Berndinar Fall 2025

Mumford picture from paper about abelian varieties and Springer lecture notes on toroidal embeddings. There are two ways to construct toric varieties: fans or polytopes. If you start with the fan, you get a toric variety. If you start with a polytope, you get a toric variety AND a bundle \(\mathcal O(1)\) . If you add to the fan data a choice of strictly convex PL-function \(\varphi:|\Sigma|\to \mathbb R\), then this correspondence is one-to-one.

\[(\Sigma, \varphi) \mapsto \operatorname{Newt}(\varphi)\]

How do you get degenerations of toric varieties from fans? They correspond to polyhedral decompositions of \(N_{\mathbb R}\). Embedding this decomposition as \(N_{\mathbb R} \times \{1\}\subset N_{\mathbb R} \times \mathbb R\) and letting \(\Sigma\) be the fan over this, you get a map \(\alpha: X_\Sigma \to \mathbb A^1\) by restricting \(N_{\mathbb R}\times \mathbb R\to \mathbb R\) to the fan \(\Sigma\). The image of \(\Sigma\) is \(\mathbb R_{\geq 0}\), the fan of \(\mathbb A^1\). In this setup there are two interesting fibers, the fiber over \(0\) and the fiber of any point in \(\mathbb A^1\setminus \{0\}\).

(include sketch)

This is called the fan picture, this seemed natural, seemed like you didn’t need a polarization at this point.

MSLD2: Hodge theory + deformation theory

’06-’07: Scattering + wall structures

Wall structures and canonical scattering paper

Let \(\pi:\widetilde B\to B\) be a universal cover. Then there exists an etale affine map \(\delta:\widetilde B\to M_{\mathbb R}\) called a developing map. Then \(\pi_1(B)\) acts on \(\widetilde B\) by deck transformations \(\Psi_\gamma \) \(\gamma \in \pi_1(B)\). This gives something called the holonomy representation \[\rho:\pi_1(B) \to \text{Aff}(M_{\mathbb R})\] which satisfies \(\rho(\gamma)\circ \delta \circ \Psi_\gamma = \delta\).

You can also write something gross down which we just call \(\text{Trans}:M_{\mathbb R} \rtimes \GL_n(\mathbb R) = \text{Aff}(M_{\mathbb R}) \to M_{\mathbb R}\), and composing this with \(\rho\) above gives something called the radiance obstruction which measure some sort of obstructions.

\(K:\widetilde B\to \mathbb R\). Then we can define a class \(\alpha \in H^1(\pi_1(B), \text{Aff}(M_{\mathbb R}, \mathbb R))\) so that \(K - K\circ \Psi_{\gamma} = \alpha(\gamma)\) is called the class of the Hessian metric. Remember that \(\gamma \in \pi_1(B)\), and notice that we’re taking group cohomology here.

Lots of scary symplectic stuff here.

Key Example: Let \(\Xi \subset M_{\mathbb R} \cong \mathbb R^n\) be a polytope containing \(0\) in its interior. Let \(B = \partial \Xi\), then \(B\) is an affine manifold with singularities. Let \(\text{Bary}(B)\) be the first barycentric subdivision of \(\partial \Xi\). Let \(\Delta\) be the union of simplices in \(\text{Bary}(B)\) not containing any vertex of \(\Xi\) and not containing any barycenter of \(n-1\) face of \(\Xi\). Note that if \(\sigma\) is an \(n-1\) dimensional face of \(\Xi\) then \(W_\sigma := \text{int}(\sigma)\) has an affine structure.

For each vertex \(v\) of \(\Xi\) , let \(\check v\) denote the union of all simplices of \(\text{Bary}(B)\) containing \(v\). \(W_v = \text{int}(\check v)\), and \(\varphi_v:W_v\to M_{\mathbb R}/\mathbb R v\).

Some observations.

1. The exponential map is a local homeomorphism onto its image, is integral on each \(\text{int}(\tau)\) for \(\tau \in \mathcal P_v\) and is integral affine on a neighborhood of the origin.
2. For all top dimensional \(\widetilde \sigma\), \(\exp(\text{int}(\widetilde \sigma)) \cap \Delta = \emptyset\) and \(\text{int}(\widetilde \sigma)\) integral affine, \(\exp(\widetilde \tau) \in \mathcal P\) for all \(\widetilde \tau \in \mathcal P_v\).

There are 3 other observations.

We end up choosing a fan structure at each vertex \(v\in \mathcal P\).

Given an affine manifold \(B\) with a polyhedral decomposition \(\mathcal P\) and a multivalued function \(\varphi\) encoding a fan structure at each vertex \(v\in \mathcal P\), we’re going to get a new set of the same data called \((\check B, \check{\mathcal P}, \check \varphi)\).

Q1: Where do I get polytopes for \(\check B\)? Q2: How do I assemble these polytopes? Q3: How do I put an affine structure on the assembled thing?

Defining \(\varphi\) Given \((B, \mathcal P)\) data, there are two sheaves we care about. The first is \(\Lambda_{\mathcal P, \mathbb R} \subseteq \Lambda_{\mathbb R}\) defined \[\Gamma(U,\Lambda_{\mathcal P, \mathbb R}) = \{v\in \Gamma(U, \Lambda_{\mathbb R}) ~| ~ \forall y\in U, \sigma \in \mathcal P, y\in \sigma, v\text{ is tangent to } \sigma. \}\] It looks sort of like the log tangent sheaf. The second is \(Q_{\mathcal P, \mathbb R}\), a subsheaf of \(i_*\left(\Lambda_{\mathbb R} / \Lambda_{\mathcal P, \mathbb R}\right)\). A section \(s \in \Gamma(U, i_*(\Lambda_{\mathbb R}/\Lambda_{\mathcal P, \mathbb R}))\) is a section of \(Q_{\mathcal P, \mathbb R} \) if for all \(\gamma:[0,1]\to B\) the pull back \(\gamma^*s \in \Gamma([0,1], \gamma^*(\Lambda_{\mathbb R}/\Lambda_{\mathcal P,\mathbb R}))\) is induced by a section \(t \in \Gamma([0,1], \gamma^*\Gamma_{\mathbb R})\).

If reading this later, one should really look at the reference paper. Stuff became quite rushed at this point. We basically get a complete polyhedral fan \(\Sigma_\tau\) for each \(\tau \in \mathcal P\). It’s supposed to be the normal fan

Given a reflexive polytope, we can do standard toric geometry to get a pair \((\mathbb P_\Delta, -K_\Delta)\) of a projective variety with a polarization. Here \(-K_{\Delta}\) is the anticanonical divisor of \(\mathbb P_\Delta\), and it corresponds to the sum of the codimension 1 faces of \(\Delta\).

Given any \(a_0,...,a_k\in (\mathbb C^*)^n\) and \(m_0,...,m_k\in \Delta \cap \mathbb Z^n\), \(V = a_0t^{m_0} + ... + a_kt^{m_k} = 0\) defines a hypersurface in \((\mathbb C^*)^n\), and taking the closure gives a hypersurface \(H \subset \mathbb P_\Delta\). It is a fact that when \(\Delta\) is reflexive, \(H\) constructed this way is in \(|-K_{\Delta}|.\)

We can resolve singularities super easily because toric geometry. It turns out that whenever you’re a reflexive polytope you get a crepant resolution of singularities.

Nice example from Ogus: consider a smooth proper morphism \(f|_U:U\to V\) coming from the restriction of a morphism \(f:X\to Y\) where both \(X\) and \(Y\) are compact and \(U\subset X\) is an open dense subset. We “want to make \(f|_{X\setminus U}:X\setminus U\to Y\setminus V\) a smooth morphism”.

Not that \(\mathbb H^q(X, \Omega^\bullet_{X}(\log D)) = H^q(X, \underline{\mathbb C})\).

Daniel then defines log structures and does the example of \(\mathbb A^2_k = \Spec k[x,y]\) with divisorial log structure given by the divisor \(D = V(xy).\)

He then defines log morphisms.

Next, Daniel talks about charts.