Nontriviality of homology classes [04EZ]

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Nontriviality of homology classes

Corollary 5.14.

(Nontriviality of homology classes) There is a lower bound $\int_{L}\text{Re}\Omega\geq C^{-1}$ depending only on the ambient Calabi-Yau and the almost calibratedness constant $\epsilon$ . Here we do not a priori specify the homology class of $L$ .

Proof.

The asymptotic hypothesis on the regularity scale implies in particular that the regularity scale has a global lower bound on $X$ . This gives a uniform lower bound on $\text{Vol}(L)$ , which by the quantitative almost calibratedness gives a lower bound on the homological integral $\int_{L}\text{Re}\Omega$ . ∎

The significance is that in a variational setup to find special Lagrangians among certain Lagrangian currents in the fixed homology class $[L_{0}]$ , it may happen that the Lagrangian breaks into several connected components, each given by a closed Lagrangian current $L_{i}$ with $\sum_{i}\int_{L_{i}}\text{Re}\Omega=\int_{L}\text{Re}{\Omega}$ . The nontriviality of each homology class then puts an upper bound on the number of connected components. Morever, each $L_{i}$ would inherit a volume upper bound from $L$ . Since all Lagrangians are contained in a fixed compact region, Poincaré duality easily gives an upper bound on the homology classes of $L_{i}$ :

\left\lVert[L_{i}]\right\rVert_{H_{n}(X)}\leq C\text{Vol}(L_{i})\leq C.

Combining the above, only finitely many homology classes, multiplicities, and number of components can appear in a given variational problem.