54 problems

Let $M$ be a compact manifold. The analytic torsion is the invariant defined from the zeta-regularized determinants of the Laplacians on differential forms, while the R-torsion is…

Let $\pi: M \rightarrow S$ be a smooth fibration, with total space $M$, and let $F\to M$ be a flat complex vector bundle. The Wagoner conjecture. Reidemeister–Franz torsion and Ray…

Burghelea–Haller conjecture. The equality

Let $Y$ be a Calabi–Yau orbifold and let $\widetilde{Y}\to Y$ be a crepant resolution. Orbifold–resolution conjecture. There is a constant $C(\widetilde{Y}/Y)$, depending only on t…

General gluing conjecture. With the assumption of product structures, the following identity holds in $Q^{S}/Q^{S,0}$:

Let $M$ be a fixed closed hyperbolic $3$-manifold, and let $M_n$ be covers of $M$ that Benjamini–Schramm converge to hyperbolic $3$-space $\mathbb{H}^3$. The analytic torsion conve…

Vanishing conjecture. We have

Let $F$ be a unitary flat vector bundle on a closed Riemannian manifold $X$. Ray and Singer's analytic torsion is the torsion invariant constructed from the spectrum of the Laplaci…

Let $E$ be an acyclic flat complex vector bundle over a closed manifold $M$, so that $H^\bullet(M,E)=0$, and suppose that $E$ admits a non-degenerate complex-valued symmetric bilin…

Let $M$ be a closed connected smooth manifold with vanishing Euler–Poincaré characteristic. Let $E$ be a flat complex vector bundle over $M$, and let $b$ be a fiberwise non-degener…

Let $W$ be a Ricci-flat Calabi–Yau threefold in the class considered by Harvey and Moore, and let $\tau_{\rm BCOV}(W)$ denote its BCOV torsion. Let $\Phi$ be the Borcherds $\Phi$-f…

Let $F_1$ denote the genus-one Gromov–Witten generating function of a Calabi–Yau threefold, and let its mirror Calabi–Yau threefolds carry Ray–Singer analytic torsions. Bershadsky–…

Let $C$ be a smooth closed curve with canonical constant-curvature metric $g_{0}^{C}$, and let $l$ be a holomorphic line bundle with induced canonical metric $g_{0}^{l}$. For a Her…

Let $C$ be a compact complex curve with Hermitian metric $g^{C}$, and let $l$ be a line bundle over $C$ with associated Hermitian metric $g^{l}$. Write $A(g^{C},g^{l})$ for the log…

Fibre-wise Morse function conjecture. The theorem holds without the assumption that such a function exists.

Let $M\to B$ be a family of compact manifolds, and let $(F,\nabla^F)\to M$ be a flat complex vector bundle with a parallel Hermitian metric such that $(H,\nabla^H)$ also admits a p…

Supersymmetric explanation conjecture. The observed behavior of the torsion may be understandable using some form of supersymmetry.

A flat Hilbertian bundle of determinant class is a flat Hilbertian vector bundle whose relevant operators are of determinant class. The Poincaré–Reidemeister metric is the metric o…

Torsion equality conjecture. The $L_2$-analytic and $L_2$-Reidemeister torsions of $M$ are equal.

Let $\mu:Y\to X$ be the birational modification in a semistable reduction, let $f:Y\to T$ be the induced family, let $F^{*}\xi$ be the pulled-back coefficient bundle, and let…

Let $\pi:X\to S$ be the degeneration under consideration, with central fibre $X_{0}$, relative canonical bundle $K_{X/S}$, and let $H$ be an ample line bundle on $X$. Write…

Let $L_2$ be the Beauville–Bogomolov lattice of a manifold of $K3^{[2]}$-type. Let $M$ be an admissible sublattice of $L_2$, let…

Let $Y$ be a compact manifold, let $Z$ be an Anosov vector field on $Y$, and let $F$ be a flat vector bundle. Let $R_{Z,F}(\sigma)$ denote the Fried zeta function, and let…

Let $X$ be a Calabi–Yau manifold. The BCOV torsion is the $C^{\infty}$ quantity obtained from the BCOV formula and the period-domain metric; in the K3 setup it is defined by … Let…

Let $M$ be a closed Riemannian manifold and let $F\to M$ be a flat complex vector bundle with a hermitian metric. The Ray–Singer metric is the product of the Ray–Singer analytic to…