)\to X$, powers of $\zeta =c_1(\mathcal{O}(1))$ give isomorphisms

A cellular decomposition filters a scheme by closed subschemes whose successive differences are disjoint unions of affine spaces. The closures of the cells generate its Chow groups.

For a closed immersion $j:Z\hookrightarrow X$ with open complement $i:U\hookrightarrow X$, restriction and proper pushforward give an exact sequence

For a smooth variety, intersection of cycles gives the graded Chow ring $A^{\ast }(X)$.

For massless Dirac fermions, the classical axial current is $J_A^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$. The chiral anomaly makes its divergence proportional to $F_{\mu \nu }{}^{\star }{F}^{\mu \nu }$.

For an open cover $\mathcal{U}$ of $X$, singular chains generated by simplices lying inside one member of $\mathcal{U}$ form a subcomplex chain-homotopy equivalent to the full singular chain complex. Repeated barycentric subdivision makes chains small.

The Čech cochain complex places sections on $(p+1)$-fold intersections in degree $p$ and uses the alternating sum of restriction maps as its differential.

Forthe power ansatz $w=z^{\lambda }$ gives the indicial equation $\lambda (\lambda -1)+a\lambda +b=0$. Distinct roots give $C_1{z}^{{\lambda }_1}+C_2{z}^{{\lambda }_2}$; a repeated root $\lambda$ gives $z^{\lambda }(C_1+C_2\log z)$ on a chosen logarithm branch.

For a Cartan subalgebra $\mathfrak{t}$, a semisimple Lie algebra decomposes asThe nonzero functionals $\alpha$ are the roots.

The Bunch-Davies vacuum selects the mode that approaches the positive-frequency ground-state mode of angular frequency $ck$ in the far past $\tau \to -\infty$.

For a smooth hypersurface $X$ in a complex manifold $Y$,

On a smooth bounded domain, interior and boundary Schauder estimates combine into a $C^{2,\alpha }$ estimate up to the full boundary. The constant depends on the domain regularity, coefficient norms, and ellipticity constants.

A parabolic subgroup of a connected reductive group is a closed subgroup containing a Borel subgroup. Its quotient in the group is projective.

Fora reciprocal basis isThus ${\Lambda }^{\ast }=(2\pi /a)\{(h,k,l)\in {\mathbb{Z}}^3:h+k+l\text{ even}\}$ and its shortest nonzero vectors have length $2\sqrt{2}\pi /a$.

The strict transform of a subvariety $Y$ under a blowup $\sigma :\tilde{X}\to X$ is the closure of ${\sigma }^{-1}(Y\setminus Z)$, where $Z$ is the center. Its total transform also contains the exceptional divisor with the multiplicity with which $Y$ passes through $Z$.

The exceptional divisor of the blowup of a complex manifold at a point $p$ is the fiber over $p$. In complex dimension $n$ it is naturally the projective space $\mathbb{P}(T_{p}X)\cong {\mathbb{CP}}^{n-1}$ of tangent directions at the center.