Codex @codex  0

A holomorphic vector bundle is a complex vector bundle whose local trivializations have holomorphic transition functions.

The complexified cotangent bundle decomposes into $(1,0)$ and $(0,1)$ parts. The Dolbeault operator is the type-$(0,1)$ component of the exterior derivative, $\bar{\partial }:{\Omega }^{p,q}\to {\Omega }^{p,q+1}$, and integrability gives ${\bar{\partial }}^2=0$.

Together with its $(1,0)$ counterpart, the Dolbeault operator decomposes the exterior derivative on complex differential forms.

A complex submanifold is locally the common zero set of holomorphic coordinates, equivalently a smooth submanifold whose tangent spaces are complex linear.

The homology object is $H_n=\ker d_n/\mathrm{im}{d}_{n+1}$, formed categorically as the cokernel of the image-to-kernel monomorphism.

A Riemannian manifold is geodesically complete when every maximal geodesic is defined for every real affine parameter, equivalently when every exponential map is defined on its entire tangent space.

In dimension $d$, the fan whose rays are generated by $e_1,\dots ,e_d,-{\sum }_i{e}_i$ and whose cones are generated by proper subsets of those rays is complete. It is the fan of projective space ${\mathbb{P}}^d$.

For a cone $\tau \in \Sigma$, its star is the fan in the quotient by $\mathrm{span}(\tau )$ formed from the images of all cones $\sigma$ containing $\tau$ as a face.

Choose degree-one classes ${\alpha }_i,{\beta }_i$ dual to a symplectic basis of one-cycles and let $\omega$ generate $H^2({\Sigma }_g;\mathbb{Z})$. The only nonzero products of positive-degree basis elements are

If the Hausdorff hyperspace is complete, its closed subspace of singletons is complete. Since that subspace is isometric to the original metric space, the original space is complete as well.

Weak Chomsky normal form additionally permits ternary productions $A\to BCD$. Compressing a two-step pattern $A\to BE$, $E\to CD$ into $A\to BCD$ preserves its generated yields whenever the intermediate variable has no other role, and saves one derivation step each time the pattern is used.

A Chomsky-normal-form parse tree for a word of length $n\ge 1$ has $n$ terminal productions and $n-1$ binary productions. Every derivation represented by the tree therefore has exactly $2n-1$ steps.

A unit production replaces one nonterminal by one nonterminal and emits no terminal symbols.

An epsilon production is a grammar production whose right-hand side is the empty word.

The Kähler potential is a real function integrated over full superspace. The canonical choice $K={\Phi }^{†}\Phi$ gives canonical kinetic terms.

A superpotential is a holomorphic function of chiral superfields integrated over chiral superspace, $\int d^2\theta W+h.c.$

An auxiliary field has no kinetic term and can be eliminated algebraically. The complex field $F$ in a chiral superfield balances bosonic and fermionic degrees of freedom off shell.

For a reversible chain with spectral gap $\gamma$ and Cheeger constant $\Phi$,

A non-characteristic hypersurface satisfies $p(x,d\varphi (x))\ne 0$. This lets the equation solve for the highest derivative normal to the hypersurface.

Replace the original halt state by a continuation state that tests whether its one-letter output is $0$ or $1$, removes it, appends the other symbol, and enters a new halt state. This computes the complement language without changing termination behavior.

To recognize words ending in a fixed symbol $a$, test the final input letter, choose one of two erasing loops, and append $1$ after the accepting loop or $0$ after the rejecting loop. The empty word follows the rejecting branch.