\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
K\"ahler cone
definition 0.0.1 [Mirror symmetry motivation notes]
Let \(X\) be a compact Kähler manifold. Under the Hodge decomposition, the real \((1,1)\)-cohomology is
\[
H^{1,1}(X;\RR):=H^{1,1}(X;\CC)\cap H^2(X;\RR).
\]
The K\"ahler cone of \(X\) is
\[
\mathcal K_X
:=\{[\omega]_{\mathrm{dR}}\in H^{1,1}(X;\RR)\mid \omega \text{ is a Kähler form on }X\}.
\]
It is an open convex cone in \(H^{1,1}(X;\RR)\). A point of \(\mathcal K_X\) is the de Rham cohomology class of a compatible symplectic form on the fixed complex manifold \(X\).
In mirror symmetry one often uses the complexified K\"ahler parameter
\[
[B]+i[\omega]\in H^2(X;\CC),
\]
where \([B]\in H^2(X;\RR)\) is usually considered modulo integral classes.
Connections
Uses
No direct links found.