kollar divisor demo

kollar_divisors/kollar.ipynb

@@ -0,0 +1,357 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "24f6e8d5-6eb7-4d85-9a3e-da8239324338",
+   "metadata": {},
+   "source": [
+    "This notebook shows a simple method to identify Kollar divisors using CYTools.\n",
+    "\n",
+    "Recall, Kollar divisors are those, $D\\in K(X)\\cap\\mathbb{Z}^{h^{1,1}}$, for which $\\kappa_{ijk} D^i D^j D^k = 0$ but $\\kappa_{ijk} D^i D^j \\neq 0$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ed933502-59e5-4419-b910-5c7372e4dd5d",
+   "metadata": {},
+   "source": [
+    "# Setup"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "862a107c-8be9-415d-9753-08754c16816d",
+   "metadata": {},
+   "source": [
+    "## Imports"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d32db58d-55a3-4dc4-a167-4e089e3eba6a",
+   "metadata": {},
+   "source": [
+    "Here, we import some useful packages..."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c38f449f-9217-4abe-af85-6722a26795f7",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# generic Python imports\n",
+    "import itertools\n",
+    "import numpy as np\n",
+    "\n",
+    "# CYTools imports\n",
+    "from cytools import Cone, fetch_polytopes, Polytope\n",
+    "\n",
+    "# custom impotys\n",
+    "import sys; sys.path.append('../cytools_utilities/')\n",
+    "from ntfe_frsts import ntfe_frsts\n",
+    "from gvs import gvs"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c4c62bfe-fd1b-4a0d-8e69-5723d3f58ea3",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "Filter warnings"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5bb4c741-2431-4cde-b4b1-581f002af713",
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "import warnings\n",
+    "warnings.filterwarnings(\"ignore\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "31588402-adc8-46d8-be2e-303da691bc4d",
+   "metadata": {},
+   "source": [
+    "## Custom functions"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4794d7f6-a749-41c7-aec6-6863431137be",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "Here, we define some useful functions..."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d5b03c78-a738-49b6-800a-cd45fa3f8704",
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def read_wati_polys(inds, fname=\"fiber-free.txt\"):\n",
+    "    \"\"\"\n",
+    "    Read polytopes from Wati's list.\n",
+    "    \n",
+    "    From http://ctp.lns.mit.edu/wati/data.html\n",
+    "    \"\"\"\n",
+    "    all_polys = []\n",
+    "\n",
+    "    # load the points\n",
+    "    with open(fname, \"r\") as f:\n",
+    "        for i_poly in range(max(poly_inds)+1):\n",
+    "            # read the points\n",
+    "            pts = []\n",
+    "\n",
+    "            f.readline()\n",
+    "            for i_coords in range(4):\n",
+    "                pts.append([int(x) for x in f.readline().split()])\n",
+    "            pts = np.array(pts).T\n",
+    "\n",
+    "            # this is a polytope of interest\n",
+    "            if i_poly in poly_inds:\n",
+    "                all_polys.append(Polytope(pts))\n",
+    "    \n",
+    "    return all_polys"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3410489c-70bd-44e4-bba3-66a7b9e04bd8",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def intersection_form(kappa, x, order=3):\n",
+    "    # get the indices to contract\n",
+    "    inds = 'abc,' + ','.join([chr(ord('a')+i) for i in range(order)])\n",
+    "    \n",
+    "    # contract them!\n",
+    "    return np.einsum(inds, kappa, *(order*[x]))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ae6d9963-1834-45c2-b6d6-9fd769039cc6",
+   "metadata": {},
+   "source": [
+    "# The scan!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f9763dda-6e53-4aec-890a-c4f26f3ae96a",
+   "metadata": {},
+   "source": [
+    "Select polytopes to scan over"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7550e056-742a-465e-a745-cd43e0675ba6",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "scan_over_wati = False"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5c15955a-251d-41d5-afc2-f5fcea37cc36",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "if scan_over_wati:\n",
+    "    # indices of the polytopes to scan over http://ctp.lns.mit.edu/wati/data.html\n",
+    "    poly_inds = list(range(20,600))\n",
+    "    \n",
+    "    # read them from the file\n",
+    "    polys = read_wati_polys(poly_inds)\n",
+    "else:\n",
+    "    # read from KS\n",
+    "    polys = fetch_polytopes(h11=6,lattice=\"N\",limit=50,as_list=True)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "428039b4-7bb8-48b3-a3b4-9504ea37ab54",
+   "metadata": {},
+   "source": [
+    "Do the scan!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c444b39b-f363-4542-8a61-aaa1765e7679",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "verbosity = 0"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ece519b0-8155-45bc-9015-db116d810e70",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "kollar_divisors = []\n",
+    "found_divisors = 0\n",
+    "\n",
+    "for i,p in enumerate(polys):\n",
+    "    # get the Hodge numbers, dualize to have h11<h21\n",
+    "    h11 = p.h11(lattice=\"N\")\n",
+    "    h21 = p.h21(lattice=\"N\")\n",
+    "    if h11>h21:\n",
+    "        p = p.dual()\n",
+    "        \n",
+    "    # reject non-favorables\n",
+    "    if not p.is_favorable(lattice=\"N\"):\n",
+    "        continue\n",
+    "\n",
+    "    print(f\"poly #{i}, (h11,h21)={(p.h11(lattice='N'),p.h21(lattice='N'))}, total found (with duplicates)={found_divisors}\",end='\\r')\n",
+    "\n",
+    "    kollar_divisors.append([])\n",
+    "    \n",
+    "    # iterate over NTFEs\n",
+    "    # ------------------\n",
+    "    ts = p.ntfe_frsts()\n",
+    "    if verbosity>0:\n",
+    "        print(f\"There are {len(ts)} NTFEs\")\n",
+    "\n",
+    "    for t in ts:\n",
+    "        kollar_divisors[-1].append([])\n",
+    "\n",
+    "        # construct the TV, CY\n",
+    "        # --------------------\n",
+    "        cy = t.cy()\n",
+    "        tv = cy.ambient_variety()\n",
+    "        if verbosity>1:\n",
+    "            print('Constructing CY, TV...')\n",
+    "\n",
+    "        # get the intersection numbers\n",
+    "        # ----------------------------\n",
+    "        if verbosity>1:\n",
+    "            print('Constructing kappa...')\n",
+    "        kappa = cy.intersection_numbers(format='dense',in_basis=True)\n",
+    "\n",
+    "        # get the Kahler cone\n",
+    "        # -------------------\n",
+    "        if verbosity>1:\n",
+    "            print('Constructing Kähler cones...')\n",
+    "        if False:\n",
+    "            # get the CY kahler cone\n",
+    "            print('with GVs...')\n",
+    "            max_deg = 10\n",
+    "            \n",
+    "            while True:\n",
+    "                try:\n",
+    "                    gvs = cy.compute_gvs(max_deg=max_deg)\n",
+    "                    break\n",
+    "                except:\n",
+    "                    max_deg += 2\n",
+    "    \n",
+    "            mori = Cone(rays=list(gvs.keys()))\n",
+    "            kahler = mori.dual()\n",
+    "            kahler = Cone(rays=kahler.extremal_rays())\n",
+    "        else:\n",
+    "            # get the toric kahler cone\n",
+    "            kahler = cy.toric_kahler_cone()\n",
+    "\n",
+    "        # find the possible Kollar divisors\n",
+    "        # ---------------------------------\n",
+    "        # (lattice points in the Kahler cone...)\n",
+    "        min_points = 200\n",
+    "        if verbosity>1:\n",
+    "            print(f\"Finding >={min_points} points...\")\n",
+    "        possible_pts = kahler.find_lattice_points(min_points=min_points, verbose=False)\n",
+    "        if verbosity>1:\n",
+    "            print(f\"Found {len(possible_pts)} points...\")\n",
+    "        \n",
+    "        # check them!\n",
+    "        # -----------\n",
+    "        for kollar_plz in possible_pts:\n",
+    "            # skip the origin... trivial divisor\n",
+    "            if sum(np.abs(kollar_plz))==0:\n",
+    "                continue\n",
+    "\n",
+    "            # check if cubic is 0\n",
+    "            if intersection_form(kappa, kollar_plz, order=3) == 0:\n",
+    "                # get the quadratic\n",
+    "                quadratic = intersection_form(kappa, kollar_plz, order=2)\n",
+    "\n",
+    "                # check if the quadratic is nonzero\n",
+    "                if any(quadratic!=0) and all(quadratic>=0):\n",
+    "                    kollar_divisors[-1][-1].append(kollar_plz)\n",
+    "                    found_divisors += 1\n",
+    "                    \n",
+    "                    # extra printing\n",
+    "                    if verbosity>0:\n",
+    "                        print(100*\":)\")\n",
+    "                        print(f\"x={kollar_plz}\")\n",
+    "                        print()\n",
+    "\n",
+    "        # extra printing\n",
+    "        if verbosity>1:\n",
+    "            print('all done!')\n",
+    "            print()"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
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+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}